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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 3795 | . . 3 ⊢ (Top ∩ Fin) ⊆ Top | |
2 | 1 | sseli 3564 | . 2 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Top) |
3 | inss2 3796 | . . . 4 ⊢ (Top ∩ Fin) ⊆ Fin | |
4 | 3 | sseli 3564 | . . 3 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Fin) |
5 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
6 | 5 | pwid 4122 | . . . . 5 ⊢ 𝑦 ∈ 𝒫 𝑦 |
7 | selpw 4115 | . . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝐽 ↔ 𝑦 ⊆ 𝐽) | |
8 | ssfi 8065 | . . . . . 6 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ⊆ 𝐽) → 𝑦 ∈ Fin) | |
9 | 7, 8 | sylan2b 491 | . . . . 5 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → 𝑦 ∈ Fin) |
10 | elin 3758 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝑦 ∧ 𝑦 ∈ Fin)) | |
11 | unieq 4380 | . . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ∪ 𝑧 = ∪ 𝑦) | |
12 | 11 | eqeq2d 2620 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∪ 𝐽 = ∪ 𝑧 ↔ ∪ 𝐽 = ∪ 𝑦)) |
13 | 12 | rspcev 3282 | . . . . . . 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 224 | . . . . 5 ⊢ ((𝑦 ∈ 𝒫 𝑦 ∧ 𝑦 ∈ Fin) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
16 | 6, 9, 15 | sylancr 694 | . . . 4 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
17 | 16 | ralrimiva 2949 | . . 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 2610 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
20 | 19 | iscmp 21001 | . 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧))) |
21 | 2, 18, 20 | sylanbrc 695 | 1 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 Fincfn 7841 Topctop 20517 Compccmp 20999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-om 6958 df-er 7629 df-en 7842 df-fin 7845 df-cmp 21000 |
This theorem is referenced by: 0cmp 21007 discmp 21011 1stckgenlem 21166 ptcmpfi 21426 kelac2lem 36652 |
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