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Theorem kgeni 22904
Description: Property of the open sets in the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
kgeni ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐴 ∩ 𝐾) ∈ (𝐽 β†Ύt 𝐾))

Proof of Theorem kgeni
Dummy variables 𝑦 π‘₯ 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inass 4184 . . . . 5 ((𝐴 ∩ 𝐾) ∩ βˆͺ 𝐽) = (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽))
2 in32 4186 . . . . 5 ((𝐴 ∩ 𝐾) ∩ βˆͺ 𝐽) = ((𝐴 ∩ βˆͺ 𝐽) ∩ 𝐾)
31, 2eqtr3i 2767 . . . 4 (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)) = ((𝐴 ∩ βˆͺ 𝐽) ∩ 𝐾)
4 df-kgen 22901 . . . . . . . . . . 11 π‘˜Gen = (𝑗 ∈ Top ↦ {π‘₯ ∈ 𝒫 βˆͺ 𝑗 ∣ βˆ€π‘¦ ∈ 𝒫 βˆͺ 𝑗((𝑗 β†Ύt 𝑦) ∈ Comp β†’ (π‘₯ ∩ 𝑦) ∈ (𝑗 β†Ύt 𝑦))})
54mptrcl 6962 . . . . . . . . . 10 (𝐴 ∈ (π‘˜Genβ€˜π½) β†’ 𝐽 ∈ Top)
65adantr 482 . . . . . . . . 9 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ 𝐽 ∈ Top)
7 toptopon2 22283 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
86, 7sylib 217 . . . . . . . 8 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
9 simpl 484 . . . . . . . 8 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ 𝐴 ∈ (π‘˜Genβ€˜π½))
10 elkgen 22903 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) β†’ (𝐴 ∈ (π‘˜Genβ€˜π½) ↔ (𝐴 βŠ† βˆͺ 𝐽 ∧ βˆ€π‘¦ ∈ 𝒫 βˆͺ 𝐽((𝐽 β†Ύt 𝑦) ∈ Comp β†’ (𝐴 ∩ 𝑦) ∈ (𝐽 β†Ύt 𝑦)))))
1110biimpa 478 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ 𝐴 ∈ (π‘˜Genβ€˜π½)) β†’ (𝐴 βŠ† βˆͺ 𝐽 ∧ βˆ€π‘¦ ∈ 𝒫 βˆͺ 𝐽((𝐽 β†Ύt 𝑦) ∈ Comp β†’ (𝐴 ∩ 𝑦) ∈ (𝐽 β†Ύt 𝑦))))
128, 9, 11syl2anc 585 . . . . . . 7 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐴 βŠ† βˆͺ 𝐽 ∧ βˆ€π‘¦ ∈ 𝒫 βˆͺ 𝐽((𝐽 β†Ύt 𝑦) ∈ Comp β†’ (𝐴 ∩ 𝑦) ∈ (𝐽 β†Ύt 𝑦))))
1312simpld 496 . . . . . 6 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ 𝐴 βŠ† βˆͺ 𝐽)
14 df-ss 3932 . . . . . 6 (𝐴 βŠ† βˆͺ 𝐽 ↔ (𝐴 ∩ βˆͺ 𝐽) = 𝐴)
1513, 14sylib 217 . . . . 5 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐴 ∩ βˆͺ 𝐽) = 𝐴)
1615ineq1d 4176 . . . 4 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ ((𝐴 ∩ βˆͺ 𝐽) ∩ 𝐾) = (𝐴 ∩ 𝐾))
173, 16eqtrid 2789 . . 3 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)) = (𝐴 ∩ 𝐾))
18 cmptop 22762 . . . . . . . 8 ((𝐽 β†Ύt 𝐾) ∈ Comp β†’ (𝐽 β†Ύt 𝐾) ∈ Top)
1918adantl 483 . . . . . . 7 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐽 β†Ύt 𝐾) ∈ Top)
20 restrcl 22524 . . . . . . . 8 ((𝐽 β†Ύt 𝐾) ∈ Top β†’ (𝐽 ∈ V ∧ 𝐾 ∈ V))
2120simprd 497 . . . . . . 7 ((𝐽 β†Ύt 𝐾) ∈ Top β†’ 𝐾 ∈ V)
2219, 21syl 17 . . . . . 6 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ 𝐾 ∈ V)
23 eqid 2737 . . . . . . 7 βˆͺ 𝐽 = βˆͺ 𝐽
2423restin 22533 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐾 ∈ V) β†’ (𝐽 β†Ύt 𝐾) = (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)))
256, 22, 24syl2anc 585 . . . . 5 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐽 β†Ύt 𝐾) = (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)))
26 simpr 486 . . . . 5 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐽 β†Ύt 𝐾) ∈ Comp)
2725, 26eqeltrrd 2839 . . . 4 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)) ∈ Comp)
28 oveq2 7370 . . . . . . 7 (𝑦 = (𝐾 ∩ βˆͺ 𝐽) β†’ (𝐽 β†Ύt 𝑦) = (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)))
2928eleq1d 2823 . . . . . 6 (𝑦 = (𝐾 ∩ βˆͺ 𝐽) β†’ ((𝐽 β†Ύt 𝑦) ∈ Comp ↔ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)) ∈ Comp))
30 ineq2 4171 . . . . . . 7 (𝑦 = (𝐾 ∩ βˆͺ 𝐽) β†’ (𝐴 ∩ 𝑦) = (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)))
3130, 28eleq12d 2832 . . . . . 6 (𝑦 = (𝐾 ∩ βˆͺ 𝐽) β†’ ((𝐴 ∩ 𝑦) ∈ (𝐽 β†Ύt 𝑦) ↔ (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)) ∈ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽))))
3229, 31imbi12d 345 . . . . 5 (𝑦 = (𝐾 ∩ βˆͺ 𝐽) β†’ (((𝐽 β†Ύt 𝑦) ∈ Comp β†’ (𝐴 ∩ 𝑦) ∈ (𝐽 β†Ύt 𝑦)) ↔ ((𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)) ∈ Comp β†’ (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)) ∈ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)))))
3312simprd 497 . . . . 5 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ βˆ€π‘¦ ∈ 𝒫 βˆͺ 𝐽((𝐽 β†Ύt 𝑦) ∈ Comp β†’ (𝐴 ∩ 𝑦) ∈ (𝐽 β†Ύt 𝑦)))
34 inss2 4194 . . . . . 6 (𝐾 ∩ βˆͺ 𝐽) βŠ† βˆͺ 𝐽
35 inex1g 5281 . . . . . . 7 (𝐾 ∈ V β†’ (𝐾 ∩ βˆͺ 𝐽) ∈ V)
36 elpwg 4568 . . . . . . 7 ((𝐾 ∩ βˆͺ 𝐽) ∈ V β†’ ((𝐾 ∩ βˆͺ 𝐽) ∈ 𝒫 βˆͺ 𝐽 ↔ (𝐾 ∩ βˆͺ 𝐽) βŠ† βˆͺ 𝐽))
3722, 35, 363syl 18 . . . . . 6 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ ((𝐾 ∩ βˆͺ 𝐽) ∈ 𝒫 βˆͺ 𝐽 ↔ (𝐾 ∩ βˆͺ 𝐽) βŠ† βˆͺ 𝐽))
3834, 37mpbiri 258 . . . . 5 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐾 ∩ βˆͺ 𝐽) ∈ 𝒫 βˆͺ 𝐽)
3932, 33, 38rspcdva 3585 . . . 4 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ ((𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)) ∈ Comp β†’ (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)) ∈ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽))))
4027, 39mpd 15 . . 3 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)) ∈ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)))
4117, 40eqeltrrd 2839 . 2 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐴 ∩ 𝐾) ∈ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)))
4241, 25eleqtrrd 2841 1 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐴 ∩ 𝐾) ∈ (𝐽 β†Ύt 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410  Vcvv 3448   ∩ cin 3914   βŠ† wss 3915  π’« cpw 4565  βˆͺ cuni 4870  β€˜cfv 6501  (class class class)co 7362   β†Ύt crest 17309  Topctop 22258  TopOnctopon 22275  Compccmp 22753  π‘˜Genckgen 22900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-rest 17311  df-top 22259  df-topon 22276  df-cmp 22754  df-kgen 22901
This theorem is referenced by:  kgentopon  22905  kgencmp  22912  kgenidm  22914  llycmpkgen2  22917  1stckgen  22921  kgencn3  22925  txkgen  23019
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