Theorem mpoeq3dva 5835

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 1182	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = D )
3	2	eqeq2d 2151	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( z = C <-> z = D ) )
4	3	pm5.32da 447	. . 3 |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. A /\ y e. B ) /\ z = D ) ) )
5	4	oprabbidv 5825	. 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 5779	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
7		df-mpo 5779	. 2 |- ( x e. A , y e. B |-> D ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = D ) }
8	5, 6, 7	3eqtr4g 2197	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   {coprab 5775   e. cmpo 5776

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-oprab 5778  df-mpo 5779

This theorem is referenced by:  mpoeq3ia  5836  mpoeq3dv  5837  ofeq  5984  fmpoco  6113  mapxpen  6742  seqeq2  10225  seqeq3  10226  cnmpt2t  12465  cnmpt22  12466  cnmptcom  12470