Theorem creftop 33877

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 2735	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscref 33875	. 2 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
3	2	simplbi 497	1 ⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   ∩ cin 3925  𝒫 cpw 4575  ∪ cuni 4883   class class class wbr 5119  Topctop 22831  Refcref 23440  CovHasRefccref 33873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-in 3933  df-ss 3943  df-pw 4577  df-uni 4884  df-cref 33874

This theorem is referenced by: (None)