Theorem mpteq1 4020

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Assertion

Ref	Expression
mpteq1	|- ( A = B -> ( x e. A |-> C ) = ( x e. B |-> C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem mpteq1

Step	Hyp	Ref	Expression
1		eqidd 2141	. . 3 |- ( x e. A -> C = C )
2	1	rgen 2488	. 2 |- A. x e. A C = C
3		mpteq12 4019	. 2 |- ( ( A = B /\ A. x e. A C = C ) -> ( x e. A |-> C ) = ( x e. B |-> C ) )
4	2, 3	mpan2 422	1 |- ( A = B -> ( x e. A |-> C ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   A.wral 2417   |-> cmpt 3997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-opab 3998  df-mpt 3999

This theorem is referenced by:  mpteq1d  4021  fmptap  5618  mpompt  5871  mpomptsx  6103  mpompts  6104  tposf12  6174  restco  12382  cnmpt1t  12493  cnmpt2t  12501