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Theorem elkgen 21748
Description: Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
elkgen (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘)))))
Distinct variable groups:   𝐴,𝑘   𝑘,𝐽   𝑘,𝑋

Proof of Theorem elkgen
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 kgenval 21747 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))})
21eleq2d 2845 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ 𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))}))
3 ineq1 4030 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑘) = (𝐴𝑘))
43eleq1d 2844 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝑘) ∈ (𝐽t 𝑘) ↔ (𝐴𝑘) ∈ (𝐽t 𝑘)))
54imbi2d 332 . . . . 5 (𝑥 = 𝐴 → (((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘))))
65ralbidv 3168 . . . 4 (𝑥 = 𝐴 → (∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘))))
76elrab 3572 . . 3 (𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))} ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘))))
8 toponmax 21138 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
9 elpw2g 5061 . . . . 5 (𝑋𝐽 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
108, 9syl 17 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1110anbi1d 623 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘))) ↔ (𝐴𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘)))))
127, 11syl5bb 275 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))} ↔ (𝐴𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘)))))
132, 12bitrd 271 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wral 3090  {crab 3094  cin 3791  wss 3792  𝒫 cpw 4379  cfv 6135  (class class class)co 6922  t crest 16467  TopOnctopon 21122  Compccmp 21598  𝑘Genckgen 21745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-top 21106  df-topon 21123  df-kgen 21746
This theorem is referenced by:  kgeni  21749  kgentopon  21750  kgenss  21755  kgenidm  21759  iskgen3  21761  kgen2ss  21767  kgencn  21768  kgencn3  21770  txkgen  21864
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