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Theorem creftop 32484
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 2733 . . 3 𝐽 = 𝐽
21iscref 32482 . 2 (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
32simplbi 499 1 (𝐽 ∈ CovHasRef𝐴𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wral 3061  wrex 3070  cin 3910  𝒫 cpw 4561   cuni 4866   class class class wbr 5106  Topctop 22258  Refcref 22869  CovHasRefccref 32480
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-in 3918  df-ss 3928  df-pw 4563  df-uni 4867  df-cref 32481
This theorem is referenced by: (None)
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