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Metamath Proof Explorer

Theorem rnmptssf

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021)

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

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