Theorem tmachlem-tpitem

Description: Topology lemma. (Contributed by Ender Ting, 27-Jul-2026)

	Ref	Expression
Hypotheses	tmach.finalph	⊢ ( 𝜑 → 𝑈 ∈ Fin )
	tmach.exindex	⊢ ( 𝜑 → 𝐼 ∈ V )
	tmach.tapelist	⊢ ( 𝜑 → 𝑇 = ( 𝑈 ↑m 𝐼 ) )
	tmach.scanmap	⊢ ( 𝜑 → 𝑆 : 𝑇 ⟶ ( 𝒫 𝐼 ∩ Fin ) )
	tmach.agreemap	⊢ ( 𝜑 → 𝐴 = ( 𝑧 ∈ 𝑇 ↦ { 𝑦 ∈ 𝑇 ∣ ( 𝑦 ↾ ( 𝑆 ‘ 𝑧 ) ) = ( 𝑧 ↾ ( 𝑆 ‘ 𝑧 ) ) } ) )
	tmach.agreement	⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝑇 ∀ 𝑦 ∈ ( 𝐴 ‘ 𝑧 ) ( 𝑆 ‘ 𝑦 ) = ( 𝑆 ‘ 𝑧 ) )
Assertion	tmachlem-tpitem	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐼 ) → ( ( 𝑖 ∈ 𝐼 ↦ 𝒫 𝑈 ) ‘ 𝑎 ) = 𝒫 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		tmach.finalph	⊢ ( 𝜑 → 𝑈 ∈ Fin )
2		tmach.exindex	⊢ ( 𝜑 → 𝐼 ∈ V )
3		tmach.tapelist	⊢ ( 𝜑 → 𝑇 = ( 𝑈 ↑m 𝐼 ) )
4		tmach.scanmap	⊢ ( 𝜑 → 𝑆 : 𝑇 ⟶ ( 𝒫 𝐼 ∩ Fin ) )
5		tmach.agreemap	⊢ ( 𝜑 → 𝐴 = ( 𝑧 ∈ 𝑇 ↦ { 𝑦 ∈ 𝑇 ∣ ( 𝑦 ↾ ( 𝑆 ‘ 𝑧 ) ) = ( 𝑧 ↾ ( 𝑆 ‘ 𝑧 ) ) } ) )
6		tmach.agreement	⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝑇 ∀ 𝑦 ∈ ( 𝐴 ‘ 𝑧 ) ( 𝑆 ‘ 𝑦 ) = ( 𝑆 ‘ 𝑧 ) )
7		eqid	⊢ ( 𝑖 ∈ 𝐼 ↦ 𝒫 𝑈 ) = ( 𝑖 ∈ 𝐼 ↦ 𝒫 𝑈 )
8		eqidd	⊢ ( 𝑖 = 𝑎 → 𝒫 𝑈 = 𝒫 𝑈 )
9		simpr	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐼 ) → 𝑎 ∈ 𝐼 )
10	1	pwexd	⊢ ( 𝜑 → 𝒫 𝑈 ∈ V )
11	10	adantr	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐼 ) → 𝒫 𝑈 ∈ V )
12	7 8 9 11	fvmptd3	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐼 ) → ( ( 𝑖 ∈ 𝐼 ↦ 𝒫 𝑈 ) ‘ 𝑎 ) = 𝒫 𝑈 )