Theorem funcsetcestrclem4

Description: Lemma 4 for funcsetcestrc . (Contributed by AV, 27-Mar-2020)

	Ref	Expression
Hypotheses	funcsetcestrc.s	⊢ 𝑆 = ( SetCat ‘ 𝑈 )
	funcsetcestrc.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
	funcsetcestrc.f	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐶 ↦ { ⟨ ( Base ‘ ndx ) , 𝑥 ⟩ } ) )
	funcsetcestrc.u	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	funcsetcestrc.o	⊢ ( 𝜑 → ω ∈ 𝑈 )
	funcsetcestrc.g	⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ) )
Assertion	funcsetcestrclem4	⊢ ( 𝜑 → 𝐺 Fn ( 𝐶 × 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		funcsetcestrc.s	⊢ 𝑆 = ( SetCat ‘ 𝑈 )
2		funcsetcestrc.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
3		funcsetcestrc.f	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐶 ↦ { ⟨ ( Base ‘ ndx ) , 𝑥 ⟩ } ) )
4		funcsetcestrc.u	⊢ ( 𝜑 → 𝑈 ∈ WUni )
5		funcsetcestrc.o	⊢ ( 𝜑 → ω ∈ 𝑈 )
6		funcsetcestrc.g	⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ) )
7		eqid	⊢ ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ) = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) )
8		ovex	⊢ ( 𝑦 ↑m 𝑥 ) ∈ V
9		resiexg	⊢ ( ( 𝑦 ↑m 𝑥 ) ∈ V → ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ∈ V )
10	8 9	ax-mp	⊢ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ∈ V
11	7 10	fnmpoi	⊢ ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ) Fn ( 𝐶 × 𝐶 )
12	6	fneq1d	⊢ ( 𝜑 → ( 𝐺 Fn ( 𝐶 × 𝐶 ) ↔ ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ) Fn ( 𝐶 × 𝐶 ) ) )
13	11 12	mpbiri	⊢ ( 𝜑 → 𝐺 Fn ( 𝐶 × 𝐶 ) )