Theorem creftop 34359

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 2760	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscref 34357	. 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 2145  ∀wral 3076  ∃wrex 3086   ∩ cin 3898  𝒫 cpw 4557  ∪ cuni 4867   class class class wbr 5103  Topctop 23121  Refcref 23731  CovHasRefccref 34355

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-in 3906  df-ss 3916  df-pw 4559  df-uni 4868  df-cref 34356

This theorem is used by: (None)