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  9298  ccatfvalfi  11376  swrdval  11436  odzval  13043  restval  13652  qusval  13697  grpinvfvalg  13900  grpinvpropdg  13933  cntzex  14144  cntzfval  14146  prdsex  14256  prdsval  14257  opprnegg  14473  lspfval  14809  lsppropd  14853  sraval  14858  aspval  15099  asclfval  15105  ressascl  15123  psrval  15134  psrmulvalfi  15160  ntrfval  15292  clsfval  15293  neifval  15332  cnpfval  15387  cnprcl2k  15398  reldvg  15871  dvfvalap  15873  eldvap  15874  vtxdgfval  16695  vtxdgop  16699  vtxdeqd  16703