Theorem funcestrcsetclem3

Description: Lemma 3 for funcestrcsetc . (Contributed by AV, 22-Mar-2020)

	Ref	Expression
Hypotheses	funcestrcsetc.e	$$⊢E=\mathrm{ExtStrCat}\left(U\right)$$
	funcestrcsetc.s	$$⊢S=\mathrm{SetCat}\left(U\right)$$
	funcestrcsetc.b	$$⊢B={\mathrm{Base}}_E$$
	funcestrcsetc.c	$$⊢C={\mathrm{Base}}_S$$
	funcestrcsetc.u	$$⊢\phi \to U\in \mathrm{WUni}$$
	funcestrcsetc.f	$$⊢\phi \to F=\left(x\in B⟼{\mathrm{Base}}_x\right)$$
Assertion	funcestrcsetclem3	$$⊢\phi \to F:B⟶C$$

Proof

Step	Hyp	Ref	Expression
1		funcestrcsetc.e	$$⊢E=\mathrm{ExtStrCat}\left(U\right)$$
2		funcestrcsetc.s	$$⊢S=\mathrm{SetCat}\left(U\right)$$
3		funcestrcsetc.b	$$⊢B={\mathrm{Base}}_E$$
4		funcestrcsetc.c	$$⊢C={\mathrm{Base}}_S$$
5		funcestrcsetc.u	$$⊢\phi \to U\in \mathrm{WUni}$$
6		funcestrcsetc.f	$$⊢\phi \to F=\left(x\in B⟼{\mathrm{Base}}_x\right)$$
7	1 3 5	estrcbasbas	$$⊢\left(\phi \wedge x\in B\right)\to {\mathrm{Base}}_x\in U$$
8	2 5	setcbas	$$⊢\phi \to U={\mathrm{Base}}_S$$
9	8	eqcomd	$$⊢\phi \to {\mathrm{Base}}_S=U$$
10	9	adantr	$$⊢\left(\phi \wedge x\in B\right)\to {\mathrm{Base}}_S=U$$
11	7 10	eleqtrrd	$$⊢\left(\phi \wedge x\in B\right)\to {\mathrm{Base}}_x\in {\mathrm{Base}}_S$$
12	11 4	eleqtrrdi	$$⊢\left(\phi \wedge x\in B\right)\to {\mathrm{Base}}_x\in C$$
13	6 12	fmpt3d	$$⊢\phi \to F:B⟶C$$