Theorem iscref

Description: The property that every open cover has an A refinement for the topological space J . (Contributed by Thierry Arnoux, 7-Jan-2020)

	Ref	Expression
Hypothesis	iscref.x	⊢ 𝑋 = ∪ 𝐽
Assertion	iscref	⊢ ( 𝐽 ∈ CovHasRef 𝐴 ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑦 ∈ 𝒫 𝐽 ( 𝑋 = ∪ 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝐽 ∩ 𝐴 ) 𝑧 Ref 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		iscref.x	⊢ 𝑋 = ∪ 𝐽
2		pweq	⊢ ( 𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽 )
3		unieq	⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽 )
4	3 1	eqtr4di	⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = 𝑋 )
5	4	eqeq1d	⊢ ( 𝑗 = 𝐽 → ( ∪ 𝑗 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦 ) )
6	2	ineq1d	⊢ ( 𝑗 = 𝐽 → ( 𝒫 𝑗 ∩ 𝐴 ) = ( 𝒫 𝐽 ∩ 𝐴 ) )
7	6	rexeqdv	⊢ ( 𝑗 = 𝐽 → ( ∃ 𝑧 ∈ ( 𝒫 𝑗 ∩ 𝐴 ) 𝑧 Ref 𝑦 ↔ ∃ 𝑧 ∈ ( 𝒫 𝐽 ∩ 𝐴 ) 𝑧 Ref 𝑦 ) )
8	5 7	imbi12d	⊢ ( 𝑗 = 𝐽 → ( ( ∪ 𝑗 = ∪ 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝑗 ∩ 𝐴 ) 𝑧 Ref 𝑦 ) ↔ ( 𝑋 = ∪ 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝐽 ∩ 𝐴 ) 𝑧 Ref 𝑦 ) ) )
9	2 8	raleqbidv	⊢ ( 𝑗 = 𝐽 → ( ∀ 𝑦 ∈ 𝒫 𝑗 ( ∪ 𝑗 = ∪ 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝑗 ∩ 𝐴 ) 𝑧 Ref 𝑦 ) ↔ ∀ 𝑦 ∈ 𝒫 𝐽 ( 𝑋 = ∪ 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝐽 ∩ 𝐴 ) 𝑧 Ref 𝑦 ) ) )
10		df-cref	⊢ CovHasRef 𝐴 = { 𝑗 ∈ Top ∣ ∀ 𝑦 ∈ 𝒫 𝑗 ( ∪ 𝑗 = ∪ 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝑗 ∩ 𝐴 ) 𝑧 Ref 𝑦 ) }
11	9 10	elrab2	⊢ ( 𝐽 ∈ CovHasRef 𝐴 ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑦 ∈ 𝒫 𝐽 ( 𝑋 = ∪ 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝐽 ∩ 𝐴 ) 𝑧 Ref 𝑦 ) ) )