Theorem mpteq12f 3837

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 1434	. . . 4 |- F/ x A. x A = C
2		nfra1 2355	. . . 4 |- F/ x A. x e. A B = D
3	1, 2	nfan 1457	. . 3 |- F/ x ( A. x A = C /\ A. x e. A B = D )
4		nfv 1421	. . 3 |- F/ y ( A. x A = C /\ A. x e. A B = D )
5		rsp 2369	. . . . . . 7 |- ( A. x e. A B = D -> ( x e. A -> B = D ) )
6	5	imp 115	. . . . . 6 |- ( ( A. x e. A B = D /\ x e. A ) -> B = D )
7	6	eqeq2d 2051	. . . . 5 |- ( ( A. x e. A B = D /\ x e. A ) -> ( y = B <-> y = D ) )
8	7	pm5.32da 425	. . . 4 |- ( A. x e. A B = D -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ y = D ) ) )
9		sp 1401	. . . . . 6 |- ( A. x A = C -> A = C )
10	9	eleq2d 2107	. . . . 5 |- ( A. x A = C -> ( x e. A <-> x e. C ) )
11	10	anbi1d 438	. . . 4 |- ( A. x A = C -> ( ( x e. A /\ y = D ) <-> ( x e. C /\ y = D ) ) )
12	8, 11	sylan9bbr 436	. . 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 3822	. 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 3820	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
15		df-mpt 3820	. 2 |- ( x e. C |-> D ) = { <. x , y >. | ( x e. C /\ y = D ) }
16	13, 14, 15	3eqtr4g 2097	1 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   e. wcel 1393   A.wral 2306   {copab 3817   |-> cmpt 3818

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-opab 3819  df-mpt 3820

This theorem is referenced by:  mpteq12dva  3838  mpteq12  3840  mpteq2ia  3843  mpteq2da  3846