Theorem funcsetcestrclem5

Description: Lemma 5 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	funcsetcestrclem5	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑌 ↑m 𝑋 ) ) )

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	6	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ) → 𝐺 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ) )
8		oveq12	⊢ ( ( 𝑦 = 𝑌 ∧ 𝑥 = 𝑋 ) → ( 𝑦 ↑m 𝑥 ) = ( 𝑌 ↑m 𝑋 ) )
9	8	ancoms	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑦 ↑m 𝑥 ) = ( 𝑌 ↑m 𝑋 ) )
10	9	reseq2d	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( I ↾ ( 𝑦 ↑m 𝑥 ) ) = ( I ↾ ( 𝑌 ↑m 𝑋 ) ) )
11	10	adantl	⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ) ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( I ↾ ( 𝑦 ↑m 𝑥 ) ) = ( I ↾ ( 𝑌 ↑m 𝑋 ) ) )
12		simprl	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ) → 𝑋 ∈ 𝐶 )
13		simprr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ) → 𝑌 ∈ 𝐶 )
14		ovexd	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ) → ( 𝑌 ↑m 𝑋 ) ∈ V )
15	14	resiexd	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ) → ( I ↾ ( 𝑌 ↑m 𝑋 ) ) ∈ V )
16	7 11 12 13 15	ovmpod	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑌 ↑m 𝑋 ) ) )