Theorem mpteq1 4168

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 2230	. . 3 |- ( x e. A -> C = C )
2	1	rgen 2583	. 2 |- A. x e. A C = C
3		mpteq12 4167	. 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 1395   e. wcel 2200   A.wral 2508   |-> cmpt 4145

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ral 2513  df-opab 4146  df-mpt 4147

This theorem is referenced by:  mpteq1d  4169  fmptap  5829  mpompt  6096  mpomptsx  6343  mpompts  6344  tposf12  6415  restco  14848  cnmpt1t  14959  cnmpt2t  14967