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Mirrors > Home > MPE Home > Th. List > iskgen3 | Structured version Visualization version GIF version |
Description: Derive the usual definition of "compactly generated". A topology is compactly generated if every subset of 𝑋 that is open in every compact subset is open. (Contributed by Mario Carneiro, 20-Mar-2015.) |
Ref | Expression |
---|---|
iskgen3.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
iskgen3 | ⊢ (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → 𝑥 ∈ 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iskgen2 22153 | . 2 ⊢ (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽)) | |
2 | iskgen3.1 | . . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | toptopon 21522 | . . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
4 | elkgen 22141 | . . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))) | |
5 | 3, 4 | sylbi 220 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))) |
6 | vex 3444 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
7 | 6 | elpw 4501 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 𝑋 ↔ 𝑥 ⊆ 𝑋) |
8 | 7 | anbi1i 626 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))) |
9 | 5, 8 | syl6bbr 292 | . . . . . . 7 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥 ∈ 𝒫 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))) |
10 | 9 | imbi1d 345 | . . . . . 6 ⊢ (𝐽 ∈ Top → ((𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ 𝐽) ↔ ((𝑥 ∈ 𝒫 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))) → 𝑥 ∈ 𝐽))) |
11 | impexp 454 | . . . . . 6 ⊢ (((𝑥 ∈ 𝒫 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))) → 𝑥 ∈ 𝐽) ↔ (𝑥 ∈ 𝒫 𝑋 → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → 𝑥 ∈ 𝐽))) | |
12 | 10, 11 | syl6bb 290 | . . . . 5 ⊢ (𝐽 ∈ Top → ((𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ 𝐽) ↔ (𝑥 ∈ 𝒫 𝑋 → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → 𝑥 ∈ 𝐽)))) |
13 | 12 | albidv 1921 | . . . 4 ⊢ (𝐽 ∈ Top → (∀𝑥(𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ 𝐽) ↔ ∀𝑥(𝑥 ∈ 𝒫 𝑋 → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → 𝑥 ∈ 𝐽)))) |
14 | dfss2 3901 | . . . 4 ⊢ ((𝑘Gen‘𝐽) ⊆ 𝐽 ↔ ∀𝑥(𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ 𝐽)) | |
15 | df-ral 3111 | . . . 4 ⊢ (∀𝑥 ∈ 𝒫 𝑋(∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → 𝑥 ∈ 𝐽) ↔ ∀𝑥(𝑥 ∈ 𝒫 𝑋 → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → 𝑥 ∈ 𝐽))) | |
16 | 13, 14, 15 | 3bitr4g 317 | . . 3 ⊢ (𝐽 ∈ Top → ((𝑘Gen‘𝐽) ⊆ 𝐽 ↔ ∀𝑥 ∈ 𝒫 𝑋(∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → 𝑥 ∈ 𝐽))) |
17 | 16 | pm5.32i 578 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → 𝑥 ∈ 𝐽))) |
18 | 1, 17 | bitri 278 | 1 ⊢ (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → 𝑥 ∈ 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 ran crn 5520 ‘cfv 6324 (class class class)co 7135 ↾t crest 16686 Topctop 21498 TopOnctopon 21515 Compccmp 21991 𝑘Genckgen 22138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-oadd 8089 df-er 8272 df-en 8493 df-fin 8496 df-fi 8859 df-rest 16688 df-topgen 16709 df-top 21499 df-topon 21516 df-bases 21551 df-cmp 21992 df-kgen 22139 |
This theorem is referenced by: (None) |
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