Theorem tmachlem-tpopen2

Description: Variable-renaming lemma connecting tmachlem-agreeprod and tmachlem-tpopen . (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-tpopen2	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → ( 𝐴 ‘ 𝑎 ) ∈ ( ∏t ‘ ( 𝑏 ∈ 𝐼 ↦ 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	1 2 3 4 5 6	tmachlem-agreeprod	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → ( 𝐴 ‘ 𝑎 ) = X 𝑖 ∈ 𝐼 if ( 𝑖 ∈ ( 𝑆 ‘ 𝑎 ) , { ( 𝑎 ‘ 𝑖 ) } , 𝑈 ) )
8	1 2 3 4 5 6	tmachlem-tpopen	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → X 𝑖 ∈ 𝐼 if ( 𝑖 ∈ ( 𝑆 ‘ 𝑎 ) , { ( 𝑎 ‘ 𝑖 ) } , 𝑈 ) ∈ ( ∏t ‘ ( 𝑏 ∈ 𝐼 ↦ 𝒫 𝑈 ) ) )
9	7 8	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → ( 𝐴 ‘ 𝑎 ) ∈ ( ∏t ‘ ( 𝑏 ∈ 𝐼 ↦ 𝒫 𝑈 ) ) )