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Mirrors > Home > MPE Home > Th. List > cmpcov | Structured version Visualization version GIF version |
Description: An open cover of a compact topology has a finite subcover. (Contributed by Jeff Hankins, 29-Jun-2009.) |
Ref | Expression |
---|---|
iscmp.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
cmpcov | ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 4681 | . . . . 5 ⊢ (𝑟 = 𝑆 → ∪ 𝑟 = ∪ 𝑆) | |
2 | 1 | eqeq2d 2788 | . . . 4 ⊢ (𝑟 = 𝑆 → (𝑋 = ∪ 𝑟 ↔ 𝑋 = ∪ 𝑆)) |
3 | pweq 4382 | . . . . . 6 ⊢ (𝑟 = 𝑆 → 𝒫 𝑟 = 𝒫 𝑆) | |
4 | 3 | ineq1d 4036 | . . . . 5 ⊢ (𝑟 = 𝑆 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑆 ∩ Fin)) |
5 | 4 | rexeqdv 3341 | . . . 4 ⊢ (𝑟 = 𝑆 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠)) |
6 | 2, 5 | imbi12d 336 | . . 3 ⊢ (𝑟 = 𝑆 → ((𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠) ↔ (𝑋 = ∪ 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠))) |
7 | iscmp.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
8 | 7 | iscmp 21604 | . . . . 5 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠))) |
9 | 8 | simprbi 492 | . . . 4 ⊢ (𝐽 ∈ Comp → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠)) |
10 | 9 | adantr 474 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠)) |
11 | simpr 479 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ⊆ 𝐽) | |
12 | ssexg 5043 | . . . . . 6 ⊢ ((𝑆 ⊆ 𝐽 ∧ 𝐽 ∈ Comp) → 𝑆 ∈ V) | |
13 | 12 | ancoms 452 | . . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ∈ V) |
14 | elpwg 4387 | . . . . 5 ⊢ (𝑆 ∈ V → (𝑆 ∈ 𝒫 𝐽 ↔ 𝑆 ⊆ 𝐽)) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → (𝑆 ∈ 𝒫 𝐽 ↔ 𝑆 ⊆ 𝐽)) |
16 | 11, 15 | mpbird 249 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ∈ 𝒫 𝐽) |
17 | 6, 10, 16 | rspcdva 3517 | . 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → (𝑋 = ∪ 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠)) |
18 | 17 | 3impia 1106 | 1 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ∃wrex 3091 Vcvv 3398 ∩ cin 3791 ⊆ wss 3792 𝒫 cpw 4379 ∪ cuni 4673 Fincfn 8243 Topctop 21109 Compccmp 21602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 ax-sep 5019 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-in 3799 df-ss 3806 df-pw 4381 df-uni 4674 df-cmp 21603 |
This theorem is referenced by: cmpcov2 21606 cncmp 21608 discmp 21614 cmpcld 21618 sscmp 21621 comppfsc 21748 alexsubALTlem1 22263 ptcmplem3 22270 lebnum 23175 heibor1 34238 |
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