Theorem mpteq1 4012

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 2140	. . 3 |- ( x e. A -> C = C )
2	1	rgen 2485	. 2 |- A. x e. A C = C
3		mpteq12 4011	. 2 |- ( ( A = B /\ A. x e. A C = C ) -> ( x e. A |-> C ) = ( x e. B |-> C ) )
4	2, 3	mpan2 421	1 |- ( A = B -> ( x e. A |-> C ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   A.wral 2416   |-> cmpt 3989

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-opab 3990  df-mpt 3991

This theorem is referenced by:  mpteq1d  4013  fmptap  5610  mpompt  5863  mpomptsx  6095  mpompts  6096  tposf12  6166  restco  12343  cnmpt1t  12454  cnmpt2t  12462