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Theorem creftop 31796
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 2738 . . 3 𝐽 = 𝐽
21iscref 31794 . 2 (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
32simplbi 498 1 (𝐽 ∈ CovHasRef𝐴𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  wral 3064  wrex 3065  cin 3886  𝒫 cpw 4533   cuni 4839   class class class wbr 5074  Topctop 22042  Refcref 22653  CovHasRefccref 31792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-pw 4535  df-uni 4840  df-cref 31793
This theorem is referenced by: (None)
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