Theorem mpteq12d 4734

Description: An equality inference for the maps to notation. Compare mpteq12dv 4733. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
mpteq12d.1	|- F/ x ph
mpteq12d.3	|- ( ph -> A = C )
mpteq12d.4	|- ( ph -> B = D )

Assertion

Ref	Expression
mpteq12d	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Proof of Theorem mpteq12d

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpteq12d.1	. . 3 |- F/ x ph
2		nfv 1843	. . 3 |- F/ y ph
3		mpteq12d.3	. . . . 5 |- ( ph -> A = C )
4	3	eleq2d 2687	. . . 4 |- ( ph -> ( x e. A <-> x e. C ) )
5		mpteq12d.4	. . . . 5 |- ( ph -> B = D )
6	5	eqeq2d 2632	. . . 4 |- ( ph -> ( y = B <-> y = D ) )
7	4, 6	anbi12d 747	. . 3 |- ( ph -> ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = D ) ) )
8	1, 2, 7	opabbid 4715	. 2 |- ( ph -> { <. x , y >. | ( x e. A /\ y = B ) } = { <. x , y >. | ( x e. C /\ y = D ) } )
9		df-mpt 4730	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
10		df-mpt 4730	. 2 |- ( x e. C |-> D ) = { <. x , y >. | ( x e. C /\ y = D ) }
11	8, 9, 10	3eqtr4g 2681	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   {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-opab 4713  df-mpt 4730

This theorem is referenced by:  smflimmpt  41016