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Theorem kgeni 21263
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 3806 . . . . 5 ((𝐴𝐾) ∩ 𝐽) = (𝐴 ∩ (𝐾 𝐽))
2 in32 3808 . . . . 5 ((𝐴𝐾) ∩ 𝐽) = ((𝐴 𝐽) ∩ 𝐾)
31, 2eqtr3i 2645 . . . 4 (𝐴 ∩ (𝐾 𝐽)) = ((𝐴 𝐽) ∩ 𝐾)
4 df-kgen 21260 . . . . . . . . . . . 12 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ∀𝑦 ∈ 𝒫 𝑗((𝑗t 𝑦) ∈ Comp → (𝑥𝑦) ∈ (𝑗t 𝑦))})
54dmmptss 5595 . . . . . . . . . . 11 dom 𝑘Gen ⊆ Top
6 elfvdm 6182 . . . . . . . . . . 11 (𝐴 ∈ (𝑘Gen‘𝐽) → 𝐽 ∈ dom 𝑘Gen)
75, 6sseldi 3585 . . . . . . . . . 10 (𝐴 ∈ (𝑘Gen‘𝐽) → 𝐽 ∈ Top)
87adantr 481 . . . . . . . . 9 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐽 ∈ Top)
9 eqid 2621 . . . . . . . . . 10 𝐽 = 𝐽
109toptopon 20657 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
118, 10sylib 208 . . . . . . . 8 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐽 ∈ (TopOn‘ 𝐽))
12 simpl 473 . . . . . . . 8 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐴 ∈ (𝑘Gen‘𝐽))
13 elkgen 21262 . . . . . . . . 9 (𝐽 ∈ (TopOn‘ 𝐽) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)))))
1413biimpa 501 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐴 ∈ (𝑘Gen‘𝐽)) → (𝐴 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦))))
1511, 12, 14syl2anc 692 . . . . . . 7 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦))))
1615simpld 475 . . . . . 6 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐴 𝐽)
17 df-ss 3573 . . . . . 6 (𝐴 𝐽 ↔ (𝐴 𝐽) = 𝐴)
1816, 17sylib 208 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 𝐽) = 𝐴)
1918ineq1d 3796 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ((𝐴 𝐽) ∩ 𝐾) = (𝐴𝐾))
203, 19syl5eq 2667 . . 3 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 𝐽)) = (𝐴𝐾))
21 inss2 3817 . . . . 5 (𝐾 𝐽) ⊆ 𝐽
22 cmptop 21121 . . . . . . . 8 ((𝐽t 𝐾) ∈ Comp → (𝐽t 𝐾) ∈ Top)
2322adantl 482 . . . . . . 7 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Top)
24 restrcl 20884 . . . . . . . 8 ((𝐽t 𝐾) ∈ Top → (𝐽 ∈ V ∧ 𝐾 ∈ V))
2524simprd 479 . . . . . . 7 ((𝐽t 𝐾) ∈ Top → 𝐾 ∈ V)
2623, 25syl 17 . . . . . 6 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐾 ∈ V)
27 inex1g 4766 . . . . . 6 (𝐾 ∈ V → (𝐾 𝐽) ∈ V)
28 elpwg 4143 . . . . . 6 ((𝐾 𝐽) ∈ V → ((𝐾 𝐽) ∈ 𝒫 𝐽 ↔ (𝐾 𝐽) ⊆ 𝐽))
2926, 27, 283syl 18 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ((𝐾 𝐽) ∈ 𝒫 𝐽 ↔ (𝐾 𝐽) ⊆ 𝐽))
3021, 29mpbiri 248 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐾 𝐽) ∈ 𝒫 𝐽)
3115simprd 479 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)))
329restin 20893 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐾 ∈ V) → (𝐽t 𝐾) = (𝐽t (𝐾 𝐽)))
338, 26, 32syl2anc 692 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) = (𝐽t (𝐾 𝐽)))
34 simpr 477 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Comp)
3533, 34eqeltrrd 2699 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t (𝐾 𝐽)) ∈ Comp)
36 oveq2 6618 . . . . . . 7 (𝑦 = (𝐾 𝐽) → (𝐽t 𝑦) = (𝐽t (𝐾 𝐽)))
3736eleq1d 2683 . . . . . 6 (𝑦 = (𝐾 𝐽) → ((𝐽t 𝑦) ∈ Comp ↔ (𝐽t (𝐾 𝐽)) ∈ Comp))
38 ineq2 3791 . . . . . . 7 (𝑦 = (𝐾 𝐽) → (𝐴𝑦) = (𝐴 ∩ (𝐾 𝐽)))
3938, 36eleq12d 2692 . . . . . 6 (𝑦 = (𝐾 𝐽) → ((𝐴𝑦) ∈ (𝐽t 𝑦) ↔ (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽))))
4037, 39imbi12d 334 . . . . 5 (𝑦 = (𝐾 𝐽) → (((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)) ↔ ((𝐽t (𝐾 𝐽)) ∈ Comp → (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽)))))
4140rspcv 3294 . . . 4 ((𝐾 𝐽) ∈ 𝒫 𝐽 → (∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)) → ((𝐽t (𝐾 𝐽)) ∈ Comp → (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽)))))
4230, 31, 35, 41syl3c 66 . . 3 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽)))
4320, 42eqeltrrd 2699 . 2 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴𝐾) ∈ (𝐽t (𝐾 𝐽)))
4443, 33eleqtrrd 2701 1 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴𝐾) ∈ (𝐽t 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3189  cin 3558  wss 3559  𝒫 cpw 4135   cuni 4407  dom cdm 5079  cfv 5852  (class class class)co 6610  t crest 16013  Topctop 20630  TopOnctopon 20647  Compccmp 21112  𝑘Genckgen 21259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-rest 16015  df-top 20631  df-topon 20648  df-cmp 21113  df-kgen 21260
This theorem is referenced by:  kgentopon  21264  kgencmp  21271  kgenidm  21273  llycmpkgen2  21276  1stckgen  21280  kgencn3  21284  txkgen  21378
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