Theorem mpteq1 4210

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 2239	. . 3 |- ( x e. A -> C = C )
2	1	rgen 2603	. 2 |- A. x e. A C = C
3		mpteq12 4209	. 2 |- ( ( A = B /\ A. x e. A C = C ) -> ( x e. A |-> C ) = ( x e. B |-> C ) )
4	2, 3	mpan2 429	1 |- ( A = B -> ( x e. A |-> C ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   A.wral 2528   |-> cmpt 4187

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-opab 4188  df-mpt 4189

This theorem is referenced by:  mpteq1d  4211  fmptap  5896  mpompt  6170  mpomptsx  6423  mpompts  6424  tposf12  6530  gsumzfi  14135  gsummptfidmadd  14138  gsumconstcmn  14143  gsumfsum  14895  restco  15198  cnmpt1t  15309  cnmpt2t  15317