Theorem creftop 34589

Description: A space where every open cover has an 𝐴 refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Assertion

Ref	Expression
creftop	⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top)

Proof of Theorem creftop

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscref 34587	. 2 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
3	2	simplbi 502	1 ⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ∩ cin 4092  𝒫 cpw 4560  ∪ cuni 4872   class class class wbr 5108  Topctop 23266  Refcref 23876  CovHasRefccref 34585

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-pw 4562  df-uni 4873  df-cref 34586

This theorem is used by: (None)