Theorem rnmptssdff

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	rnmptssdff.1	$$⊢Ⅎx\phi$$
	rnmptssdff.2	$$⊢\underset{\_}{Ⅎ}xA$$
	rnmptssdff.3	$$⊢\underset{\_}{Ⅎ}xC$$
	rnmptssdff.4	$$⊢F=\left(x\in A⟼B\right)$$
	rnmptssdff.5	$$⊢\left(\phi \wedge x\in A\right)\to B\in C$$
Assertion	rnmptssdff	$$⊢\phi \to \mathrm{ran}F\subseteq C$$

Proof

Step	Hyp	Ref	Expression
1		rnmptssdff.1	$$⊢Ⅎx\phi$$
2		rnmptssdff.2	$$⊢\underset{\_}{Ⅎ}xA$$
3		rnmptssdff.3	$$⊢\underset{\_}{Ⅎ}xC$$
4		rnmptssdff.4	$$⊢F=\left(x\in A⟼B\right)$$
5		rnmptssdff.5	$$⊢\left(\phi \wedge x\in A\right)\to B\in C$$
6	1 5	ralrimia	$$⊢\phi \to \forall x\in AB\in C$$
7	2 3 4	rnmptssff	$$⊢\forall x\in AB\in C\to \mathrm{ran}F\subseteq C$$
8	6 7	syl	$$⊢\phi \to \mathrm{ran}F\subseteq C$$