Article GRFUNC_1, MML version 4.99.1005

:: GRFUNC_1:th 6
theorem
for b1 being Relation-like Function-like set
for b2 being set
      st b2 c= b1
   holds b2 is Relation-like Function-like set;

:: GRFUNC_1:th 8
theorem
for b1, b2 being Relation-like Function-like set holds
   b1 c= b2
iff
   proj1 b1 c= proj1 b2 &
    (for b3 being set
          st b3 in proj1 b1
       holds b1 . b3 = b2 . b3);

:: GRFUNC_1:th 9
theorem
for b1, b2 being Relation-like Function-like set
      st proj1 b1 = proj1 b2 & b1 c= b2
   holds b1 = b2;

:: GRFUNC_1:th 12
theorem
for b1, b2 being set
for b3, b4 being Relation-like Function-like set
      st [b1,b2] in b4 * b3
   holds [b1,b4 . b1] in b4 & [b4 . b1,b2] in b3;

:: GRFUNC_1:th 13
theorem
for b1, b2, b3 being Relation-like Function-like set
      st b1 c= b2
   holds b1 * b3 c= b2 * b3 & b3 * b1 c= b3 * b2;

:: GRFUNC_1:th 15
theorem
for b1, b2 being set holds
{[b1,b2]} is Relation-like Function-like set;

:: GRFUNC_1:th 16
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set
      st b3 = {[b1,b2]}
   holds b3 . b1 = b2;

:: GRFUNC_1:th 18
theorem
for b1 being set
for b2 being Relation-like Function-like set
      st proj1 b2 = {b1}
   holds b2 = {[b1,b2 . b1]};

:: GRFUNC_1:th 19
theorem
for b1, b2, b3, b4 being set holds
   {[b1,b2],[b3,b4]} is Relation-like Function-like set
iff
   (b1 = b3 implies b2 = b4);

:: GRFUNC_1:th 25
theorem
for b1 being Relation-like Function-like set holds
      b1 is one-to-one
   iff
      for b2, b3, b4 being set
            st [b2,b4] in b1 & [b3,b4] in b1
         holds b2 = b3;

:: GRFUNC_1:th 26
theorem
for b1, b2 being Relation-like Function-like set
      st b1 c= b2 & b2 is one-to-one
   holds b1 is one-to-one;

:: GRFUNC_1:th 27
theorem
for b1 being set
for b2 being Relation-like Function-like set holds
   b2 /\ b1 is Relation-like Function-like set;

:: GRFUNC_1:th 28
theorem
for b1, b2, b3 being Relation-like Function-like set
      st b1 = b2 /\ b3
   holds proj1 b1 c= (proj1 b2) /\ proj1 b3 & proj2 b1 c= (proj2 b2) /\ proj2 b3;

:: GRFUNC_1:th 29
theorem
for b1 being set
for b2, b3, b4 being Relation-like Function-like set
      st b2 = b3 /\ b4 & b1 in proj1 b2
   holds b2 . b1 = b3 . b1 & b2 . b1 = b4 . b1;

:: GRFUNC_1:th 30
theorem
for b1, b2, b3 being Relation-like Function-like set
      st (b1 is one-to-one or b2 is one-to-one) & b3 = b1 /\ b2
   holds b3 is one-to-one;

:: GRFUNC_1:th 31
theorem
for b1, b2 being Relation-like Function-like set
      st proj1 b1 misses proj1 b2
   holds b1 \/ b2 is Relation-like Function-like set;

:: GRFUNC_1:th 32
theorem
for b1, b2, b3 being Relation-like Function-like set
      st b1 c= b2 & b3 c= b2
   holds b1 \/ b3 is Relation-like Function-like set;

:: GRFUNC_1:th 33
theorem
for b1, b2, b3 being Relation-like Function-like set
      st b1 = b2 \/ b3
   holds proj1 b1 = (proj1 b2) \/ proj1 b3 & proj2 b1 = (proj2 b2) \/ proj2 b3;

:: GRFUNC_1:th 34
theorem
for b1 being set
for b2, b3, b4 being Relation-like Function-like set
      st b1 in proj1 b2 & b3 = b2 \/ b4
   holds b3 . b1 = b2 . b1;

:: GRFUNC_1:th 35
theorem
for b1 being set
for b2, b3, b4 being Relation-like Function-like set
      st b1 in proj1 b2 & b3 = b4 \/ b2
   holds b3 . b1 = b2 . b1;

:: GRFUNC_1:th 36
theorem
for b1 being set
for b2, b3, b4 being Relation-like Function-like set
      st b1 in proj1 b2 & b2 = b3 \/ b4 & b2 . b1 <> b3 . b1
   holds b2 . b1 = b4 . b1;

:: GRFUNC_1:th 37
theorem
for b1, b2, b3 being Relation-like Function-like set
      st b1 is one-to-one & b2 is one-to-one & b3 = b1 \/ b2 & proj2 b1 misses proj2 b2
   holds b3 is one-to-one;

:: GRFUNC_1:th 38
theorem
for b1 being set
for b2 being Relation-like Function-like set holds
   b2 \ b1 is Relation-like Function-like set;

:: GRFUNC_1:th 46
theorem
for b1 being Relation-like Function-like set
      st b1 = {}
   holds b1 is one-to-one;

:: GRFUNC_1:th 47
theorem
for b1 being Relation-like Function-like set
   st b1 is one-to-one
for b2, b3 being set holds
   [b3,b2] in b1 "
iff
   [b2,b3] in b1;

:: GRFUNC_1:th 49
theorem
for b1 being Relation-like Function-like set
      st b1 = {}
   holds b1 " = {};

:: GRFUNC_1:th 52
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set holds
      b1 in proj1 b3 & b1 in b2
   iff
      [b1,b3 . b1] in b3 | b2;

:: GRFUNC_1:th 64
theorem
for b1, b2 being Relation-like Function-like set
      st b1 c= b2
   holds b2 | proj1 b1 = b1;

:: GRFUNC_1:th 67
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set holds
      b1 in proj1 b3 & b3 . b1 in b2
   iff
      [b1,b3 . b1] in b2 | b3;

:: GRFUNC_1:th 79
theorem
for b1, b2 being Relation-like Function-like set
      st b1 c= b2 & b2 is one-to-one
   holds (proj2 b1) | b2 = b1;

:: GRFUNC_1:th 87
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set holds
      b1 in b3 " b2
   iff
      [b1,b3 . b1] in b3 & b3 . b1 in b2;

:: GRFUNC_1:th 88
theorem
for b1 being set
for b2, b3 being Relation-like Function-like set
      st b1 c= proj1 b2 & b2 c= b3
   holds b2 | b1 = b3 | b1;

:: GRFUNC_1:th 89
theorem
for b1 being Relation-like Function-like set
for b2 being set
      st b2 in proj1 b1
   holds b1 | {b2} = {[b2,b1 . b2]};

:: GRFUNC_1:th 90
theorem
for b1, b2 being Relation-like Function-like set
for b3 being set
      st proj1 b1 = proj1 b2 & b1 . b3 = b2 . b3
   holds b1 | {b3} = b2 | {b3};

:: GRFUNC_1:th 91
theorem
for b1, b2 being Relation-like Function-like set
for b3, b4 being set
      st proj1 b1 = proj1 b2 & b1 . b3 = b2 . b3 & b1 . b4 = b2 . b4
   holds b1 | {b3,b4} = b2 | {b3,b4};

:: GRFUNC_1:th 92
theorem
for b1, b2 being Relation-like Function-like set
for b3, b4, b5 being set
      st proj1 b1 = proj1 b2 & b1 . b3 = b2 . b3 & b1 . b4 = b2 . b4 & b1 . b5 = b2 . b5
   holds b1 | {b3,b4,b5} = b2 | {b3,b4,b5};

:: GRFUNC_1:funcreg 1
registration
  let a1 be Relation-like Function-like set;
  let a2 be set;
  cluster a1 \ a2 -> Function-like;
end;

:: GRFUNC_1:th 93
theorem
for b1 being set
for b2, b3 being Relation-like Function-like set
      st b1 in (proj1 b2) \ proj1 b3
   holds (b2 \ b3) . b1 = b2 . b1;

:: GRFUNC_1:th 94
theorem
for b1, b2, b3 being Relation-like Function-like set
      st b1 c= b2 & b1 c= b3
   holds b2 | proj1 b1 = b3 | proj1 b1;