Theorem funcestrcsetclem1

Description: Lemma 1 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	funcestrcsetclem1	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) = ( Base ‘ 𝑋 ) )

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	6	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
8		fveq2	⊢ ( 𝑥 = 𝑋 → ( Base ‘ 𝑥 ) = ( Base ‘ 𝑋 ) )
9	8	adantl	⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑥 = 𝑋 ) → ( Base ‘ 𝑥 ) = ( Base ‘ 𝑋 ) )
10		simpr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
11		fvexd	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( Base ‘ 𝑋 ) ∈ V )
12	7 9 10 11	fvmptd	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) = ( Base ‘ 𝑋 ) )