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	$$⊢F=\left(x\in A⟼B\right)$$
Assertion	rnmptss	$$⊢\forall x\in AB\in C\to \mathrm{ran}F\subseteq C$$

Proof

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