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Mirrors > Home > MPE Home > Th. List > elkgen | Structured version Visualization version GIF version |
Description: Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.) |
Ref | Expression |
---|---|
elkgen | ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kgenval 23459 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))}) | |
2 | 1 | eleq2d 2815 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ 𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))})) |
3 | ineq1 4207 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑘) = (𝐴 ∩ 𝑘)) | |
4 | 3 | eleq1d 2814 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘) ↔ (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))) |
5 | 4 | imbi2d 339 | . . . . 5 ⊢ (𝑥 = 𝐴 → (((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) ↔ ((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))) |
6 | 5 | ralbidv 3175 | . . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))) |
7 | 6 | elrab 3684 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))} ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))) |
8 | toponmax 22848 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
9 | elpw2g 5350 | . . . . 5 ⊢ (𝑋 ∈ 𝐽 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
11 | 10 | anbi1d 629 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))) |
12 | 7, 11 | bitrid 282 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))} ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))) |
13 | 2, 12 | bitrd 278 | 1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 {crab 3430 ∩ cin 3948 ⊆ wss 3949 𝒫 cpw 4606 ‘cfv 6553 (class class class)co 7426 ↾t crest 17409 TopOnctopon 22832 Compccmp 23310 𝑘Genckgen 23457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-top 22816 df-topon 22833 df-kgen 23458 |
This theorem is referenced by: kgeni 23461 kgentopon 23462 kgenss 23467 kgenidm 23471 iskgen3 23473 kgen2ss 23479 kgencn 23480 kgencn3 23482 txkgen 23576 |
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