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Mirrors > Home > MPE Home > Th. List > alexsubALTlem1 | Structured version Visualization version GIF version |
Description: Lemma for alexsubALT 23929. A compact space has a subbase such that every cover taken from it has a finite subcover. (Contributed by Jeff Hankins, 27-Jan-2010.) |
Ref | Expression |
---|---|
alexsubALT.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
alexsubALTlem1 | ⊢ (𝐽 ∈ Comp → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmptop 23273 | . . 3 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | |
2 | fitop 22776 | . . . . 5 ⊢ (𝐽 ∈ Top → (fi‘𝐽) = 𝐽) | |
3 | 2 | fveq2d 6895 | . . . 4 ⊢ (𝐽 ∈ Top → (topGen‘(fi‘𝐽)) = (topGen‘𝐽)) |
4 | tgtop 22850 | . . . 4 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
5 | 3, 4 | eqtr2d 2768 | . . 3 ⊢ (𝐽 ∈ Top → 𝐽 = (topGen‘(fi‘𝐽))) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝐽 ∈ Comp → 𝐽 = (topGen‘(fi‘𝐽))) |
7 | velpw 4603 | . . . 4 ⊢ (𝑐 ∈ 𝒫 𝐽 ↔ 𝑐 ⊆ 𝐽) | |
8 | alexsubALT.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
9 | 8 | cmpcov 23267 | . . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) |
10 | 9 | 3exp 1117 | . . . 4 ⊢ (𝐽 ∈ Comp → (𝑐 ⊆ 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))) |
11 | 7, 10 | biimtrid 241 | . . 3 ⊢ (𝐽 ∈ Comp → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))) |
12 | 11 | ralrimiv 3140 | . 2 ⊢ (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)) |
13 | 2fveq3 6896 | . . . . 5 ⊢ (𝑥 = 𝐽 → (topGen‘(fi‘𝑥)) = (topGen‘(fi‘𝐽))) | |
14 | 13 | eqeq2d 2738 | . . . 4 ⊢ (𝑥 = 𝐽 → (𝐽 = (topGen‘(fi‘𝑥)) ↔ 𝐽 = (topGen‘(fi‘𝐽)))) |
15 | pweq 4612 | . . . . 5 ⊢ (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽) | |
16 | 15 | raleqdv 3320 | . . . 4 ⊢ (𝑥 = 𝐽 → (∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))) |
17 | 14, 16 | anbi12d 630 | . . 3 ⊢ (𝑥 = 𝐽 → ((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)) ↔ (𝐽 = (topGen‘(fi‘𝐽)) ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))) |
18 | 17 | spcegv 3582 | . 2 ⊢ (𝐽 ∈ Comp → ((𝐽 = (topGen‘(fi‘𝐽)) ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)) → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))) |
19 | 6, 12, 18 | mp2and 698 | 1 ⊢ (𝐽 ∈ Comp → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 ∩ cin 3943 ⊆ wss 3944 𝒫 cpw 4598 ∪ cuni 4903 ‘cfv 6542 Fincfn 8953 ficfi 9419 topGenctg 17404 Topctop 22769 Compccmp 23264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7863 df-1o 8478 df-er 8716 df-en 8954 df-fin 8957 df-fi 9420 df-topgen 17410 df-top 22770 df-cmp 23265 |
This theorem is referenced by: alexsubALT 23929 |
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