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Metamath Proof Explorer

Theorem fvmptelcdm

Description: The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021)

		Ref	Expression
	Hypothesis	fvmptelcdm.1	$⊢ φ \to (x \in A ⟼ B) : A ⟶ C$
	Assertion	fvmptelcdm	$⊢ (φ \land x \in A) \to B \in C$

Step	Hyp	Ref	Expression
1		fvmptelcdm.1	$⊢ φ \to (x \in A ⟼ B) : A ⟶ C$
2		eqid	$⊢ (x \in A ⟼ B) = (x \in A ⟼ B)$
3	2	fmpt	$⊢ \forall x \in A B \in C \leftrightarrow (x \in A ⟼ B) : A ⟶ C$
4	1 3	sylibr	$⊢ φ \to \forall x \in A B \in C$
5	4	r19.21bi	$⊢ (φ \land x \in A) \to B \in C$