Theorem mpteq12f 3858

Description: An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Assertion

Ref	Expression
mpteq12f	|- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Proof of Theorem mpteq12f

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfa1 1474	. . . 4 |- F/ x A. x A = C
2		nfra1 2397	. . . 4 |- F/ x A. x e. A B = D
3	1, 2	nfan 1497	. . 3 |- F/ x ( A. x A = C /\ A. x e. A B = D )
4		nfv 1461	. . 3 |- F/ y ( A. x A = C /\ A. x e. A B = D )
5		rsp 2411	. . . . . . 7 |- ( A. x e. A B = D -> ( x e. A -> B = D ) )
6	5	imp 122	. . . . . 6 |- ( ( A. x e. A B = D /\ x e. A ) -> B = D )
7	6	eqeq2d 2092	. . . . 5 |- ( ( A. x e. A B = D /\ x e. A ) -> ( y = B <-> y = D ) )
8	7	pm5.32da 439	. . . 4 |- ( A. x e. A B = D -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ y = D ) ) )
9		sp 1441	. . . . . 6 |- ( A. x A = C -> A = C )
10	9	eleq2d 2148	. . . . 5 |- ( A. x A = C -> ( x e. A <-> x e. C ) )
11	10	anbi1d 452	. . . 4 |- ( A. x A = C -> ( ( x e. A /\ y = D ) <-> ( x e. C /\ y = D ) ) )
12	8, 11	sylan9bbr 450	. . 3 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = D ) ) )
13	3, 4, 12	opabbid 3843	. 2 |- ( ( A. x A = C /\ A. x e. A B = D ) -> { <. x , y >. | ( x e. A /\ y = B ) } = { <. x , y >. | ( x e. C /\ y = D ) } )
14		df-mpt 3841	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
15		df-mpt 3841	. 2 |- ( x e. C |-> D ) = { <. x , y >. | ( x e. C /\ y = D ) }
16	13, 14, 15	3eqtr4g 2138	1 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   = wceq 1284   e. wcel 1433   A.wral 2348   {copab 3838   |-> cmpt 3839

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-opab 3840  df-mpt 3841

This theorem is referenced by:  mpteq12dva  3859  mpteq12  3861  mpteq2ia  3864  mpteq2da  3867