![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 2728 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | iscref 33478 | . 2 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))) |
3 | 2 | simplbi 496 | 1 ⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ∃wrex 3067 ∩ cin 3948 𝒫 cpw 4606 ∪ cuni 4912 class class class wbr 5152 Topctop 22815 Refcref 23426 CovHasRefccref 33476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-in 3956 df-ss 3966 df-pw 4608 df-uni 4913 df-cref 33477 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |