Theorem mpteq2ia 4120

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 2196	. . 3 |- A = A
2	1	ax-gen 1463	. 2 |- A. x A = A
3		mpteq2ia.1	. . 3 |- ( x e. A -> B = C )
4	3	rgen 2550	. 2 |- A. x e. A B = C
5		mpteq12f 4114	. 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 1362   = wceq 1364   e. wcel 2167   A.wral 2475   |-> cmpt 4095

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-opab 4096  df-mpt 4097

This theorem is referenced by:  mpteq2i  4121  feqresmpt  5616  elfvmptrab  5658  fmptap  5753  offres  6193  cnrecnv  11077  ege2le3  11838  eirraplem  11944  cnmpt1st  14534  cnmpt2nd  14535  expcn  14815  expcncf  14855  dvexp  14957  dveflem  14972  dvef  14973  elply2  14981  plyid  14992