Metamath Proof Explorer

Theorem mpteq12da

Description: An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021) Remove dependency on ax-10 . (Revised by SN, 11-Nov-2024)

		Ref	Expression
	Hypotheses	mpteq12da.1	$⊢ Ⅎ x φ$
		mpteq12da.2	$⊢ φ \to A = C$
		mpteq12da.3	$⊢ (φ \land x \in A) \to B = D$
	Assertion	mpteq12da	$⊢ φ \to (x \in A ⟼ B) = (x \in C ⟼ D)$

Proof

Step	Hyp	Ref	Expression
1		mpteq12da.1	$⊢ Ⅎ x φ$
2		mpteq12da.2	$⊢ φ \to A = C$
3		mpteq12da.3	$⊢ (φ \land x \in A) \to B = D$
4		nfv	$⊢ Ⅎ y φ$
5	3	eqeq2d	$⊢ (φ \land x \in A) \to (y = B \leftrightarrow y = D)$
6	5	pm5.32da	$⊢ φ \to ((x \in A \land y = B) \leftrightarrow (x \in A \land y = D))$
7	2	eleq2d	$⊢ φ \to (x \in A \leftrightarrow x \in C)$
8	7	anbi1d	$⊢ φ \to ((x \in A \land y = D) \leftrightarrow (x \in C \land y = D))$
9	6 8	bitrd	$⊢ φ \to ((x \in A \land y = B) \leftrightarrow (x \in C \land y = D))$
10	1 4 9	opabbid	$⊢ φ \to \{⟨x, y⟩ \| (x \in A \land y = B)\} = \{⟨x, y⟩ \| (x \in C \land y = D)\}$
11		df-mpt	$⊢ (x \in A ⟼ B) = \{⟨x, y⟩ \| (x \in A \land y = B)\}$
12		df-mpt	$⊢ (x \in C ⟼ D) = \{⟨x, y⟩ \| (x \in C \land y = D)\}$
13	10 11 12	3eqtr4g	$⊢ φ \to (x \in A ⟼ B) = (x \in C ⟼ D)$