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	$$⊢\underset{\_}{Ⅎ}xA$$
	rnmptssff.2	$$⊢\underset{\_}{Ⅎ}xC$$
	rnmptssff.3	$$⊢F=\left(x\in A⟼B\right)$$
Assertion	rnmptssff	$$⊢\forall x\in AB\in C\to \mathrm{ran}F\subseteq C$$

Proof

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