Theorem mpteq1d 4129

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 4128	. 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 1373   |-> cmpt 4105

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-ral 2489  df-opab 4106  df-mpt 4107

This theorem is referenced by:  mptimass  5035  fmptapd  5775  offval  6166  swrd00g  11102  swrdlend  11111  swrd0g  11113  qusex  13157  mulgnn0gsum  13464  gsumfzconst  13677  gsumfzsnfd  13681  gsumfzfsumlem0  14348  gsumfzfsumlemm  14349