Theorem rnmptssdf

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

Proof

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