Theorem mpteq2ia 4212

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 2238	. . 3 |- A = A
2	1	ax-gen 1502	. 2 |- A. x A = A
3		mpteq2ia.1	. . 3 |- ( x e. A -> B = C )
4	3	rgen 2603	. 2 |- A. x e. A B = C
5		mpteq12f 4206	. 2 |- ( ( A. x A = A /\ A. x e. A B = C ) -> ( x e. A |-> B ) = ( x e. A |-> C ) )
6	2, 4, 5	mp2an 430	1 |- ( x e. A |-> B ) = ( x e. A |-> C )

Syntax hints:   -> wi 4   A.wal 1400   = 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:  mpteq2i  4213  feqresmpt  5751  elfvmptrab  5795  fmptap  5896  offres  6358  cnrecnv  11654  ege2le3  12416  eirraplem  12522  cnmpt1st  15312  cnmpt2nd  15313  expcn  15593  expcncf  15633  dvexp  15735  dveflem  15750  dvef  15751  elply2  15759  plyid  15770