Theorem funcestrcsetclem4

Description: Lemma 4 for funcestrcsetc . (Contributed by AV, 22-Mar-2020)

	Ref	Expression
Hypotheses	funcestrcsetc.e	⊢ 𝐸 = ( ExtStrCat ‘ 𝑈 )
	funcestrcsetc.s	⊢ 𝑆 = ( SetCat ‘ 𝑈 )
	funcestrcsetc.b	⊢ 𝐵 = ( Base ‘ 𝐸 )
	funcestrcsetc.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
	funcestrcsetc.u	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	funcestrcsetc.f	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
	funcestrcsetc.g	⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) )
Assertion	funcestrcsetclem4	⊢ ( 𝜑 → 𝐺 Fn ( 𝐵 × 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		funcestrcsetc.e	⊢ 𝐸 = ( ExtStrCat ‘ 𝑈 )
2		funcestrcsetc.s	⊢ 𝑆 = ( SetCat ‘ 𝑈 )
3		funcestrcsetc.b	⊢ 𝐵 = ( Base ‘ 𝐸 )
4		funcestrcsetc.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
5		funcestrcsetc.u	⊢ ( 𝜑 → 𝑈 ∈ WUni )
6		funcestrcsetc.f	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
7		funcestrcsetc.g	⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) )
8		eqid	⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )
9		ovex	⊢ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ∈ V
10		resiexg	⊢ ( ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ∈ V → ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ∈ V )
11	9 10	ax-mp	⊢ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ∈ V
12	8 11	fnmpoi	⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) Fn ( 𝐵 × 𝐵 )
13	7	fneq1d	⊢ ( 𝜑 → ( 𝐺 Fn ( 𝐵 × 𝐵 ) ↔ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) Fn ( 𝐵 × 𝐵 ) ) )
14	12 13	mpbiri	⊢ ( 𝜑 → 𝐺 Fn ( 𝐵 × 𝐵 ) )