Theorem mpteq1df 39443

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

Hypotheses

Ref	Expression
mpteq1df.1	|- F/ x ph
mpteq1df.2	|- ( ph -> A = B )

Assertion

Ref	Expression
mpteq1df	|- ( ph -> ( x e. A |-> C ) = ( x e. B |-> C ) )

Proof of Theorem mpteq1df

Step	Hyp	Ref	Expression
1		mpteq1df.1	. . 3 |- F/ x ph
2		mpteq1df.2	. . 3 |- ( ph -> A = B )
3	1, 2	alrimi 2082	. 2 |- ( ph -> A. x A = B )
4		eqid 2622	. . . 4 |- C = C
5	4	rgenw 2924	. . 3 |- A. x e. A C = C
6	5	a1i 11	. 2 |- ( ph -> A. x e. A C = C )
7		mpteq12f 4731	. 2 |- ( ( A. x A = B /\ A. x e. A C = C ) -> ( x e. A |-> C ) = ( x e. B |-> C ) )
8	3, 6, 7	syl2anc 693	1 |- ( ph -> ( x e. A |-> C ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   F/wnf 1708   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:  smfliminflem  41036