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Mirrors > Home > MPE Home > Th. List > alexsubALTlem1 | Structured version Visualization version GIF version |
Description: Lemma for alexsubALT 22653. 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 21997 | . . 3 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | |
2 | fitop 21502 | . . . . 5 ⊢ (𝐽 ∈ Top → (fi‘𝐽) = 𝐽) | |
3 | 2 | fveq2d 6668 | . . . 4 ⊢ (𝐽 ∈ Top → (topGen‘(fi‘𝐽)) = (topGen‘𝐽)) |
4 | tgtop 21575 | . . . 4 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
5 | 3, 4 | eqtr2d 2857 | . . 3 ⊢ (𝐽 ∈ Top → 𝐽 = (topGen‘(fi‘𝐽))) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝐽 ∈ Comp → 𝐽 = (topGen‘(fi‘𝐽))) |
7 | velpw 4546 | . . . 4 ⊢ (𝑐 ∈ 𝒫 𝐽 ↔ 𝑐 ⊆ 𝐽) | |
8 | alexsubALT.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
9 | 8 | cmpcov 21991 | . . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) |
10 | 9 | 3exp 1115 | . . . 4 ⊢ (𝐽 ∈ Comp → (𝑐 ⊆ 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))) |
11 | 7, 10 | syl5bi 244 | . . 3 ⊢ (𝐽 ∈ Comp → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))) |
12 | 11 | ralrimiv 3181 | . 2 ⊢ (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)) |
13 | 2fveq3 6669 | . . . . 5 ⊢ (𝑥 = 𝐽 → (topGen‘(fi‘𝑥)) = (topGen‘(fi‘𝐽))) | |
14 | 13 | eqeq2d 2832 | . . . 4 ⊢ (𝑥 = 𝐽 → (𝐽 = (topGen‘(fi‘𝑥)) ↔ 𝐽 = (topGen‘(fi‘𝐽)))) |
15 | pweq 4541 | . . . . 5 ⊢ (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽) | |
16 | 15 | raleqdv 3415 | . . . 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 632 | . . 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 3596 | . 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 697 | 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 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 ∪ cuni 4831 ‘cfv 6349 Fincfn 8503 ficfi 8868 topGenctg 16705 Topctop 21495 Compccmp 21988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-fin 8507 df-fi 8869 df-topgen 16711 df-top 21496 df-cmp 21989 |
This theorem is referenced by: alexsubALT 22653 |
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