Theorem mpoeq3dva 6080

Description: Slightly more general equality inference for the maps-to notation. (Contributed by NM, 17-Oct-2013.)

Hypothesis

Ref	Expression
mpoeq3dva.1	|- ( ( ph /\ x e. A /\ y e. B ) -> C = D )

Assertion

Ref	Expression
mpoeq3dva	|- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )

Distinct variable groups:   ph, x   ph, y

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

Proof of Theorem mpoeq3dva

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpoeq3dva.1	. . . . . 6 |- ( ( ph /\ x e. A /\ y e. B ) -> C = D )
2	1	3expb 1228	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = D )
3	2	eqeq2d 2241	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( z = C <-> z = D ) )
4	3	pm5.32da 452	. . 3 |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. A /\ y e. B ) /\ z = D ) ) )
5	4	oprabbidv 6070	. 2 |- ( ph -> { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = D ) } )
6		df-mpo 6018	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
7		df-mpo 6018	. 2 |- ( x e. A , y e. B |-> D ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = D ) }
8	5, 6, 7	3eqtr4g 2287	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   {coprab 6014   e. cmpo 6015

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-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-oprab 6017  df-mpo 6018

This theorem is referenced by:  mpoeq3ia  6081  mpoeq3dv  6082  ofeq  6233  fmpoco  6376  mapxpen  7029  seqeq2  10703  seqeq3  10704  grpsubpropd2  13678  mulgpropdg  13741  cnmpt2t  15007  cnmpt22  15008  cnmptcom  15012