Theorem setsv

Description: The value of the structure replacement function is a set. (Contributed by AV, 10-Nov-2021)

	Ref	Expression
Assertion	setsv	$$⊢\left(S\in V\wedge B\in W\right)\to S\mathrm{sSet}⟨A,B⟩\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		setsval	$$⊢\left(S\in V\wedge B\in W\right)\to S\mathrm{sSet}⟨A,B⟩=\left({S\upharpoonright }_{\left(\mathrm{V}\setminus \left\{A\right\}\right)}\right)\cup \left\{⟨A,B⟩\right\}$$
2		resexg	$$⊢S\in V\to {S\upharpoonright }_{\left(\mathrm{V}\setminus \left\{A\right\}\right)}\in \mathrm{V}$$
3		snex	$$⊢\left\{⟨A,B⟩\right\}\in \mathrm{V}$$
4	3	a1i	$$⊢\left(S\in V\wedge B\in W\right)\to \left\{⟨A,B⟩\right\}\in \mathrm{V}$$
5		unexg	$$⊢\left({S\upharpoonright }_{\left(\mathrm{V}\setminus \left\{A\right\}\right)}\in \mathrm{V}\wedge \left\{⟨A,B⟩\right\}\in \mathrm{V}\right)\to \left({S\upharpoonright }_{\left(\mathrm{V}\setminus \left\{A\right\}\right)}\right)\cup \left\{⟨A,B⟩\right\}\in \mathrm{V}$$
6	2 4 5	syl2an2r	$$⊢\left(S\in V\wedge B\in W\right)\to \left({S\upharpoonright }_{\left(\mathrm{V}\setminus \left\{A\right\}\right)}\right)\cup \left\{⟨A,B⟩\right\}\in \mathrm{V}$$
7	1 6	eqeltrd	$$⊢\left(S\in V\wedge B\in W\right)\to S\mathrm{sSet}⟨A,B⟩\in \mathrm{V}$$