Theorem mptcnv 5146

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 4160	. 2 |- ( ph -> { <. y , x >. | ( x e. A /\ y = B ) } = { <. y , x >. | ( y e. C /\ x = D ) } )
3		df-mpt 4157	. . . 4 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
4	3	cnveqi 4911	. . 3 |- `' ( x e. A |-> B ) = `' { <. x , y >. | ( x e. A /\ y = B ) }
5		cnvopab 5145	. . 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 4157	. 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 1398   e. wcel 2202   {copab 4154   |-> cmpt 4155   `'ccnv 4730

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-mpt 4157  df-xp 4737  df-rel 4738  df-cnv 4739

This theorem is referenced by: (None)