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Theorem creftop 33845
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 ∈ CovHasRef𝐴𝐽 ∈ Top)

Proof of Theorem creftop
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 𝐽 = 𝐽
21iscref 33843 . 2 (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
32simplbi 497 1 (𝐽 ∈ CovHasRef𝐴𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cin 3950  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  Topctop 22899  Refcref 23510  CovHasRefccref 33841
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 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602  df-uni 4908  df-cref 33842
This theorem is referenced by: (None)
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