Theorem funcringcsetcALTV2lem1

Description: Lemma 1 for funcringcsetcALTV2 . (Contributed by AV, 15-Feb-2020) (New usage is discouraged.)

	Ref	Expression
Hypotheses	funcringcsetcALTV2.r	⊢ 𝑅 = ( RingCat ‘ 𝑈 )
	funcringcsetcALTV2.s	⊢ 𝑆 = ( SetCat ‘ 𝑈 )
	funcringcsetcALTV2.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	funcringcsetcALTV2.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
	funcringcsetcALTV2.u	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	funcringcsetcALTV2.f	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
Assertion	funcringcsetcALTV2lem1	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) = ( Base ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		funcringcsetcALTV2.r	⊢ 𝑅 = ( RingCat ‘ 𝑈 )
2		funcringcsetcALTV2.s	⊢ 𝑆 = ( SetCat ‘ 𝑈 )
3		funcringcsetcALTV2.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
4		funcringcsetcALTV2.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
5		funcringcsetcALTV2.u	⊢ ( 𝜑 → 𝑈 ∈ WUni )
6		funcringcsetcALTV2.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 ) )