Theorem mpteq2ia 4131

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 4125	. 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 4106

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 4107  df-mpt 4108

This theorem is referenced by:  mpteq2i  4132  feqresmpt  5635  elfvmptrab  5677  fmptap  5776  offres  6222  cnrecnv  11254  ege2le3  12015  eirraplem  12121  cnmpt1st  14793  cnmpt2nd  14794  expcn  15074  expcncf  15114  dvexp  15216  dveflem  15231  dvef  15232  elply2  15240  plyid  15251