Theorem mpteq12dv 4165

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 4164	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   |-> cmpt 4144

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 4145  df-mpt 4146

This theorem is referenced by:  mpteq12i  4171  offval  6224  offval3  6277  ccatfvalfi  11122  swrdval  11175  odzval  12759  restval  13273  prdsex  13297  prdsval  13301  qusval  13351  grpinvfvalg  13570  grpinvpropdg  13603  opprnegg  14041  lspfval  14346  lsppropd  14390  sraval  14395  psrval  14624  ntrfval  14768  clsfval  14769  neifval  14808  cnpfval  14863  cnprcl2k  14874  reldvg  15347  dvfvalap  15349  eldvap  15350