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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 4134 | . 2 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Top) | |
2 | elinel2 4135 | . . 3 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Fin) | |
3 | vex 3435 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 3 | pwid 4563 | . . . . 5 ⊢ 𝑦 ∈ 𝒫 𝑦 |
5 | velpw 4544 | . . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝐽 ↔ 𝑦 ⊆ 𝐽) | |
6 | ssfi 8938 | . . . . . 6 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ⊆ 𝐽) → 𝑦 ∈ Fin) | |
7 | 5, 6 | sylan2b 594 | . . . . 5 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → 𝑦 ∈ Fin) |
8 | elin 3908 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝑦 ∧ 𝑦 ∈ Fin)) | |
9 | unieq 4856 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → ∪ 𝑧 = ∪ 𝑦) | |
10 | 9 | rspceeqv 3576 | . . . . . . 7 ⊢ ((𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∪ 𝐽 = ∪ 𝑦) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) |
11 | 10 | ex 413 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
12 | 8, 11 | sylbir 234 | . . . . 5 ⊢ ((𝑦 ∈ 𝒫 𝑦 ∧ 𝑦 ∈ Fin) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
13 | 4, 7, 12 | sylancr 587 | . . . 4 ⊢ ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) |
14 | 13 | ralrimiva 3110 | . . 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 2740 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
17 | 16 | iscmp 22537 | . 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧))) |
18 | 1, 15, 17 | sylanbrc 583 | 1 ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ∃wrex 3067 ∩ cin 3891 ⊆ wss 3892 𝒫 cpw 4539 ∪ cuni 4845 Fincfn 8716 Topctop 22040 Compccmp 22535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-om 7707 df-1o 8288 df-en 8717 df-fin 8720 df-cmp 22536 |
This theorem is referenced by: 0cmp 22543 discmp 22547 1stckgenlem 22702 ptcmpfi 22962 kelac2lem 40886 |
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