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Theorem kgeni 22121
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 4171 . . . . 5 ((𝐴𝐾) ∩ 𝐽) = (𝐴 ∩ (𝐾 𝐽))
2 in32 4173 . . . . 5 ((𝐴𝐾) ∩ 𝐽) = ((𝐴 𝐽) ∩ 𝐾)
31, 2eqtr3i 2846 . . . 4 (𝐴 ∩ (𝐾 𝐽)) = ((𝐴 𝐽) ∩ 𝐾)
4 df-kgen 22118 . . . . . . . . . . 11 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ∀𝑦 ∈ 𝒫 𝑗((𝑗t 𝑦) ∈ Comp → (𝑥𝑦) ∈ (𝑗t 𝑦))})
54mptrcl 6750 . . . . . . . . . 10 (𝐴 ∈ (𝑘Gen‘𝐽) → 𝐽 ∈ Top)
65adantr 484 . . . . . . . . 9 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐽 ∈ Top)
7 toptopon2 21502 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
86, 7sylib 221 . . . . . . . 8 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐽 ∈ (TopOn‘ 𝐽))
9 simpl 486 . . . . . . . 8 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐴 ∈ (𝑘Gen‘𝐽))
10 elkgen 22120 . . . . . . . . 9 (𝐽 ∈ (TopOn‘ 𝐽) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)))))
1110biimpa 480 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐴 ∈ (𝑘Gen‘𝐽)) → (𝐴 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦))))
128, 9, 11syl2anc 587 . . . . . . 7 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 𝐽 ∧ ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦))))
1312simpld 498 . . . . . 6 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐴 𝐽)
14 df-ss 3927 . . . . . 6 (𝐴 𝐽 ↔ (𝐴 𝐽) = 𝐴)
1513, 14sylib 221 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 𝐽) = 𝐴)
1615ineq1d 4163 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ((𝐴 𝐽) ∩ 𝐾) = (𝐴𝐾))
173, 16syl5eq 2868 . . 3 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 𝐽)) = (𝐴𝐾))
18 cmptop 21979 . . . . . . . 8 ((𝐽t 𝐾) ∈ Comp → (𝐽t 𝐾) ∈ Top)
1918adantl 485 . . . . . . 7 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Top)
20 restrcl 21741 . . . . . . . 8 ((𝐽t 𝐾) ∈ Top → (𝐽 ∈ V ∧ 𝐾 ∈ V))
2120simprd 499 . . . . . . 7 ((𝐽t 𝐾) ∈ Top → 𝐾 ∈ V)
2219, 21syl 17 . . . . . 6 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → 𝐾 ∈ V)
23 eqid 2821 . . . . . . 7 𝐽 = 𝐽
2423restin 21750 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐾 ∈ V) → (𝐽t 𝐾) = (𝐽t (𝐾 𝐽)))
256, 22, 24syl2anc 587 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) = (𝐽t (𝐾 𝐽)))
26 simpr 488 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Comp)
2725, 26eqeltrrd 2913 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t (𝐾 𝐽)) ∈ Comp)
28 oveq2 7138 . . . . . . 7 (𝑦 = (𝐾 𝐽) → (𝐽t 𝑦) = (𝐽t (𝐾 𝐽)))
2928eleq1d 2896 . . . . . 6 (𝑦 = (𝐾 𝐽) → ((𝐽t 𝑦) ∈ Comp ↔ (𝐽t (𝐾 𝐽)) ∈ Comp))
30 ineq2 4158 . . . . . . 7 (𝑦 = (𝐾 𝐽) → (𝐴𝑦) = (𝐴 ∩ (𝐾 𝐽)))
3130, 28eleq12d 2906 . . . . . 6 (𝑦 = (𝐾 𝐽) → ((𝐴𝑦) ∈ (𝐽t 𝑦) ↔ (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽))))
3229, 31imbi12d 348 . . . . 5 (𝑦 = (𝐾 𝐽) → (((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)) ↔ ((𝐽t (𝐾 𝐽)) ∈ Comp → (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽)))))
3312simprd 499 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ∀𝑦 ∈ 𝒫 𝐽((𝐽t 𝑦) ∈ Comp → (𝐴𝑦) ∈ (𝐽t 𝑦)))
34 inss2 4181 . . . . . 6 (𝐾 𝐽) ⊆ 𝐽
35 inex1g 5196 . . . . . . 7 (𝐾 ∈ V → (𝐾 𝐽) ∈ V)
36 elpwg 4515 . . . . . . 7 ((𝐾 𝐽) ∈ V → ((𝐾 𝐽) ∈ 𝒫 𝐽 ↔ (𝐾 𝐽) ⊆ 𝐽))
3722, 35, 363syl 18 . . . . . 6 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ((𝐾 𝐽) ∈ 𝒫 𝐽 ↔ (𝐾 𝐽) ⊆ 𝐽))
3834, 37mpbiri 261 . . . . 5 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐾 𝐽) ∈ 𝒫 𝐽)
3932, 33, 38rspcdva 3602 . . . 4 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → ((𝐽t (𝐾 𝐽)) ∈ Comp → (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽))))
4027, 39mpd 15 . . 3 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴 ∩ (𝐾 𝐽)) ∈ (𝐽t (𝐾 𝐽)))
4117, 40eqeltrrd 2913 . 2 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴𝐾) ∈ (𝐽t (𝐾 𝐽)))
4241, 25eleqtrrd 2915 1 ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴𝐾) ∈ (𝐽t 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3126  {crab 3130  Vcvv 3471  cin 3909  wss 3910  𝒫 cpw 4512   cuni 4811  cfv 6328  (class class class)co 7130  t crest 16673  Topctop 21477  TopOnctopon 21494  Compccmp 21970  𝑘Genckgen 22117
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-rest 16675  df-top 21478  df-topon 21495  df-cmp 21971  df-kgen 22118
This theorem is referenced by:  kgentopon  22122  kgencmp  22129  kgenidm  22131  llycmpkgen2  22134  1stckgen  22138  kgencn3  22142  txkgen  22236
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