Theorem mpoeq3dva 6032

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 1207	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = D )
3	2	eqeq2d 2219	. . . 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 6022	. 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 5972	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
7		df-mpo 5972	. 2 |- ( x e. A , y e. B |-> D ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = D ) }
8	5, 6, 7	3eqtr4g 2265	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2178   {coprab 5968   e. cmpo 5969

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-oprab 5971  df-mpo 5972

This theorem is referenced by:  mpoeq3ia  6033  mpoeq3dv  6034  ofeq  6184  fmpoco  6325  mapxpen  6970  seqeq2  10633  seqeq3  10634  grpsubpropd2  13552  mulgpropdg  13615  cnmpt2t  14880  cnmpt22  14881  cnmptcom  14885