Theorem mpoeq3dva 5917

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 1199	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = D )
3	2	eqeq2d 2182	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( z = C <-> z = D ) )
4	3	pm5.32da 449	. . 3 |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. A /\ y e. B ) /\ z = D ) ) )
5	4	oprabbidv 5907	. 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 5858	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
7		df-mpo 5858	. 2 |- ( x e. A , y e. B |-> D ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = D ) }
8	5, 6, 7	3eqtr4g 2228	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   {coprab 5854   e. cmpo 5855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-oprab 5857  df-mpo 5858

This theorem is referenced by:  mpoeq3ia  5918  mpoeq3dv  5919  ofeq  6063  fmpoco  6195  mapxpen  6826  seqeq2  10405  seqeq3  10406  cnmpt2t  13087  cnmpt22  13088  cnmptcom  13092