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Theorem elkgen 22685
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 22684 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))})
21eleq2d 2826 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ 𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))}))
3 ineq1 4145 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑘) = (𝐴𝑘))
43eleq1d 2825 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝑘) ∈ (𝐽t 𝑘) ↔ (𝐴𝑘) ∈ (𝐽t 𝑘)))
54imbi2d 341 . . . . 5 (𝑥 = 𝐴 → (((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘))))
65ralbidv 3123 . . . 4 (𝑥 = 𝐴 → (∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘))))
76elrab 3626 . . 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 22073 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
9 elpw2g 5272 . . . . 5 (𝑋𝐽 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
108, 9syl 17 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1110anbi1d 630 . . 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 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 278 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 205  wa 396   = wceq 1542  wcel 2110  wral 3066  {crab 3070  cin 3891  wss 3892  𝒫 cpw 4539  cfv 6432  (class class class)co 7271  t crest 17129  TopOnctopon 22057  Compccmp 22535  𝑘Genckgen 22682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-iota 6390  df-fun 6434  df-fv 6440  df-ov 7274  df-top 22041  df-topon 22058  df-kgen 22683
This theorem is referenced by:  kgeni  22686  kgentopon  22687  kgenss  22692  kgenidm  22696  iskgen3  22698  kgen2ss  22704  kgencn  22705  kgencn3  22707  txkgen  22801
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