Theorem funcestrcsetclem3

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

	Ref	Expression
Hypotheses	funcestrcsetc.e	⊢ 𝐸 = ( ExtStrCat ‘ 𝑈 )
	funcestrcsetc.s	⊢ 𝑆 = ( SetCat ‘ 𝑈 )
	funcestrcsetc.b	⊢ 𝐵 = ( Base ‘ 𝐸 )
	funcestrcsetc.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
	funcestrcsetc.u	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	funcestrcsetc.f	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
Assertion	funcestrcsetclem3	⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		funcestrcsetc.e	⊢ 𝐸 = ( ExtStrCat ‘ 𝑈 )
2		funcestrcsetc.s	⊢ 𝑆 = ( SetCat ‘ 𝑈 )
3		funcestrcsetc.b	⊢ 𝐵 = ( Base ‘ 𝐸 )
4		funcestrcsetc.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
5		funcestrcsetc.u	⊢ ( 𝜑 → 𝑈 ∈ WUni )
6		funcestrcsetc.f	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
7	1 3 5	estrcbasbas	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ 𝑥 ) ∈ 𝑈 )
8	2 5	setcbas	⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝑆 ) )
9	8	eqcomd	⊢ ( 𝜑 → ( Base ‘ 𝑆 ) = 𝑈 )
10	9	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ 𝑆 ) = 𝑈 )
11	7 10	eleqtrrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ 𝑥 ) ∈ ( Base ‘ 𝑆 ) )
12	11 4	eleqtrrdi	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ 𝑥 ) ∈ 𝐶 )
13	6 12	fmpt3d	⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝐶 )