Theorem mpteq2ia 4130

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 2205	. . 3 |- A = A
2	1	ax-gen 1472	. 2 |- A. x A = A
3		mpteq2ia.1	. . 3 |- ( x e. A -> B = C )
4	3	rgen 2559	. 2 |- A. x e. A B = C
5		mpteq12f 4124	. 2 |- ( ( A. x A = A /\ A. x e. A B = C ) -> ( x e. A |-> B ) = ( x e. A |-> C ) )
6	2, 4, 5	mp2an 426	1 |- ( x e. A |-> B ) = ( x e. A |-> C )

Syntax hints:   -> wi 4   A.wal 1371   = wceq 1373   e. wcel 2176   A.wral 2484   |-> cmpt 4105

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-ral 2489  df-opab 4106  df-mpt 4107

This theorem is referenced by:  mpteq2i  4131  feqresmpt  5633  elfvmptrab  5675  fmptap  5774  offres  6220  cnrecnv  11221  ege2le3  11982  eirraplem  12088  cnmpt1st  14760  cnmpt2nd  14761  expcn  15041  expcncf  15081  dvexp  15183  dveflem  15198  dvef  15199  elply2  15207  plyid  15218