Theorem mpteq2ia 4217

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 4211	. 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 )

This proof depends on syntax axioms:   -> wi 4   A.wal 1400   = wceq 1402   e. wcel 2209   A.wral 2528   |-> cmpt 4192

This proof depends on 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 proof 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 4193  df-mpt 4194

This theorem is used by:  mpteq2i  4218  feqresmpt  5757  elfvmptrab  5802  fmptap  5905  offres  6368  cnrecnv  11676  ege2le3  12438  eirraplem  12544  cnmpt1st  15389  cnmpt2nd  15390  expcn  15670  expcncf  15710  dvexp  15812  dveflem  15827  dvef  15828  elply2  15836  plyid  15847