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Metamath Proof Explorer

Theorem rnmptssff

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 24-Jan-2025)

	Ref	Expression
Hypotheses	rnmptssff.1	$⊢ \underline{Ⅎ} x A$
	rnmptssff.2	$⊢ \underline{Ⅎ} x C$
	rnmptssff.3	$⊢ F = (x \in A ⟼ B)$
Assertion	rnmptssff	$⊢ \forall x \in A B \in C \to ran F \subseteq C$

Step	Hyp	Ref	Expression
1		rnmptssff.1	$⊢ \underline{Ⅎ} x A$
2		rnmptssff.2	$⊢ \underline{Ⅎ} x C$
3		rnmptssff.3	$⊢ F = (x \in A ⟼ B)$
4	1 2 3	fmptff	$⊢ \forall x \in A B \in C \leftrightarrow F : A ⟶ C$
5		frn	$⊢ F : A ⟶ C \to ran F \subseteq C$
6	4 5	sylbi	$⊢ \forall x \in A B \in C \to ran F \subseteq C$