Article PZFMISC1, MML version 4.99.1005

:: PZFMISC1:th 1
theorem
for b1, b2, b3 being set
for b4 being ManySortedSet of b1
      st b2 in b1
   holds proj1 (b4 +* (b2 .--> b3)) = b1;

:: PZFMISC1:th 2
theorem
for b1 being Relation-like Function-like set
      st b1 = {}
   holds b1 is ManySortedSet of {};

:: PZFMISC1:th 3
theorem
for b1 being set
   st b1 is not empty
for b2 being ManySortedSet of b1
      st b2 is empty-yielding
   holds b2 is not non-empty;

:: PZFMISC1:funcnot 1 => PZFMISC1:func 1
definition
  let a1 be set;
  let a2 be ManySortedSet of a1;
  func {A2} -> ManySortedSet of a1 means
    for b1 being set
          st b1 in a1
       holds it . b1 = {a2 . b1};
end;

:: PZFMISC1:def 1
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1 holds
   b3 = {b2}
iff
   for b4 being set
         st b4 in b1
      holds b3 . b4 = {b2 . b4};

:: PZFMISC1:funcreg 1
registration
  let a1 be set;
  let a2 be ManySortedSet of a1;
  cluster {a2} -> non-empty finite-yielding;
end;

:: PZFMISC1:funcnot 2 => PZFMISC1:func 2
definition
  let a1 be set;
  let a2, a3 be ManySortedSet of a1;
  func {A2,A3} -> ManySortedSet of a1 means
    for b1 being set
          st b1 in a1
       holds it . b1 = {a2 . b1,a3 . b1};
  commutativity;
::  for a1 being set
::  for a2, a3 being ManySortedSet of a1 holds
::  {a2,a3} = {a3,a2};
end;

:: PZFMISC1:def 2
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1 holds
   b4 = {b2,b3}
iff
   for b5 being set
         st b5 in b1
      holds b4 . b5 = {b2 . b5,b3 . b5};

:: PZFMISC1:funcreg 2
registration
  let a1 be set;
  let a2, a3 be ManySortedSet of a1;
  cluster {a2,a3} -> non-empty finite-yielding;
end;

:: PZFMISC1:th 4
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1 holds
   b2 = {b3}
iff
   for b4 being ManySortedSet of b1 holds
         b4 in b2
      iff
         b4 = b3;

:: PZFMISC1:th 5
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st for b5 being ManySortedSet of b1 holds
              b5 in b2
           iff
              (b5 = b3 or b5 = b4)
   holds b2 = {b3,b4};

:: PZFMISC1:th 6
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
   st b2 = {b3,b4}
for b5 being ManySortedSet of b1
      st (b5 = b3 or b5 = b4)
   holds b5 in b2;

:: PZFMISC1:th 8
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b2 in {b3}
   holds b2 = b3;

:: PZFMISC1:th 9
theorem
for b1 being set
for b2 being ManySortedSet of b1 holds
   b2 in {b2};

:: PZFMISC1:th 10
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st (b2 = b3 or b2 = b4)
   holds b2 in {b3,b4};

:: PZFMISC1:th 11
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1 holds
{b2} \/ {b3} = {b2,b3};

:: PZFMISC1:th 12
theorem
for b1 being set
for b2 being ManySortedSet of b1 holds
   {b2,b2} = {b2};

:: PZFMISC1:th 14
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st {b2} c= {b3}
   holds b2 = b3;

:: PZFMISC1:th 15
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st {b2} = {b3}
   holds b2 = b3;

:: PZFMISC1:th 16
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st {b2} = {b3,b4}
   holds b2 = b3 & b2 = b4;

:: PZFMISC1:th 17
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st {b2} = {b3,b4}
   holds b3 = b4;

:: PZFMISC1:th 18
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1 holds
{b2} c= {b2,b3} & {b3} c= {b2,b3};

:: PZFMISC1:th 19
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st ({b2} \/ {b3} = {b2} or {b2} \/ {b3} = {b3})
   holds b2 = b3;

:: PZFMISC1:th 20
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1 holds
{b2} \/ {b2,b3} = {b2,b3};

:: PZFMISC1:th 22
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b1 is not empty &
         {b2} /\ {b3} = [0] b1
   holds b2 <> b3;

:: PZFMISC1:th 23
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st ({b2} /\ {b3} = {b2} or {b2} /\ {b3} = {b3})
   holds b2 = b3;

:: PZFMISC1:th 24
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1 holds
{b2} /\ {b2,b3} = {b2} &
 {b3} /\ {b2,b3} = {b3};

:: PZFMISC1:th 25
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b1 is not empty &
         {b2} \ {b3} = {b2}
   holds b2 <> b3;

:: PZFMISC1:th 26
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st {b2} \ {b3} = [0] b1
   holds b2 = b3;

:: PZFMISC1:th 27
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1 holds
{b2} \ {b2,b3} = [0] b1 &
 {b3} \ {b2,b3} = [0] b1;

:: PZFMISC1:th 28
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st {b2} c= {b3}
   holds {b2} = {b3};

:: PZFMISC1:th 29
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st {b2,b3} c= {b4}
   holds b2 = b4 & b3 = b4;

:: PZFMISC1:th 30
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st {b2,b3} c= {b4}
   holds {b2,b3} = {b4};

:: PZFMISC1:th 31
theorem
for b1 being set
for b2 being ManySortedSet of b1 holds
   bool {b2} = {[0] b1,{b2}};

:: PZFMISC1:th 32
theorem
for b1 being set
for b2 being ManySortedSet of b1 holds
   {b2} c= bool b2;

:: PZFMISC1:th 33
theorem
for b1 being set
for b2 being ManySortedSet of b1 holds
   union {b2} = b2;

:: PZFMISC1:th 34
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1 holds
union {{b2},{b3}} = {b2,b3};

:: PZFMISC1:th 35
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1 holds
union {b2,b3} = b2 \/ b3;

:: PZFMISC1:th 36
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1 holds
   {b2} c= b3
iff
   b2 in b3;

:: PZFMISC1:th 37
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1 holds
   {b2,b3} c= b4
iff
   b2 in b4 & b3 in b4;

:: PZFMISC1:th 38
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st (not b2 = [0] b1 & not b2 = {b3} & not b2 = {b4} implies b2 = {b3,b4})
   holds b2 c= {b3,b4};

:: PZFMISC1:th 39
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st (b2 in b3 or b2 = b4)
   holds b2 in b3 \/ {b4};

:: PZFMISC1:th 40
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1 holds
   b2 \/ {b3} c= b4
iff
   b3 in b4 & b2 c= b4;

:: PZFMISC1:th 41
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st {b2} \/ b3 = b3
   holds b2 in b3;

:: PZFMISC1:th 42
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b2 in b3
   holds {b2} \/ b3 = b3;

:: PZFMISC1:th 43
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1 holds
   {b2,b3} \/ b4 = b4
iff
   b2 in b4 & b3 in b4;

:: PZFMISC1:th 44
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b1 is not empty
   holds {b2} \/ b3 <> [0] b1;

:: PZFMISC1:th 45
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st b1 is not empty
   holds {b2,b3} \/ b4 <> [0] b1;

:: PZFMISC1:th 46
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b2 /\ {b3} = {b3}
   holds b3 in b2;

:: PZFMISC1:th 47
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b2 in b3
   holds b3 /\ {b2} = {b2};

:: PZFMISC1:th 48
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1 holds
   b2 in b3 & b4 in b3
iff
   {b2,b4} /\ b3 = {b2,b4};

:: PZFMISC1:th 49
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b1 is not empty & {b2} /\ b3 = [0] b1
   holds not b2 in b3;

:: PZFMISC1:th 50
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st b1 is not empty & {b2,b3} /\ b4 = [0] b1
   holds not b2 in b4 & not b3 in b4;

:: PZFMISC1:th 51
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st b2 in b3 \ {b4}
   holds b2 in b3;

:: PZFMISC1:th 52
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st b1 is not empty & b2 in b3 \ {b4}
   holds b2 <> b4;

:: PZFMISC1:th 53
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b1 is not empty & b2 \ {b3} = b2
   holds not b3 in b2;

:: PZFMISC1:th 54
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b1 is not empty &
         {b2} \ b3 = {b2}
   holds not b2 in b3;

:: PZFMISC1:th 55
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1 holds
   {b2} \ b3 = [0] b1
iff
   b2 in b3;

:: PZFMISC1:th 56
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st b1 is not empty &
         {b2,b3} \ b4 = {b2}
   holds not b2 in b4;

:: PZFMISC1:th 58
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st b1 is not empty &
         {b2,b3} \ b4 = {b2,b3}
   holds not b2 in b4 & not b3 in b4;

:: PZFMISC1:th 59
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1 holds
   {b2,b3} \ b4 = [0] b1
iff
   b2 in b4 & b3 in b4;

:: PZFMISC1:th 60
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st (not b2 = [0] b1 & not b2 = {b3} & not b2 = {b4} implies b2 = {b3,b4})
   holds b2 \ {b3,b4} = [0] b1;

:: PZFMISC1:th 61
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st (b2 = [0] b1 or b3 = [0] b1)
   holds [|b2,b3|] = [0] b1;

:: PZFMISC1:th 62
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b2 is non-empty & b3 is non-empty & [|b2,b3|] = [|b3,b2|]
   holds b2 = b3;

:: PZFMISC1:th 63
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st [|b2,b2|] = [|b3,b3|]
   holds b2 = b3;

:: PZFMISC1:th 64
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st b2 is non-empty &
         ([|b3,b2|] c= [|b4,b2|] or [|b2,b3|] c= [|b2,b4|])
   holds b3 c= b4;

:: PZFMISC1:th 65
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1
      st b2 c= b3
   holds [|b2,b4|] c= [|b3,b4|] & [|b4,b2|] c= [|b4,b3|];

:: PZFMISC1:th 66
theorem
for b1 being set
for b2, b3, b4, b5 being ManySortedSet of b1
      st b2 c= b3 & b4 c= b5
   holds [|b2,b4|] c= [|b3,b5|];

:: PZFMISC1:th 67
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1 holds
[|b2 \/ b3,b4|] = [|b2,b4|] \/ [|b3,b4|] &
 [|b4,b2 \/ b3|] = [|b4,b2|] \/ [|b4,b3|];

:: PZFMISC1:th 68
theorem
for b1 being set
for b2, b3, b4, b5 being ManySortedSet of b1 holds
[|b2 \/ b3,b4 \/ b5|] = (([|b2,b4|] \/ [|b2,b5|]) \/ [|b3,b4|]) \/ [|b3,b5|];

:: PZFMISC1:th 69
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1 holds
[|b2 /\ b3,b4|] = [|b2,b4|] /\ [|b3,b4|] &
 [|b4,b2 /\ b3|] = [|b4,b2|] /\ [|b4,b3|];

:: PZFMISC1:th 70
theorem
for b1 being set
for b2, b3, b4, b5 being ManySortedSet of b1 holds
[|b2 /\ b3,b4 /\ b5|] = [|b2,b4|] /\ [|b3,b5|];

:: PZFMISC1:th 71
theorem
for b1 being set
for b2, b3, b4, b5 being ManySortedSet of b1
      st b2 c= b3 & b4 c= b5
   holds [|b2,b5|] /\ [|b3,b4|] = [|b2,b4|];

:: PZFMISC1:th 72
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1 holds
[|b2 \ b3,b4|] = [|b2,b4|] \ [|b3,b4|] &
 [|b4,b2 \ b3|] = [|b4,b2|] \ [|b4,b3|];

:: PZFMISC1:th 73
theorem
for b1 being set
for b2, b3, b4, b5 being ManySortedSet of b1 holds
[|b2,b3|] \ [|b4,b5|] = [|b2 \ b4,b3|] \/ [|b2,b3 \ b5|];

:: PZFMISC1:th 74
theorem
for b1 being set
for b2, b3, b4, b5 being ManySortedSet of b1
      st (b2 /\ b3 = [0] b1 or b4 /\ b5 = [0] b1)
   holds [|b2,b4|] /\ [|b3,b5|] = [0] b1;

:: PZFMISC1:th 75
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b2 is non-empty
   holds [|{b3},b2|] is non-empty & [|b2,{b3}|] is non-empty;

:: PZFMISC1:th 76
theorem
for b1 being set
for b2, b3, b4 being ManySortedSet of b1 holds
[|{b2,b3},b4|] = [|{b2},b4|] \/ [|{b3},b4|] &
 [|b4,{b2,b3}|] = [|b4,{b2}|] \/ [|b4,{b3}|];

:: PZFMISC1:th 77
theorem
for b1 being set
for b2, b3, b4, b5 being ManySortedSet of b1
      st b2 is non-empty & b3 is non-empty & [|b2,b3|] = [|b4,b5|]
   holds b2 = b4 & b3 = b5;

:: PZFMISC1:th 78
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st (b2 c= [|b2,b3|] or b2 c= [|b3,b2|])
   holds b2 = [0] b1;

:: PZFMISC1:th 79
theorem
for b1 being set
for b2, b3, b4, b5, b6 being ManySortedSet of b1
      st b2 in [|b3,b4|] & b2 in [|b5,b6|]
   holds b2 in [|b3 /\ b5,b4 /\ b6|];

:: PZFMISC1:th 80
theorem
for b1 being set
for b2, b3, b4, b5 being ManySortedSet of b1
      st [|b2,b3|] c= [|b4,b5|] & [|b2,b3|] is non-empty
   holds b2 c= b4 & b3 c= b5;

:: PZFMISC1:th 81
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b2 c= b3
   holds [|b2,b2|] c= [|b3,b3|];

:: PZFMISC1:th 82
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b2 /\ b3 = [0] b1
   holds [|b2,b3|] /\ [|b3,b2|] = [0] b1;

:: PZFMISC1:th 83
theorem
for b1 being set
for b2, b3, b4, b5 being ManySortedSet of b1
      st b2 is non-empty &
         ([|b2,b3|] c= [|b4,b5|] or [|b3,b2|] c= [|b5,b4|])
   holds b3 c= b5;

:: PZFMISC1:th 84
theorem
for b1 being set
for b2, b3, b4, b5, b6, b7 being ManySortedSet of b1
      st b2 c= [|b3,b4|] & b5 c= [|b6,b7|]
   holds b2 \/ b5 c= [|b3 \/ b6,b4 \/ b7|];

:: PZFMISC1:prednot 1 => PZFMISC1:pred 1
definition
  let a1 be set;
  let a2, a3 be ManySortedSet of a1;
  pred A2 is_transformable_to A3 means
    for b1 being set
          st b1 in a1 & a3 . b1 = {}
       holds a2 . b1 = {};
  reflexivity;
::  for a1 being set
::  for a2 being ManySortedSet of a1 holds
::     a2 is_transformable_to a2;
end;

:: PZFMISC1:dfs 3
definiens
  let a1 be set;
  let a2, a3 be ManySortedSet of a1;
To prove
     a2 is_transformable_to a3
it is sufficient to prove
  thus for b1 being set
          st b1 in a1 & a3 . b1 = {}
       holds a2 . b1 = {};

:: PZFMISC1:def 3
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1 holds
   b2 is_transformable_to b3
iff
   for b4 being set
         st b4 in b1 & b3 . b4 = {}
      holds b2 . b4 = {};