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Theorem creftop 30247
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 2770 . . 3 𝐽 = 𝐽
21iscref 30245 . 2 (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
32simplbi 479 1 (𝐽 ∈ CovHasRef𝐴𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  wral 3060  wrex 3061  cin 3720  𝒫 cpw 4295   cuni 4572   class class class wbr 4784  Topctop 20917  Refcref 21525  CovHasRefccref 30243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-in 3728  df-ss 3735  df-pw 4297  df-uni 4573  df-cref 30244
This theorem is referenced by: (None)
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