Theorem mpteq12f 4163

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 1587	. . . 4 |- F/ x A. x A = C
2		nfra1 2561	. . . 4 |- F/ x A. x e. A B = D
3	1, 2	nfan 1611	. . 3 |- F/ x ( A. x A = C /\ A. x e. A B = D )
4		nfv 1574	. . 3 |- F/ y ( A. x A = C /\ A. x e. A B = D )
5		rsp 2577	. . . . . . 7 |- ( A. x e. A B = D -> ( x e. A -> B = D ) )
6	5	imp 124	. . . . . 6 |- ( ( A. x e. A B = D /\ x e. A ) -> B = D )
7	6	eqeq2d 2241	. . . . 5 |- ( ( A. x e. A B = D /\ x e. A ) -> ( y = B <-> y = D ) )
8	7	pm5.32da 452	. . . 4 |- ( A. x e. A B = D -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ y = D ) ) )
9		sp 1557	. . . . . 6 |- ( A. x A = C -> A = C )
10	9	eleq2d 2299	. . . . 5 |- ( A. x A = C -> ( x e. A <-> x e. C ) )
11	10	anbi1d 465	. . . 4 |- ( A. x A = C -> ( ( x e. A /\ y = D ) <-> ( x e. C /\ y = D ) ) )
12	8, 11	sylan9bbr 463	. . 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 4148	. 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 4146	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
15		df-mpt 4146	. 2 |- ( x e. C |-> D ) = { <. x , y >. | ( x e. C /\ y = D ) }
16	13, 14, 15	3eqtr4g 2287	1 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1393   = wceq 1395   e. wcel 2200   A.wral 2508   {copab 4143   |-> cmpt 4144

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ral 2513  df-opab 4145  df-mpt 4146

This theorem is referenced by:  mpteq12dva  4164  mpteq12  4166  mpteq2ia  4169  mpteq2da  4172