Theorem mpteq12dv 4192

Description: An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)

Hypotheses

Ref	Expression
mpteq12dv.1	|- ( ph -> A = C )
mpteq12dv.2	|- ( ph -> B = D )

Assertion

Ref	Expression
mpteq12dv	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Distinct variable group:   ph, x

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

Proof of Theorem mpteq12dv

Step	Hyp	Ref	Expression
1		mpteq12dv.1	. 2 |- ( ph -> A = C )
2		mpteq12dv.2	. . 3 |- ( ph -> B = D )
3	2	adantr 276	. 2 |- ( ( ph /\ x e. A ) -> B = D )
4	1, 3	mpteq12dva 4191	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   |-> cmpt 4171

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-ral 2525  df-opab 4172  df-mpt 4173

This theorem is referenced by:  mpteq12i  4198  offval  6274  offval3  6327  ccatfvalfi  11280  swrdval  11340  odzval  12939  restval  13458  prdsex  13482  prdsval  13486  qusval  13536  grpinvfvalg  13755  grpinvpropdg  13788  opprnegg  14227  lspfval  14536  lsppropd  14580  sraval  14585  psrval  14814  ntrfval  14965  clsfval  14966  neifval  15005  cnpfval  15060  cnprcl2k  15071  reldvg  15544  dvfvalap  15546  eldvap  15547  vtxdgfval  16283  vtxdgop  16287  vtxdeqd  16291