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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 | inss1 3866 | . . 3 ⊢ (Top ∩ Fin) ⊆ Top | |
2 | 1 | sseli 3632 | . 2 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Top) |
3 | inss2 3867 | . . . 4 ⊢ (Top ∩ Fin) ⊆ Fin | |
4 | 3 | sseli 3632 | . . 3 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Fin) |
5 | vex 3234 | . . . . . 6 ⊢ 𝑦 ∈ V | |
6 | 5 | pwid 4207 | . . . . 5 ⊢ 𝑦 ∈ 𝒫 𝑦 |
7 | selpw 4198 | . . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝐽 ↔ 𝑦 ⊆ 𝐽) | |
8 | ssfi 8221 | . . . . . 6 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ⊆ 𝐽) → 𝑦 ∈ Fin) | |
9 | 7, 8 | sylan2b 491 | . . . . 5 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → 𝑦 ∈ Fin) |
10 | elin 3829 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝑦 ∧ 𝑦 ∈ Fin)) | |
11 | unieq 4476 | . . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ∪ 𝑧 = ∪ 𝑦) | |
12 | 11 | eqeq2d 2661 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∪ 𝐽 = ∪ 𝑧 ↔ ∪ 𝐽 = ∪ 𝑦)) |
13 | 12 | rspcev 3340 | . . . . . . 7 ⊢ ((𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∪ 𝐽 = ∪ 𝑦) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) |
14 | 13 | ex 449 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
15 | 10, 14 | sylbir 225 | . . . . 5 ⊢ ((𝑦 ∈ 𝒫 𝑦 ∧ 𝑦 ∈ Fin) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
16 | 6, 9, 15 | sylancr 696 | . . . 4 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
17 | 16 | ralrimiva 2995 | . . 3 ⊢ (𝐽 ∈ Fin → ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
18 | 4, 17 | syl 17 | . 2 ⊢ (𝐽 ∈ (Top ∩ Fin) → ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
19 | eqid 2651 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
20 | 19 | iscmp 21239 | . 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧))) |
21 | 2, 18, 20 | sylanbrc 699 | 1 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ∃wrex 2942 ∩ cin 3606 ⊆ wss 3607 𝒫 cpw 4191 ∪ cuni 4468 Fincfn 7997 Topctop 20746 Compccmp 21237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-om 7108 df-er 7787 df-en 7998 df-fin 8001 df-cmp 21238 |
This theorem is referenced by: 0cmp 21245 discmp 21249 1stckgenlem 21404 ptcmpfi 21664 kelac2lem 37951 |
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