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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > creftop | Structured version Visualization version GIF version |
Description: A space where every open cover has an 𝐴 refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
Ref | Expression |
---|---|
creftop | ⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | iscref 33805 | . 2 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))) |
3 | 2 | simplbi 497 | 1 ⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ∩ cin 3962 𝒫 cpw 4605 ∪ cuni 4912 class class class wbr 5148 Topctop 22915 Refcref 23526 CovHasRefccref 33803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-in 3970 df-ss 3980 df-pw 4607 df-uni 4913 df-cref 33804 |
This theorem is referenced by: (None) |
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