Theorem funcringcsetcALTV2lem5

Description: Lemma 5 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 ) ) )
	funcringcsetcALTV2.g	⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
Assertion	funcringcsetcALTV2lem5	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑋 RingHom 𝑌 ) ) )

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		funcringcsetcALTV2.g	⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
8	7	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
9		oveq12	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑥 RingHom 𝑦 ) = ( 𝑋 RingHom 𝑌 ) )
10	9	adantl	⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑥 RingHom 𝑦 ) = ( 𝑋 RingHom 𝑌 ) )
11	10	reseq2d	⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( I ↾ ( 𝑥 RingHom 𝑦 ) ) = ( I ↾ ( 𝑋 RingHom 𝑌 ) ) )
12		simprl	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
13		simprr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
14		ovexd	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 RingHom 𝑌 ) ∈ V )
15	14	resiexd	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( I ↾ ( 𝑋 RingHom 𝑌 ) ) ∈ V )
16	8 11 12 13 15	ovmpod	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑋 RingHom 𝑌 ) ) )