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Theorem kgeni 23041
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 4220 . . . . 5 ((𝐴 ∩ 𝐾) ∩ βˆͺ 𝐽) = (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽))
2 in32 4222 . . . . 5 ((𝐴 ∩ 𝐾) ∩ βˆͺ 𝐽) = ((𝐴 ∩ βˆͺ 𝐽) ∩ 𝐾)
31, 2eqtr3i 2763 . . . 4 (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)) = ((𝐴 ∩ βˆͺ 𝐽) ∩ 𝐾)
4 df-kgen 23038 . . . . . . . . . . 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 7008 . . . . . . . . . 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 22420 . . . . . . . . 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 23040 . . . . . . . . 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 3966 . . . . . 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 4212 . . . 4 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ ((𝐴 ∩ βˆͺ 𝐽) ∩ 𝐾) = (𝐴 ∩ 𝐾))
173, 16eqtrid 2785 . . 3 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)) = (𝐴 ∩ 𝐾))
18 cmptop 22899 . . . . . . . 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 22661 . . . . . . . 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 2733 . . . . . . 7 βˆͺ 𝐽 = βˆͺ 𝐽
2423restin 22670 . . . . . 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 2835 . . . 4 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)) ∈ Comp)
28 oveq2 7417 . . . . . . 7 (𝑦 = (𝐾 ∩ βˆͺ 𝐽) β†’ (𝐽 β†Ύt 𝑦) = (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)))
2928eleq1d 2819 . . . . . 6 (𝑦 = (𝐾 ∩ βˆͺ 𝐽) β†’ ((𝐽 β†Ύt 𝑦) ∈ Comp ↔ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)) ∈ Comp))
30 ineq2 4207 . . . . . . 7 (𝑦 = (𝐾 ∩ βˆͺ 𝐽) β†’ (𝐴 ∩ 𝑦) = (𝐴 ∩ (𝐾 ∩ βˆͺ 𝐽)))
3130, 28eleq12d 2828 . . . . . 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 4230 . . . . . 6 (𝐾 ∩ βˆͺ 𝐽) βŠ† βˆͺ 𝐽
35 inex1g 5320 . . . . . . 7 (𝐾 ∈ V β†’ (𝐾 ∩ βˆͺ 𝐽) ∈ V)
36 elpwg 4606 . . . . . . 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 3614 . . . 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 2835 . 2 ((𝐴 ∈ (π‘˜Genβ€˜π½) ∧ (𝐽 β†Ύt 𝐾) ∈ Comp) β†’ (𝐴 ∩ 𝐾) ∈ (𝐽 β†Ύt (𝐾 ∩ βˆͺ 𝐽)))
4241, 25eleqtrrd 2837 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 3062  {crab 3433  Vcvv 3475   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409   β†Ύt crest 17366  Topctop 22395  TopOnctopon 22412  Compccmp 22890  π‘˜Genckgen 23037
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-rest 17368  df-top 22396  df-topon 22413  df-cmp 22891  df-kgen 23038
This theorem is referenced by:  kgentopon  23042  kgencmp  23049  kgenidm  23051  llycmpkgen2  23054  1stckgen  23058  kgencn3  23062  txkgen  23156
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