Theorem mpteq12dv 4171

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 4170	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   |-> cmpt 4150

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2515  df-opab 4151  df-mpt 4152

This theorem is referenced by:  mpteq12i  4177  offval  6243  offval3  6296  ccatfvalfi  11173  swrdval  11233  odzval  12832  restval  13346  prdsex  13370  prdsval  13374  qusval  13424  grpinvfvalg  13643  grpinvpropdg  13676  opprnegg  14115  lspfval  14421  lsppropd  14465  sraval  14470  psrval  14699  ntrfval  14843  clsfval  14844  neifval  14883  cnpfval  14938  cnprcl2k  14949  reldvg  15422  dvfvalap  15424  eldvap  15425  vtxdgfval  16158  vtxdgop  16162  vtxdeqd  16166