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Theorem kgen2ss 21406
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 1081 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽 ∈ (TopOn‘𝑋))
2 elpwi 4201 . . . . . . . . 9 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
3 resttopon 21013 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
41, 2, 3syl2an 493 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
5 simp2 1082 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ (TopOn‘𝑋))
6 resttopon 21013 . . . . . . . . . . 11 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐾t 𝑘) ∈ (TopOn‘𝑘))
75, 2, 6syl2an 493 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐾t 𝑘) ∈ (TopOn‘𝑘))
8 toponuni 20767 . . . . . . . . . 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 6233 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (TopOn‘𝑘) = (TopOn‘ (𝐾t 𝑘)))
114, 10eleqtrd 2732 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)))
12 simpl2 1085 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑋))
13 topontop 20766 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
1412, 13syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ Top)
15 simpl3 1086 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐽𝐾)
16 ssrest 21028 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝐽𝐾) → (𝐽t 𝑘) ⊆ (𝐾t 𝑘))
1714, 15, 16syl2anc 694 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ⊆ (𝐾t 𝑘))
18 eqid 2651 . . . . . . . . . 10 (𝐾t 𝑘) = (𝐾t 𝑘)
1918sscmp 21256 . . . . . . . . 9 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐾t 𝑘) ∈ Comp ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘)) → (𝐽t 𝑘) ∈ Comp)
20193com23 1291 . . . . . . . 8 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘) ∧ (𝐾t 𝑘) ∈ Comp) → (𝐽t 𝑘) ∈ Comp)
21203expia 1286 . . . . . . 7 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘)) → ((𝐾t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Comp))
2211, 17, 21syl2anc 694 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐾t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Comp))
2317sseld 3635 . . . . . 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 2991 . . . 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 588 . . 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 21387 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)))))
28273ad2ant1 1102 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)))))
29 elkgen 21387 . . . 4 (𝐾 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘)))))
30293ad2ant2 1103 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘)))))
3126, 28, 303imtr4d 283 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐾)))
3231ssrdv 3642 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468  cfv 5926  (class class class)co 6690  t crest 16128  Topctop 20746  TopOnctopon 20763  Compccmp 21237  𝑘Genckgen 21384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-kgen 21385
This theorem is referenced by: (None)
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