Theorem funcringcsetcALTV2lem3

Description: Lemma 3 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	funcringcsetcALTV2lem3	⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝐶 )

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	1 3 5	ringcbasbas	⊢ ( ( 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	12	fmpttd	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) : 𝐵 ⟶ 𝐶 )
14	6	feq1d	⊢ ( 𝜑 → ( 𝐹 : 𝐵 ⟶ 𝐶 ↔ ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) : 𝐵 ⟶ 𝐶 ) )
15	13 14	mpbird	⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝐶 )