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Theorem kgeni 23545
Description: Property of the open sets in the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
kgeni ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴𝐾) ∈ (𝐽t 𝐾))

Proof of Theorem kgeni
Dummy variables 𝑦 𝑥 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inass 4228 . . . . 5 ((𝐴𝐾) ∩ 𝐽) = (𝐴 ∩ (𝐾 𝐽))
2 in32 4230 . . . . 5 ((𝐴𝐾) ∩ 𝐽) = ((𝐴 𝐽) ∩ 𝐾)
31, 2eqtr3i 2767 . . . 4 (𝐴 ∩ (𝐾 𝐽)) = ((𝐴 𝐽) ∩ 𝐾)
4 df-kgen 23542 . . . . . . . . . . 11 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ∀𝑦 ∈ 𝒫 𝑗((𝑗t 𝑦) ∈ Comp → (𝑥𝑦) ∈ (𝑗t 𝑦))})
54mptrcl 7025 . . . . . . . . . 10 (𝐴 ∈ (𝑘Gen‘𝐽) → 𝐽 ∈ Top)
65adantr 480 . . . . . . . . 9 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐽 ∈ Top)
7 toptopon2 22924 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
86, 7sylib 218 . . . . . . . 8 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐽 ∈ (TopOn‘ 𝐽))
9 simpl 482 . . . . . . . 8 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐴 ∈ (𝑘Gen‘𝐽))
10 elkgen 23544 . . . . . . . . 9 (𝐽 ∈ (TopOn‘ 𝐽) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)))))
1110biimpa 476 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐴 ∈ (𝑘Gen‘𝐽)) → (𝐴 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦))))
128, 9, 11syl2anc 584 . . . . . . 7 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦))))
1312simpld 494 . . . . . 6 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐴 𝐽)
14 dfss2 3969 . . . . . 6 (𝐴 𝐽 ↔ (𝐴 𝐽) = 𝐴)
1513, 14sylib 218 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 𝐽) = 𝐴)
1615ineq1d 4219 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ((𝐴 𝐽) ∩ 𝐾) = (𝐴𝐾))
173, 16eqtrid 2789 . . 3 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 𝐽)) = (𝐴𝐾))
18 cmptop 23403 . . . . . . . 8 ((𝐽t 𝐾) ∈ Comp → (𝐽t 𝐾) ∈ Top)
1918adantl 481 . . . . . . 7 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Top)
20 restrcl 23165 . . . . . . . 8 ((𝐽t 𝐾) ∈ Top → (𝐽 ∈ V ∧ 𝐾 ∈ V))
2120simprd 495 . . . . . . 7 ((𝐽t 𝐾) ∈ Top → 𝐾 ∈ V)
2219, 21syl 17 . . . . . 6 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐾 ∈ V)
23 eqid 2737 . . . . . . 7 𝐽 = 𝐽
2423restin 23174 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐾 ∈ V) → (𝐽t 𝐾) = (𝐽t (𝐾 𝐽)))
256, 22, 24syl2anc 584 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) = (𝐽t (𝐾 𝐽)))
26 simpr 484 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Comp)
2725, 26eqeltrrd 2842 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t (𝐾 𝐽)) ∈ Comp)
28 oveq2 7439 . . . . . . 7 (𝑦 = (𝐾 𝐽) → (𝐽t 𝑦) = (𝐽t (𝐾 𝐽)))
2928eleq1d 2826 . . . . . 6 (𝑦 = (𝐾 𝐽) → ((𝐽t 𝑦) ∈ Comp ↔ (𝐽t (𝐾 𝐽)) ∈ Comp))
30 ineq2 4214 . . . . . . 7 (𝑦 = (𝐾 𝐽) → (𝐴𝑦) = (𝐴 ∩ (𝐾 𝐽)))
3130, 28eleq12d 2835 . . . . . 6 (𝑦 = (𝐾 𝐽) → ((𝐴𝑦) ∈ (𝐽t 𝑦) ↔ (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽))))
3229, 31imbi12d 344 . . . . 5 (𝑦 = (𝐾 𝐽) → (((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)) ↔ ((𝐽t (𝐾 𝐽)) ∈ Comp → (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽)))))
3312simprd 495 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)))
34 inss2 4238 . . . . . 6 (𝐾 𝐽) ⊆ 𝐽
35 inex1g 5319 . . . . . . 7 (𝐾 ∈ V → (𝐾 𝐽) ∈ V)
36 elpwg 4603 . . . . . . 7 ((𝐾 𝐽) ∈ V → ((𝐾 𝐽) ∈ 𝒫 𝐽 ↔ (𝐾 𝐽) ⊆ 𝐽))
3722, 35, 363syl 18 . . . . . 6 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ((𝐾 𝐽) ∈ 𝒫 𝐽 ↔ (𝐾 𝐽) ⊆ 𝐽))
3834, 37mpbiri 258 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐾 𝐽) ∈ 𝒫 𝐽)
3932, 33, 38rspcdva 3623 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ((𝐽t (𝐾 𝐽)) ∈ Comp → (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽))))
4027, 39mpd 15 . . 3 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽)))
4117, 40eqeltrrd 2842 . 2 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴𝐾) ∈ (𝐽t (𝐾 𝐽)))
4241, 25eleqtrrd 2844 1 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴𝐾) ∈ (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  wss 3951  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-rep 5279  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-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  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-iun 4993  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-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-rest 17467  df-top 22900  df-topon 22917  df-cmp 23395  df-kgen 23542
This theorem is referenced by:  kgentopon  23546  kgencmp  23553  kgenidm  23555  llycmpkgen2  23558  1stckgen  23562  kgencn3  23566  txkgen  23660
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