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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 4916 | . . . . 5 ⊢ (𝑟 = 𝑆 → ∪ 𝑟 = ∪ 𝑆) | |
2 | 1 | eqeq2d 2737 | . . . 4 ⊢ (𝑟 = 𝑆 → (𝑋 = ∪ 𝑟 ↔ 𝑋 = ∪ 𝑆)) |
3 | pweq 4611 | . . . . . 6 ⊢ (𝑟 = 𝑆 → 𝒫 𝑟 = 𝒫 𝑆) | |
4 | 3 | ineq1d 4209 | . . . . 5 ⊢ (𝑟 = 𝑆 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑆 ∩ Fin)) |
5 | 4 | rexeqdv 3316 | . . . 4 ⊢ (𝑟 = 𝑆 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠)) |
6 | 2, 5 | imbi12d 343 | . . 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 23380 | . . . . 5 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠))) |
9 | 8 | simprbi 495 | . . . 4 ⊢ (𝐽 ∈ Comp → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠)) |
10 | 9 | adantr 479 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠)) |
11 | ssexg 5320 | . . . . 5 ⊢ ((𝑆 ⊆ 𝐽 ∧ 𝐽 ∈ Comp) → 𝑆 ∈ V) | |
12 | 11 | ancoms 457 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ∈ V) |
13 | simpr 483 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ⊆ 𝐽) | |
14 | 12, 13 | elpwd 4603 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ∈ 𝒫 𝐽) |
15 | 6, 10, 14 | rspcdva 3608 | . 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → (𝑋 = ∪ 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠)) |
16 | 15 | 3impia 1114 | 1 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 Vcvv 3462 ∩ cin 3945 ⊆ wss 3946 𝒫 cpw 4597 ∪ cuni 4905 Fincfn 8966 Topctop 22883 Compccmp 23378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5296 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-in 3953 df-ss 3963 df-pw 4599 df-uni 4906 df-cmp 23379 |
This theorem is referenced by: cmpcov2 23382 cncmp 23384 discmp 23390 cmpcld 23394 sscmp 23397 comppfsc 23524 alexsubALTlem1 24039 ptcmplem3 24046 lebnum 24978 heibor1 37524 |
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