Theorem mptcnv 5137

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 4153	. 2 |- ( ph -> { <. y , x >. | ( x e. A /\ y = B ) } = { <. y , x >. | ( y e. C /\ x = D ) } )
3		df-mpt 4150	. . . 4 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
4	3	cnveqi 4903	. . 3 |- `' ( x e. A |-> B ) = `' { <. x , y >. | ( x e. A /\ y = B ) }
5		cnvopab 5136	. . 3 |- `' { <. x , y >. | ( x e. A /\ y = B ) } = { <. y , x >. | ( x e. A /\ y = B ) }
6	4, 5	eqtri 2250	. 2 |- `' ( x e. A |-> B ) = { <. y , x >. | ( x e. A /\ y = B ) }
7		df-mpt 4150	. 2 |- ( y e. C |-> D ) = { <. y , x >. | ( y e. C /\ x = D ) }
8	2, 6, 7	3eqtr4g 2287	1 |- ( ph -> `' ( x e. A |-> B ) = ( y e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {copab 4147   |-> cmpt 4148   `'ccnv 4722

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-mpt 4150  df-xp 4729  df-rel 4730  df-cnv 4731

This theorem is referenced by: (None)