Theorem funcringcsetcALTV2lem4

Description: Lemma 4 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	funcringcsetcALTV2lem4	⊢ ( 𝜑 → 𝐺 Fn ( 𝐵 × 𝐵 ) )

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		eqid	⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) )
9		ovex	⊢ ( 𝑥 RingHom 𝑦 ) ∈ V
10		id	⊢ ( ( 𝑥 RingHom 𝑦 ) ∈ V → ( 𝑥 RingHom 𝑦 ) ∈ V )
11	10	resiexd	⊢ ( ( 𝑥 RingHom 𝑦 ) ∈ V → ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ∈ V )
12	9 11	ax-mp	⊢ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ∈ V
13	8 12	fnmpoi	⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) Fn ( 𝐵 × 𝐵 )
14	7	fneq1d	⊢ ( 𝜑 → ( 𝐺 Fn ( 𝐵 × 𝐵 ) ↔ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) Fn ( 𝐵 × 𝐵 ) ) )
15	13 14	mpbiri	⊢ ( 𝜑 → 𝐺 Fn ( 𝐵 × 𝐵 ) )