Theorem mptcnv 5139

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 4155	. 2 |- ( ph -> { <. y , x >. | ( x e. A /\ y = B ) } = { <. y , x >. | ( y e. C /\ x = D ) } )
3		df-mpt 4152	. . . 4 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
4	3	cnveqi 4905	. . 3 |- `' ( x e. A |-> B ) = `' { <. x , y >. | ( x e. A /\ y = B ) }
5		cnvopab 5138	. . 3 |- `' { <. x , y >. | ( x e. A /\ y = B ) } = { <. y , x >. | ( x e. A /\ y = B ) }
6	4, 5	eqtri 2252	. 2 |- `' ( x e. A |-> B ) = { <. y , x >. | ( x e. A /\ y = B ) }
7		df-mpt 4152	. 2 |- ( y e. C |-> D ) = { <. y , x >. | ( y e. C /\ x = D ) }
8	2, 6, 7	3eqtr4g 2289	1 |- ( ph -> `' ( x e. A |-> B ) = ( y e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   {copab 4149   |-> cmpt 4150   `'ccnv 4724

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-mpt 4152  df-xp 4731  df-rel 4732  df-cnv 4733

This theorem is referenced by: (None)