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Theorem elkgen 23560
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 23559 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))})
21eleq2d 2825 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ 𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))}))
3 ineq1 4221 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑘) = (𝐴𝑘))
43eleq1d 2824 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝑘) ∈ (𝐽t 𝑘) ↔ (𝐴𝑘) ∈ (𝐽t 𝑘)))
54imbi2d 340 . . . . 5 (𝑥 = 𝐴 → (((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘))))
65ralbidv 3176 . . . 4 (𝑥 = 𝐴 → (∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘))))
76elrab 3695 . . 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 22948 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
9 elpw2g 5339 . . . . 5 (𝑋𝐽 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
108, 9syl 17 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1110anbi1d 631 . . 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, 11bitrid 283 . 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 279 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 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  cin 3962  wss 3963  𝒫 cpw 4605  cfv 6563  (class class class)co 7431  t crest 17467  TopOnctopon 22932  Compccmp 23410  𝑘Genckgen 23557
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-top 22916  df-topon 22933  df-kgen 23558
This theorem is referenced by:  kgeni  23561  kgentopon  23562  kgenss  23567  kgenidm  23571  iskgen3  23573  kgen2ss  23579  kgencn  23580  kgencn3  23582  txkgen  23676
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