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	$$⊢\underset{\_}{Ⅎ}xC$$
	rnmptssf.2	$$⊢F=\left(x\in A⟼B\right)$$
Assertion	rnmptssf	$$⊢\forall x\in AB\in C\to \mathrm{ran}F\subseteq C$$

Proof

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