Theorem mpteq12da 39452

Description: An equality inference for the maps to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
mpteq12da.1	|- F/ x ph
mpteq12da.2	|- ( ph -> A = C )
mpteq12da.3	|- ( ( ph /\ x e. A ) -> B = D )

Assertion

Ref	Expression
mpteq12da	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Proof of Theorem mpteq12da

Step	Hyp	Ref	Expression
1		mpteq12da.1	. . 3 |- F/ x ph
2		mpteq12da.2	. . 3 |- ( ph -> A = C )
3	1, 2	alrimi 2082	. 2 |- ( ph -> A. x A = C )
4		mpteq12da.3	. . . 4 |- ( ( ph /\ x e. A ) -> B = D )
5	4	ex 450	. . 3 |- ( ph -> ( x e. A -> B = D ) )
6	1, 5	ralrimi 2957	. 2 |- ( ph -> A. x e. A B = D )
7		mpteq12f 4731	. 2 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )
8	3, 6, 7	syl2anc 693	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   |-> 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:  smflimmpt  41016  smflimsupmpt  41035  smfliminfmpt  41038