| 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 4614 | . . 3 ⊢ (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽) | |
| 2 | unieq 4918 | . . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
| 3 | iscref.x | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋) |
| 5 | 4 | eqeq1d 2739 | . . . 4 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦)) |
| 6 | 1 | ineq1d 4219 | . . . . 5 ⊢ (𝑗 = 𝐽 → (𝒫 𝑗 ∩ 𝐴) = (𝒫 𝐽 ∩ 𝐴)) |
| 7 | 6 | rexeqdv 3327 | . . . 4 ⊢ (𝑗 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)) |
| 8 | 5, 7 | imbi12d 344 | . . 3 ⊢ (𝑗 = 𝐽 → ((∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) ↔ (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))) |
| 9 | 1, 8 | raleqbidv 3346 | . 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 33842 | . 2 ⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)} | |
| 11 | 9, 10 | elrab2 3695 | 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∩ cin 3950 𝒫 cpw 4600 ∪ cuni 4907 class class class wbr 5143 Topctop 22899 Refcref 23510 CovHasRefccref 33841 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-pw 4602 df-uni 4908 df-cref 33842 |
| This theorem is referenced by: creftop 33845 crefi 33846 crefss 33848 cmpcref 33849 cmppcmp 33857 dispcmp 33858 pcmplfin 33859 |
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