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Theorem kgenval 21617
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 21616 . . 3 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘))})
21a1i 11 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘))}))
3 unieq 4601 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
4 toponuni 20997 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
54eqcomd 2770 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 = 𝑋)
63, 5sylan9eqr 2820 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑗 = 𝑋)
76pweqd 4319 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝒫 𝑗 = 𝒫 𝑋)
8 oveq1 6848 . . . . . . 7 (𝑗 = 𝐽 → (𝑗t 𝑘) = (𝐽t 𝑘))
98eleq1d 2828 . . . . . 6 (𝑗 = 𝐽 → ((𝑗t 𝑘) ∈ Comp ↔ (𝐽t 𝑘) ∈ Comp))
108eleq2d 2829 . . . . . 6 (𝑗 = 𝐽 → ((𝑥𝑘) ∈ (𝑗t 𝑘) ↔ (𝑥𝑘) ∈ (𝐽t 𝑘)))
119, 10imbi12d 335 . . . . 5 (𝑗 = 𝐽 → (((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))))
1211adantl 473 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))))
137, 12raleqbidv 3299 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))))
147, 13rabeqbidv 3343 . 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))})
15 topontop 20996 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
16 toponmax 21009 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
17 pwexg 5013 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
18 rabexg 4971 . . 3 (𝒫 𝑋 ∈ V → {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))} ∈ V)
1916, 17, 183syl 18 . 2 (𝐽 ∈ (TopOn‘𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))} ∈ V)
202, 14, 15, 19fvmptd 6476 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 197  wa 384   = wceq 1652  wcel 2155  wral 3054  {crab 3058  Vcvv 3349  cin 3730  𝒫 cpw 4314   cuni 4593  cmpt 4887  cfv 6067  (class class class)co 6841  t crest 16348  Topctop 20976  TopOnctopon 20993  Compccmp 21468  𝑘Genckgen 21615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-iota 6030  df-fun 6069  df-fv 6075  df-ov 6844  df-top 20977  df-topon 20994  df-kgen 21616
This theorem is referenced by:  elkgen  21618  kgentopon  21620
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