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Theorem kgenval 22135
 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 22134 . 2 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘))})
2 unieq 4838 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
3 toponuni 21514 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
43eqcomd 2825 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 = 𝑋)
52, 4sylan9eqr 2876 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝑗 = 𝑋)
65pweqd 4542 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝒫 𝑗 = 𝒫 𝑋)
7 oveq1 7155 . . . . . . 7 (𝑗 = 𝐽 → (𝑗t 𝑘) = (𝐽t 𝑘))
87eleq1d 2895 . . . . . 6 (𝑗 = 𝐽 → ((𝑗t 𝑘) ∈ Comp ↔ (𝐽t 𝑘) ∈ Comp))
97eleq2d 2896 . . . . . 6 (𝑗 = 𝐽 → ((𝑥𝑘) ∈ (𝑗t 𝑘) ↔ (𝑥𝑘) ∈ (𝐽t 𝑘)))
108, 9imbi12d 347 . . . . 5 (𝑗 = 𝐽 → (((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))))
1110adantl 484 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘)) ↔ ((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))))
126, 11raleqbidv 3400 . . 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 3484 . 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 21513 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15 toponmax 21526 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
16 pwexg 5270 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
17 rabexg 5225 . . 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 6769 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 208   ∧ wa 398   = wceq 1530   ∈ wcel 2107  ∀wral 3136  {crab 3140  Vcvv 3493   ∩ cin 3933  𝒫 cpw 4537  ∪ cuni 4830  ‘cfv 6348  (class class class)co 7148   ↾t crest 16686  Topctop 21493  TopOnctopon 21510  Compccmp 21986  𝑘Genckgen 22133 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-top 21494  df-topon 21511  df-kgen 22134 This theorem is referenced by:  elkgen  22136  kgentopon  22138
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