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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 | elinel1 4174 | . 2 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Top) | |
2 | elinel2 4175 | . . 3 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Fin) | |
3 | vex 3499 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 3 | pwid 4565 | . . . . 5 ⊢ 𝑦 ∈ 𝒫 𝑦 |
5 | velpw 4546 | . . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝐽 ↔ 𝑦 ⊆ 𝐽) | |
6 | ssfi 8740 | . . . . . 6 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ⊆ 𝐽) → 𝑦 ∈ Fin) | |
7 | 5, 6 | sylan2b 595 | . . . . 5 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → 𝑦 ∈ Fin) |
8 | elin 4171 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝑦 ∧ 𝑦 ∈ Fin)) | |
9 | unieq 4851 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → ∪ 𝑧 = ∪ 𝑦) | |
10 | 9 | rspceeqv 3640 | . . . . . . 7 ⊢ ((𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∪ 𝐽 = ∪ 𝑦) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) |
11 | 10 | ex 415 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
12 | 8, 11 | sylbir 237 | . . . . 5 ⊢ ((𝑦 ∈ 𝒫 𝑦 ∧ 𝑦 ∈ Fin) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
13 | 4, 7, 12 | sylancr 589 | . . . 4 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
14 | 13 | ralrimiva 3184 | . . 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 2823 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
17 | 16 | iscmp 21998 | . 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧))) |
18 | 1, 15, 17 | sylanbrc 585 | 1 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ∩ cin 3937 ⊆ wss 3938 𝒫 cpw 4541 ∪ cuni 4840 Fincfn 8511 Topctop 21503 Compccmp 21996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-om 7583 df-er 8291 df-en 8512 df-fin 8515 df-cmp 21997 |
This theorem is referenced by: 0cmp 22004 discmp 22008 1stckgenlem 22163 ptcmpfi 22423 kelac2lem 39671 |
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