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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscref | Structured version Visualization version GIF version |
Description: The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
Ref | Expression |
---|---|
iscref.x | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
iscref | ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4618 | . . 3 ⊢ (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽) | |
2 | unieq 4920 | . . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
3 | iscref.x | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 2, 3 | eqtr4di 2783 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋) |
5 | 4 | eqeq1d 2727 | . . . 4 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦)) |
6 | 1 | ineq1d 4209 | . . . . 5 ⊢ (𝑗 = 𝐽 → (𝒫 𝑗 ∩ 𝐴) = (𝒫 𝐽 ∩ 𝐴)) |
7 | 6 | rexeqdv 3315 | . . . 4 ⊢ (𝑗 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)) |
8 | 5, 7 | imbi12d 343 | . . 3 ⊢ (𝑗 = 𝐽 → ((∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) ↔ (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))) |
9 | 1, 8 | raleqbidv 3329 | . 2 ⊢ (𝑗 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) ↔ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))) |
10 | df-cref 33577 | . 2 ⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)} | |
11 | 9, 10 | elrab2 3682 | 1 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 ∩ cin 3943 𝒫 cpw 4604 ∪ cuni 4909 class class class wbr 5149 Topctop 22844 Refcref 23455 CovHasRefccref 33576 |
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 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-in 3951 df-ss 3961 df-pw 4606 df-uni 4910 df-cref 33577 |
This theorem is referenced by: creftop 33580 crefi 33581 crefss 33583 cmpcref 33584 cmppcmp 33592 dispcmp 33593 pcmplfin 33594 |
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