Theorem rnmptssd

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020)

	Ref	Expression
Hypotheses	rnmptssd.1	$$⊢Ⅎx\phi$$
	rnmptssd.2	$$⊢F=\left(x\in A⟼B\right)$$
	rnmptssd.3	$$⊢\left(\phi \wedge x\in A\right)\to B\in C$$
Assertion	rnmptssd	$$⊢\phi \to \mathrm{ran}F\subseteq C$$

Proof

Step	Hyp	Ref	Expression
1		rnmptssd.1	$$⊢Ⅎx\phi$$
2		rnmptssd.2	$$⊢F=\left(x\in A⟼B\right)$$
3		rnmptssd.3	$$⊢\left(\phi \wedge x\in A\right)\to B\in C$$
4	1 3	ralrimia	$$⊢\phi \to \forall x\in AB\in C$$
5	2	rnmptss	$$⊢\forall x\in AB\in C\to \mathrm{ran}F\subseteq C$$
6	4 5	syl	$$⊢\phi \to \mathrm{ran}F\subseteq C$$