Theorem setsstrset

Description: Relation between df-sets and df-strset . Temporary theorem kept during the transition from the former to the latter. (Contributed by BJ, 13-Feb-2022)

	Ref	Expression
Assertion	setsstrset	$$⊢\left(S\in V\wedge B\in W\right)\to \left[B/A\right.]sS=S\mathrm{sSet}⟨A\left(\mathrm{ndx}\right),B⟩$$

Proof

Step	Hyp	Ref	Expression
1		df-strset	$$⊢\left[B/A\right.]sS=\left({S\upharpoonright }_{\left(\mathrm{V}\setminus \left\{A\left(\mathrm{ndx}\right)\right\}\right)}\right)\cup \left\{⟨A\left(\mathrm{ndx}\right),B⟩\right\}$$
2		setsval	$$⊢\left(S\in V\wedge B\in W\right)\to S\mathrm{sSet}⟨A\left(\mathrm{ndx}\right),B⟩=\left({S\upharpoonright }_{\left(\mathrm{V}\setminus \left\{A\left(\mathrm{ndx}\right)\right\}\right)}\right)\cup \left\{⟨A\left(\mathrm{ndx}\right),B⟩\right\}$$
3	1 2	eqtr4id	$$⊢\left(S\in V\wedge B\in W\right)\to \left[B/A\right.]sS=S\mathrm{sSet}⟨A\left(\mathrm{ndx}\right),B⟩$$