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Theorem kgen2ss 22160
Description: The compact generator preserves the subset (fineness) relationship on topologies. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
kgen2ss ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))

Proof of Theorem kgen2ss
Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1133 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽 ∈ (TopOn‘𝑋))
2 elpwi 4506 . . . . . . . . 9 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
3 resttopon 21766 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
41, 2, 3syl2an 598 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
5 simp2 1134 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ (TopOn‘𝑋))
6 resttopon 21766 . . . . . . . . . . 11 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐾t 𝑘) ∈ (TopOn‘𝑘))
75, 2, 6syl2an 598 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐾t 𝑘) ∈ (TopOn‘𝑘))
8 toponuni 21519 . . . . . . . . . 10 ((𝐾t 𝑘) ∈ (TopOn‘𝑘) → 𝑘 = (𝐾t 𝑘))
97, 8syl 17 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 = (𝐾t 𝑘))
109fveq2d 6649 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (TopOn‘𝑘) = (TopOn‘ (𝐾t 𝑘)))
114, 10eleqtrd 2892 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)))
12 simpl2 1189 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑋))
13 topontop 21518 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
1412, 13syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ Top)
15 simpl3 1190 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐽𝐾)
16 ssrest 21781 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝐽𝐾) → (𝐽t 𝑘) ⊆ (𝐾t 𝑘))
1714, 15, 16syl2anc 587 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ⊆ (𝐾t 𝑘))
18 eqid 2798 . . . . . . . . . 10 (𝐾t 𝑘) = (𝐾t 𝑘)
1918sscmp 22010 . . . . . . . . 9 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐾t 𝑘) ∈ Comp ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘)) → (𝐽t 𝑘) ∈ Comp)
20193com23 1123 . . . . . . . 8 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘) ∧ (𝐾t 𝑘) ∈ Comp) → (𝐽t 𝑘) ∈ Comp)
21203expia 1118 . . . . . . 7 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘)) → ((𝐾t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Comp))
2211, 17, 21syl2anc 587 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐾t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Comp))
2317sseld 3914 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝑥𝑘) ∈ (𝐽t 𝑘) → (𝑥𝑘) ∈ (𝐾t 𝑘)))
2422, 23imim12d 81 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)) → ((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘))))
2524ralimdva 3144 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)) → ∀𝑘 ∈ 𝒫 𝑋((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘))))
2625anim2d 614 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → ((𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))) → (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘)))))
27 elkgen 22141 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)))))
28273ad2ant1 1130 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)))))
29 elkgen 22141 . . . 4 (𝐾 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘)))))
30293ad2ant2 1131 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘)))))
3126, 28, 303imtr4d 297 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐾)))
3231ssrdv 3921 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wral 3106  cin 3880  wss 3881  𝒫 cpw 4497   cuni 4800  cfv 6324  (class class class)co 7135  t crest 16686  Topctop 21498  TopOnctopon 21515  Compccmp 21991  𝑘Genckgen 22138
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-er 8272  df-en 8493  df-fin 8496  df-fi 8859  df-rest 16688  df-topgen 16709  df-top 21499  df-topon 21516  df-bases 21551  df-cmp 21992  df-kgen 22139
This theorem is referenced by: (None)
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