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Theorem kgen2ss 22091
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 1128 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽 ∈ (TopOn‘𝑋))
2 elpwi 4547 . . . . . . . . 9 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
3 resttopon 21697 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
41, 2, 3syl2an 595 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
5 simp2 1129 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ (TopOn‘𝑋))
6 resttopon 21697 . . . . . . . . . . 11 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐾t 𝑘) ∈ (TopOn‘𝑘))
75, 2, 6syl2an 595 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐾t 𝑘) ∈ (TopOn‘𝑘))
8 toponuni 21450 . . . . . . . . . 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 6667 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (TopOn‘𝑘) = (TopOn‘ (𝐾t 𝑘)))
114, 10eleqtrd 2912 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)))
12 simpl2 1184 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑋))
13 topontop 21449 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
1412, 13syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ Top)
15 simpl3 1185 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐽𝐾)
16 ssrest 21712 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝐽𝐾) → (𝐽t 𝑘) ⊆ (𝐾t 𝑘))
1714, 15, 16syl2anc 584 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ⊆ (𝐾t 𝑘))
18 eqid 2818 . . . . . . . . . 10 (𝐾t 𝑘) = (𝐾t 𝑘)
1918sscmp 21941 . . . . . . . . 9 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐾t 𝑘) ∈ Comp ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘)) → (𝐽t 𝑘) ∈ Comp)
20193com23 1118 . . . . . . . 8 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘) ∧ (𝐾t 𝑘) ∈ Comp) → (𝐽t 𝑘) ∈ Comp)
21203expia 1113 . . . . . . 7 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘)) → ((𝐾t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Comp))
2211, 17, 21syl2anc 584 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐾t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Comp))
2317sseld 3963 . . . . . 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 3174 . . . 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 611 . . 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 22072 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)))))
28273ad2ant1 1125 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)))))
29 elkgen 22072 . . . 4 (𝐾 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘)))))
30293ad2ant2 1126 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘)))))
3126, 28, 303imtr4d 295 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐾)))
3231ssrdv 3970 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830  cfv 6348  (class class class)co 7145  t crest 16682  Topctop 21429  TopOnctopon 21446  Compccmp 21922  𝑘Genckgen 22069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-oadd 8095  df-er 8278  df-en 8498  df-fin 8501  df-fi 8863  df-rest 16684  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cmp 21923  df-kgen 22070
This theorem is referenced by: (None)
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