Theorem mpteq1 4086

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 2178	. . 3 |- ( x e. A -> C = C )
2	1	rgen 2530	. 2 |- A. x e. A C = C
3		mpteq12 4085	. 2 |- ( ( A = B /\ A. x e. A C = C ) -> ( x e. A |-> C ) = ( x e. B |-> C ) )
4	2, 3	mpan2 425	1 |- ( A = B -> ( x e. A |-> C ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   A.wral 2455   |-> cmpt 4063

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-opab 4064  df-mpt 4065

This theorem is referenced by:  mpteq1d  4087  fmptap  5704  mpompt  5963  mpomptsx  6194  mpompts  6195  tposf12  6266  restco  13536  cnmpt1t  13647  cnmpt2t  13655