Theorem mpteq12dv 4213

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 4212	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   |-> cmpt 4192

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-opab 4193  df-mpt 4194

This theorem is used by:  mpteq12i  4219  offval  6310  offval3  6367  indv  9297  ccatfvalfi  11374  swrdval  11434  odzval  13040  restval  13648  qusval  13693  grpinvfvalg  13896  grpinvpropdg  13929  prdsex  14221  prdsval  14222  opprnegg  14438  lspfval  14774  lsppropd  14818  sraval  14823  aspval  15064  asclfval  15070  ressascl  15088  psrval  15099  ntrfval  15250  clsfval  15251  neifval  15290  cnpfval  15345  cnprcl2k  15356  reldvg  15829  dvfvalap  15831  eldvap  15832  vtxdgfval  16627  vtxdgop  16631  vtxdeqd  16635