Theorem creftop 34181

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 2769	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscref 34179	. 2 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
3	2	simplbi 501	1 ⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ∩ cin 3912  𝒫 cpw 4567  ∪ cuni 4876   class class class wbr 5113  Topctop 23019  Refcref 23628  CovHasRefccref 34177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-in 3920  df-ss 3930  df-pw 4569  df-uni 4877  df-cref 34178

This theorem is referenced by: (None)