Theorem mptcnv 5023

Description: The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)

Hypothesis

Ref	Expression
mptcnv.1	|- ( ph -> ( ( x e. A /\ y = B ) <-> ( y e. C /\ x = D ) ) )

Assertion

Ref	Expression
mptcnv	|- ( ph -> `' ( x e. A |-> B ) = ( y e. C |-> D ) )

Distinct variable groups:   x, y, ph   x, C   x, D   y, A   y, B

Allowed substitution hints:   A( x)   B( x)   C( y)   D( y)

Proof of Theorem mptcnv

Step	Hyp	Ref	Expression
1		mptcnv.1	. . 3 |- ( ph -> ( ( x e. A /\ y = B ) <-> ( y e. C /\ x = D ) ) )
2	1	opabbidv 4064	. 2 |- ( ph -> { <. y , x >. | ( x e. A /\ y = B ) } = { <. y , x >. | ( y e. C /\ x = D ) } )
3		df-mpt 4061	. . . 4 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
4	3	cnveqi 4795	. . 3 |- `' ( x e. A |-> B ) = `' { <. x , y >. | ( x e. A /\ y = B ) }
5		cnvopab 5022	. . 3 |- `' { <. x , y >. | ( x e. A /\ y = B ) } = { <. y , x >. | ( x e. A /\ y = B ) }
6	4, 5	eqtri 2196	. 2 |- `' ( x e. A |-> B ) = { <. y , x >. | ( x e. A /\ y = B ) }
7		df-mpt 4061	. 2 |- ( y e. C |-> D ) = { <. y , x >. | ( y e. C /\ x = D ) }
8	2, 6, 7	3eqtr4g 2233	1 |- ( ph -> `' ( x e. A |-> B ) = ( y e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2146   {copab 4058   |-> cmpt 4059   `'ccnv 4619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-mpt 4061  df-xp 4626  df-rel 4627  df-cnv 4628

This theorem is referenced by: (None)