Theorem mpteq12f 4731

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 2028	. . . 4 |- F/ x A. x A = C
2		nfra1 2941	. . . 4 |- F/ x A. x e. A B = D
3	1, 2	nfan 1828	. . 3 |- F/ x ( A. x A = C /\ A. x e. A B = D )
4		nfv 1843	. . 3 |- F/ y ( A. x A = C /\ A. x e. A B = D )
5		rspa 2930	. . . . . 6 |- ( ( A. x e. A B = D /\ x e. A ) -> B = D )
6	5	eqeq2d 2632	. . . . 5 |- ( ( A. x e. A B = D /\ x e. A ) -> ( y = B <-> y = D ) )
7	6	pm5.32da 673	. . . 4 |- ( A. x e. A B = D -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ y = D ) ) )
8		sp 2053	. . . . . 6 |- ( A. x A = C -> A = C )
9	8	eleq2d 2687	. . . . 5 |- ( A. x A = C -> ( x e. A <-> x e. C ) )
10	9	anbi1d 741	. . . 4 |- ( A. x A = C -> ( ( x e. A /\ y = D ) <-> ( x e. C /\ y = D ) ) )
11	7, 10	sylan9bbr 737	. . 3 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = D ) ) )
12	3, 4, 11	opabbid 4715	. 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 ) } )
13		df-mpt 4730	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
14		df-mpt 4730	. 2 |- ( x e. C |-> D ) = { <. x , y >. | ( x e. C /\ y = D ) }
15	12, 13, 14	3eqtr4g 2681	1 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   {copab 4712   |-> cmpt 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-opab 4713  df-mpt 4730

This theorem is referenced by:  mpteq12dva  4732  mpteq12  4736  mpteq2ia  4740  mpteq2da  4743  esumeq12dvaf  30093  refsum2cnlem1  39196  mpteq1df  39443  mpteq12da  39452  smfsupmpt  41021  smfinflem  41023  smfinfmpt  41025