Theorem mpteq1d 4211

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)

Hypothesis

Ref	Expression
mpteq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
mpteq1d	|- ( ph -> ( x e. A |-> C ) = ( x e. B |-> C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   C( x)

Proof of Theorem mpteq1d

Step	Hyp	Ref	Expression
1		mpteq1d.1	. 2 |- ( ph -> A = B )
2		mpteq1 4210	. 2 |- ( A = B -> ( x e. A |-> C ) = ( x e. B |-> C ) )
3	1, 2	syl 14	1 |- ( ph -> ( x e. A |-> C ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   = wceq 1402   |-> cmpt 4187

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 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 theorem 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 4188  df-mpt 4189

This theorem is referenced by:  mptimass  5134  fmptapd  5897  offval  6300  swrd00g  11399  swrdlend  11408  swrd0g  11410  qusex  13623  mulgnn0gzsum  13908  gzsumconst  14120  gzsumsnfd  14124  gzsumshift  14126  gzsumgsum  14132