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Theorem kgeni 21554
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 3972 . . . . 5 ((𝐴𝐾) ∩ 𝐽) = (𝐴 ∩ (𝐾 𝐽))
2 in32 3974 . . . . 5 ((𝐴𝐾) ∩ 𝐽) = ((𝐴 𝐽) ∩ 𝐾)
31, 2eqtr3i 2795 . . . 4 (𝐴 ∩ (𝐾 𝐽)) = ((𝐴 𝐽) ∩ 𝐾)
4 df-kgen 21551 . . . . . . . . . . . 12 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ∀𝑦 ∈ 𝒫 𝑗((𝑗t 𝑦) ∈ Comp → (𝑥𝑦) ∈ (𝑗t 𝑦))})
54dmmptss 5773 . . . . . . . . . . 11 dom 𝑘Gen ⊆ Top
6 elfvdm 6359 . . . . . . . . . . 11 (𝐴 ∈ (𝑘Gen‘𝐽) → 𝐽 ∈ dom 𝑘Gen)
75, 6sseldi 3750 . . . . . . . . . 10 (𝐴 ∈ (𝑘Gen‘𝐽) → 𝐽 ∈ Top)
87adantr 466 . . . . . . . . 9 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐽 ∈ Top)
9 eqid 2771 . . . . . . . . . 10 𝐽 = 𝐽
109toptopon 20935 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
118, 10sylib 208 . . . . . . . 8 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐽 ∈ (TopOn‘ 𝐽))
12 simpl 468 . . . . . . . 8 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐴 ∈ (𝑘Gen‘𝐽))
13 elkgen 21553 . . . . . . . . 9 (𝐽 ∈ (TopOn‘ 𝐽) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)))))
1413biimpa 462 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐴 ∈ (𝑘Gen‘𝐽)) → (𝐴 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦))))
1511, 12, 14syl2anc 573 . . . . . . 7 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦))))
1615simpld 482 . . . . . 6 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐴 𝐽)
17 df-ss 3737 . . . . . 6 (𝐴 𝐽 ↔ (𝐴 𝐽) = 𝐴)
1816, 17sylib 208 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 𝐽) = 𝐴)
1918ineq1d 3964 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ((𝐴 𝐽) ∩ 𝐾) = (𝐴𝐾))
203, 19syl5eq 2817 . . 3 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 𝐽)) = (𝐴𝐾))
21 cmptop 21412 . . . . . . . 8 ((𝐽t 𝐾) ∈ Comp → (𝐽t 𝐾) ∈ Top)
2221adantl 467 . . . . . . 7 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Top)
23 restrcl 21175 . . . . . . . 8 ((𝐽t 𝐾) ∈ Top → (𝐽 ∈ V ∧ 𝐾 ∈ V))
2423simprd 483 . . . . . . 7 ((𝐽t 𝐾) ∈ Top → 𝐾 ∈ V)
2522, 24syl 17 . . . . . 6 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐾 ∈ V)
269restin 21184 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐾 ∈ V) → (𝐽t 𝐾) = (𝐽t (𝐾 𝐽)))
278, 25, 26syl2anc 573 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) = (𝐽t (𝐾 𝐽)))
28 simpr 471 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Comp)
2927, 28eqeltrrd 2851 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t (𝐾 𝐽)) ∈ Comp)
30 oveq2 6799 . . . . . . 7 (𝑦 = (𝐾 𝐽) → (𝐽t 𝑦) = (𝐽t (𝐾 𝐽)))
3130eleq1d 2835 . . . . . 6 (𝑦 = (𝐾 𝐽) → ((𝐽t 𝑦) ∈ Comp ↔ (𝐽t (𝐾 𝐽)) ∈ Comp))
32 ineq2 3959 . . . . . . 7 (𝑦 = (𝐾 𝐽) → (𝐴𝑦) = (𝐴 ∩ (𝐾 𝐽)))
3332, 30eleq12d 2844 . . . . . 6 (𝑦 = (𝐾 𝐽) → ((𝐴𝑦) ∈ (𝐽t 𝑦) ↔ (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽))))
3431, 33imbi12d 333 . . . . 5 (𝑦 = (𝐾 𝐽) → (((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)) ↔ ((𝐽t (𝐾 𝐽)) ∈ Comp → (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽)))))
3515simprd 483 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)))
36 inss2 3982 . . . . . 6 (𝐾 𝐽) ⊆ 𝐽
37 inex1g 4935 . . . . . . 7 (𝐾 ∈ V → (𝐾 𝐽) ∈ V)
38 elpwg 4305 . . . . . . 7 ((𝐾 𝐽) ∈ V → ((𝐾 𝐽) ∈ 𝒫 𝐽 ↔ (𝐾 𝐽) ⊆ 𝐽))
3925, 37, 383syl 18 . . . . . 6 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ((𝐾 𝐽) ∈ 𝒫 𝐽 ↔ (𝐾 𝐽) ⊆ 𝐽))
4036, 39mpbiri 248 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐾 𝐽) ∈ 𝒫 𝐽)
4134, 35, 40rspcdva 3466 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ((𝐽t (𝐾 𝐽)) ∈ Comp → (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽))))
4229, 41mpd 15 . . 3 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽)))
4320, 42eqeltrrd 2851 . 2 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴𝐾) ∈ (𝐽t (𝐾 𝐽)))
4443, 27eleqtrrd 2853 1 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴𝐾) ∈ (𝐽t 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  cin 3722  wss 3723  𝒫 cpw 4297   cuni 4574  dom cdm 5249  cfv 6029  (class class class)co 6791  t crest 16282  Topctop 20911  TopOnctopon 20928  Compccmp 21403  𝑘Genckgen 21550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-1st 7313  df-2nd 7314  df-rest 16284  df-top 20912  df-topon 20929  df-cmp 21404  df-kgen 21551
This theorem is referenced by:  kgentopon  21555  kgencmp  21562  kgenidm  21564  llycmpkgen2  21567  1stckgen  21571  kgencn3  21575  txkgen  21669
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