Theorem mpteq12dv 4169

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 4168	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   |-> cmpt 4148

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-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ral 2513  df-opab 4149  df-mpt 4150

This theorem is referenced by:  mpteq12i  4175  offval  6238  offval3  6291  ccatfvalfi  11159  swrdval  11219  odzval  12804  restval  13318  prdsex  13342  prdsval  13346  qusval  13396  grpinvfvalg  13615  grpinvpropdg  13648  opprnegg  14086  lspfval  14392  lsppropd  14436  sraval  14441  psrval  14670  ntrfval  14814  clsfval  14815  neifval  14854  cnpfval  14909  cnprcl2k  14920  reldvg  15393  dvfvalap  15395  eldvap  15396  vtxdgfval  16094  vtxdgop  16098  vtxdeqd  16102