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Metamath Proof Explorer

Theorem rnmptss

Description: The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017)

		Ref	Expression
	Hypothesis	rnmptss.1	$⊢ F = (x \in A ⟼ B)$
	Assertion	rnmptss	$⊢ \forall x \in A B \in C \to ran F \subseteq C$

Step	Hyp	Ref	Expression
1		rnmptss.1	$⊢ F = (x \in A ⟼ B)$
2	1	fmpt	$⊢ \forall x \in A B \in C \leftrightarrow F : A ⟶ C$
3		frn	$⊢ F : A ⟶ C \to ran F \subseteq C$
4	2 3	sylbi	$⊢ \forall x \in A B \in C \to ran F \subseteq C$