Theorem funcsetcestrclem6

Description: Lemma 6 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	funcsetcestrclem6	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ∧ 𝐻 ∈ ( 𝑌 ↑m 𝑋 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = 𝐻 )

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	1 2 3 4 5 6	funcsetcestrclem5	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑌 ↑m 𝑋 ) ) )
8	7	3adant3	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ∧ 𝐻 ∈ ( 𝑌 ↑m 𝑋 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑌 ↑m 𝑋 ) ) )
9	8	fveq1d	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ∧ 𝐻 ∈ ( 𝑌 ↑m 𝑋 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = ( ( I ↾ ( 𝑌 ↑m 𝑋 ) ) ‘ 𝐻 ) )
10		fvresi	⊢ ( 𝐻 ∈ ( 𝑌 ↑m 𝑋 ) → ( ( I ↾ ( 𝑌 ↑m 𝑋 ) ) ‘ 𝐻 ) = 𝐻 )
11	10	3ad2ant3	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ∧ 𝐻 ∈ ( 𝑌 ↑m 𝑋 ) ) → ( ( I ↾ ( 𝑌 ↑m 𝑋 ) ) ‘ 𝐻 ) = 𝐻 )
12	9 11	eqtrd	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ∧ 𝐻 ∈ ( 𝑌 ↑m 𝑋 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = 𝐻 )