![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fincmp | Structured version Visualization version GIF version |
Description: A finite topology is compact. (Contributed by FL, 22-Dec-2008.) |
Ref | Expression |
---|---|
fincmp | ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elinel1 4122 | . 2 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Top) | |
2 | elinel2 4123 | . . 3 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Fin) | |
3 | vex 3444 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 3 | pwid 4521 | . . . . 5 ⊢ 𝑦 ∈ 𝒫 𝑦 |
5 | velpw 4502 | . . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝐽 ↔ 𝑦 ⊆ 𝐽) | |
6 | ssfi 8722 | . . . . . 6 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ⊆ 𝐽) → 𝑦 ∈ Fin) | |
7 | 5, 6 | sylan2b 596 | . . . . 5 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → 𝑦 ∈ Fin) |
8 | elin 3897 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝑦 ∧ 𝑦 ∈ Fin)) | |
9 | unieq 4811 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → ∪ 𝑧 = ∪ 𝑦) | |
10 | 9 | rspceeqv 3586 | . . . . . . 7 ⊢ ((𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∪ 𝐽 = ∪ 𝑦) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) |
11 | 10 | ex 416 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
12 | 8, 11 | sylbir 238 | . . . . 5 ⊢ ((𝑦 ∈ 𝒫 𝑦 ∧ 𝑦 ∈ Fin) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
13 | 4, 7, 12 | sylancr 590 | . . . 4 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
14 | 13 | ralrimiva 3149 | . . 3 ⊢ (𝐽 ∈ Fin → ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
15 | 2, 14 | syl 17 | . 2 ⊢ (𝐽 ∈ (Top ∩ Fin) → ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
16 | eqid 2798 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
17 | 16 | iscmp 21993 | . 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧))) |
18 | 1, 15, 17 | sylanbrc 586 | 1 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 Fincfn 8492 Topctop 21498 Compccmp 21991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-om 7561 df-er 8272 df-en 8493 df-fin 8496 df-cmp 21992 |
This theorem is referenced by: 0cmp 21999 discmp 22003 1stckgenlem 22158 ptcmpfi 22418 kelac2lem 40008 |
Copyright terms: Public domain | W3C validator |