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Theorem kgenval 23543
Description: Value of the compact generator. (The "k" in 𝑘Gen comes from the name "k-space" for these spaces, after the German word kompakt.) (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
kgenval (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))})
Distinct variable groups:   𝑥,𝑘,𝐽   𝑘,𝑋,𝑥

Proof of Theorem kgenval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 df-kgen 23542 . 2 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘))})
2 unieq 4918 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
3 toponuni 22920 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
43eqcomd 2743 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 = 𝑋)
52, 4sylan9eqr 2799 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑗 = 𝑋)
65pweqd 4617 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝒫 𝑗 = 𝒫 𝑋)
7 oveq1 7438 . . . . . . 7 (𝑗 = 𝐽 → (𝑗t 𝑘) = (𝐽t 𝑘))
87eleq1d 2826 . . . . . 6 (𝑗 = 𝐽 → ((𝑗t 𝑘) ∈ Comp ↔ (𝐽t 𝑘) ∈ Comp))
97eleq2d 2827 . . . . . 6 (𝑗 = 𝐽 → ((𝑥𝑘) ∈ (𝑗t 𝑘) ↔ (𝑥𝑘) ∈ (𝐽t 𝑘)))
108, 9imbi12d 344 . . . . 5 (𝑗 = 𝐽 → (((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))))
1110adantl 481 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))))
126, 11raleqbidv 3346 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))))
136, 12rabeqbidv 3455 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → {𝑥 ∈ 𝒫 𝑗 ∣ ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘))} = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))})
14 topontop 22919 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15 toponmax 22932 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
16 pwexg 5378 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
17 rabexg 5337 . . 3 (𝒫 𝑋 ∈ V → {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))} ∈ V)
1815, 16, 173syl 18 . 2 (𝐽 ∈ (TopOn‘𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))} ∈ V)
191, 13, 14, 18fvmptd2 7024 1 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  cin 3950  𝒫 cpw 4600   cuni 4907  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  TopOnctopon 22916  Compccmp 23394  𝑘Genckgen 23541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-top 22900  df-topon 22917  df-kgen 23542
This theorem is referenced by:  elkgen  23544  kgentopon  23546
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