Theorem mpteq2ia 4075

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

Hypothesis

Ref	Expression
mpteq2ia.1	|- ( x e. A -> B = C )

Assertion

Ref	Expression
mpteq2ia	|- ( x e. A |-> B ) = ( x e. A |-> C )

Proof of Theorem mpteq2ia

Step	Hyp	Ref	Expression
1		eqid 2170	. . 3 |- A = A
2	1	ax-gen 1442	. 2 |- A. x A = A
3		mpteq2ia.1	. . 3 |- ( x e. A -> B = C )
4	3	rgen 2523	. 2 |- A. x e. A B = C
5		mpteq12f 4069	. 2 |- ( ( A. x A = A /\ A. x e. A B = C ) -> ( x e. A |-> B ) = ( x e. A |-> C ) )
6	2, 4, 5	mp2an 424	1 |- ( x e. A |-> B ) = ( x e. A |-> C )

Syntax hints:   -> wi 4   A.wal 1346   = wceq 1348   e. wcel 2141   A.wral 2448   |-> cmpt 4050

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-opab 4051  df-mpt 4052

This theorem is referenced by:  mpteq2i  4076  feqresmpt  5550  elfvmptrab  5591  fmptap  5686  offres  6114  cnrecnv  10874  ege2le3  11634  eirraplem  11739  cnmpt1st  13082  cnmpt2nd  13083  expcncf  13386  dvexp  13469  dveflem  13481  dvef  13482