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Theorem kgeni 23032
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 4218 . . . . 5 ((𝐴 ∩ 𝐾) ∩ βˆͺ 𝐽) = (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽))
2 in32 4220 . . . . 5 ((𝐴 ∩ 𝐾) ∩ βˆͺ 𝐽) = ((𝐴 ∩ βˆͺ 𝐽) ∩ 𝐾)
31, 2eqtr3i 2762 . . . 4 (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)) = ((𝐴 ∩ βˆͺ 𝐽) ∩ 𝐾)
4 df-kgen 23029 . . . . . . . . . . 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 7004 . . . . . . . . . 10 (𝐴 ∈ (π‘˜Genβ€˜π½) β†’ 𝐽 ∈ Top)
65adantr 481 . . . . . . . . 9 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ 𝐽 ∈ Top)
7 toptopon2 22411 . . . . . . . . 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 483 . . . . . . . 8 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ 𝐴 ∈ (π‘˜Genβ€˜π½))
10 elkgen 23031 . . . . . . . . 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 477 . . . . . . . 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 584 . . . . . . 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 495 . . . . . 6 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ 𝐴 βŠ† βˆͺ 𝐽)
14 df-ss 3964 . . . . . 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 4210 . . . 4 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ ((𝐴 ∩ βˆͺ 𝐽) ∩ 𝐾) = (𝐴 ∩ 𝐾))
173, 16eqtrid 2784 . . 3 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)) = (𝐴 ∩ 𝐾))
18 cmptop 22890 . . . . . . . 8 ((𝐽 β†Ύt 𝐾) ∈ Comp β†’ (𝐽 β†Ύt 𝐾) ∈ Top)
1918adantl 482 . . . . . . 7 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐽 β†Ύt 𝐾) ∈ Top)
20 restrcl 22652 . . . . . . . 8 ((𝐽 β†Ύt 𝐾) ∈ Top β†’ (𝐽 ∈ V ∧ 𝐾 ∈ V))
2120simprd 496 . . . . . . 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 2732 . . . . . . 7 βˆͺ 𝐽 = βˆͺ 𝐽
2423restin 22661 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐾 ∈ V) β†’ (𝐽 β†Ύt 𝐾) = (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)))
256, 22, 24syl2anc 584 . . . . 5 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐽 β†Ύt 𝐾) = (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)))
26 simpr 485 . . . . 5 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐽 β†Ύt 𝐾) ∈ Comp)
2725, 26eqeltrrd 2834 . . . 4 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)) ∈ Comp)
28 oveq2 7413 . . . . . . 7 (𝑦 = (𝐾 ∩ βˆͺ 𝐽) β†’ (𝐽 β†Ύt 𝑦) = (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)))
2928eleq1d 2818 . . . . . 6 (𝑦 = (𝐾 ∩ βˆͺ 𝐽) β†’ ((𝐽 β†Ύt 𝑦) ∈ Comp ↔ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)) ∈ Comp))
30 ineq2 4205 . . . . . . 7 (𝑦 = (𝐾 ∩ βˆͺ 𝐽) β†’ (𝐴 ∩ 𝑦) = (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)))
3130, 28eleq12d 2827 . . . . . 6 (𝑦 = (𝐾 ∩ βˆͺ 𝐽) β†’ ((𝐴 ∩ 𝑦) ∈ (𝐽 β†Ύt 𝑦) ↔ (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)) ∈ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽))))
3229, 31imbi12d 344 . . . . 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 496 . . . . 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 4228 . . . . . 6 (𝐾 ∩ βˆͺ 𝐽) βŠ† βˆͺ 𝐽
35 inex1g 5318 . . . . . . 7 (𝐾 ∈ V β†’ (𝐾 ∩ βˆͺ 𝐽) ∈ V)
36 elpwg 4604 . . . . . . 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 257 . . . . 5 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐾 ∩ βˆͺ 𝐽) ∈ 𝒫 βˆͺ 𝐽)
3932, 33, 38rspcdva 3613 . . . 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 2834 . 2 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐴 ∩ 𝐾) ∈ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)))
4241, 25eleqtrrd 2836 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432  Vcvv 3474   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  βˆͺ cuni 4907  β€˜cfv 6540  (class class class)co 7405   β†Ύt crest 17362  Topctop 22386  TopOnctopon 22403  Compccmp 22881  π‘˜Genckgen 23028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-rest 17364  df-top 22387  df-topon 22404  df-cmp 22882  df-kgen 23029
This theorem is referenced by:  kgentopon  23033  kgencmp  23040  kgenidm  23042  llycmpkgen2  23045  1stckgen  23049  kgencn3  23053  txkgen  23147
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