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Theorem creftop 29241
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 2610 . . 3 𝐽 = 𝐽
21iscref 29239 . 2 (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
32simplbi 475 1 (𝐽 ∈ CovHasRef𝐴𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  wral 2896  wrex 2897  cin 3539  𝒫 cpw 4108   cuni 4372   class class class wbr 4583  Topctop 20517  Refcref 21115  CovHasRefccref 29237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-cref 29238
This theorem is referenced by: (None)
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