Theorem mpteq12dv 4125

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 4124	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175   |-> cmpt 4104

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-ral 2488  df-opab 4105  df-mpt 4106

This theorem is referenced by:  mpteq12i  4131  offval  6165  offval3  6218  ccatfvalfi  11046  odzval  12535  restval  13048  prdsex  13072  prdsval  13076  qusval  13126  grpinvfvalg  13345  grpinvpropdg  13378  opprnegg  13816  lspfval  14121  lsppropd  14165  sraval  14170  psrval  14399  ntrfval  14543  clsfval  14544  neifval  14583  cnpfval  14638  cnprcl2k  14649  reldvg  15122  dvfvalap  15124  eldvap  15125