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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 4877 | . . . . 5 ⊢ (𝑟 = 𝑆 → ∪ 𝑟 = ∪ 𝑆) | |
2 | 1 | eqeq2d 2744 | . . . 4 ⊢ (𝑟 = 𝑆 → (𝑋 = ∪ 𝑟 ↔ 𝑋 = ∪ 𝑆)) |
3 | pweq 4575 | . . . . . 6 ⊢ (𝑟 = 𝑆 → 𝒫 𝑟 = 𝒫 𝑆) | |
4 | 3 | ineq1d 4172 | . . . . 5 ⊢ (𝑟 = 𝑆 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑆 ∩ Fin)) |
5 | 4 | rexeqdv 3313 | . . . 4 ⊢ (𝑟 = 𝑆 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠)) |
6 | 2, 5 | imbi12d 345 | . . 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 22755 | . . . . 5 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠))) |
9 | 8 | simprbi 498 | . . . 4 ⊢ (𝐽 ∈ Comp → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠)) |
10 | 9 | adantr 482 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠)) |
11 | ssexg 5281 | . . . . 5 ⊢ ((𝑆 ⊆ 𝐽 ∧ 𝐽 ∈ Comp) → 𝑆 ∈ V) | |
12 | 11 | ancoms 460 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ∈ V) |
13 | simpr 486 | . . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ⊆ 𝐽) | |
14 | 12, 13 | elpwd 4567 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ∈ 𝒫 𝐽) |
15 | 6, 10, 14 | rspcdva 3581 | . 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → (𝑋 = ∪ 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠)) |
16 | 15 | 3impia 1118 | 1 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∃wrex 3070 Vcvv 3444 ∩ cin 3910 ⊆ wss 3911 𝒫 cpw 4561 ∪ cuni 4866 Fincfn 8886 Topctop 22258 Compccmp 22753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-in 3918 df-ss 3928 df-pw 4563 df-uni 4867 df-cmp 22754 |
This theorem is referenced by: cmpcov2 22757 cncmp 22759 discmp 22765 cmpcld 22769 sscmp 22772 comppfsc 22899 alexsubALTlem1 23414 ptcmplem3 23421 lebnum 24343 heibor1 36315 |
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