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 2821 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | iscref 31103 | . 2 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))) |
3 | 2 | simplbi 500 | 1 ⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ∩ cin 3934 𝒫 cpw 4538 ∪ cuni 4831 class class class wbr 5058 Topctop 21495 Refcref 22104 CovHasRefccref 31101 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 df-pw 4540 df-uni 4832 df-cref 31102 |
This theorem is referenced by: (None) |
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