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Theorem kgen2ss 21580
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 1131 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽 ∈ (TopOn‘𝑋))
2 elpwi 4312 . . . . . . . . 9 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
3 resttopon 21187 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
41, 2, 3syl2an 495 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
5 simp2 1132 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ (TopOn‘𝑋))
6 resttopon 21187 . . . . . . . . . . 11 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐾t 𝑘) ∈ (TopOn‘𝑘))
75, 2, 6syl2an 495 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐾t 𝑘) ∈ (TopOn‘𝑘))
8 toponuni 20941 . . . . . . . . . 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 6357 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (TopOn‘𝑘) = (TopOn‘ (𝐾t 𝑘)))
114, 10eleqtrd 2841 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)))
12 simpl2 1230 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑋))
13 topontop 20940 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
1412, 13syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ Top)
15 simpl3 1232 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐽𝐾)
16 ssrest 21202 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝐽𝐾) → (𝐽t 𝑘) ⊆ (𝐾t 𝑘))
1714, 15, 16syl2anc 696 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ⊆ (𝐾t 𝑘))
18 eqid 2760 . . . . . . . . . 10 (𝐾t 𝑘) = (𝐾t 𝑘)
1918sscmp 21430 . . . . . . . . 9 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐾t 𝑘) ∈ Comp ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘)) → (𝐽t 𝑘) ∈ Comp)
20193com23 1121 . . . . . . . 8 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘) ∧ (𝐾t 𝑘) ∈ Comp) → (𝐽t 𝑘) ∈ Comp)
21203expia 1115 . . . . . . 7 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘)) → ((𝐾t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Comp))
2211, 17, 21syl2anc 696 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐾t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Comp))
2317sseld 3743 . . . . . 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 3100 . . . 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 590 . . 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 21561 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)))))
28273ad2ant1 1128 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)))))
29 elkgen 21561 . . . 4 (𝐾 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘)))))
30293ad2ant2 1129 . . 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 3750 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 1072   = wceq 1632  wcel 2139  wral 3050  cin 3714  wss 3715  𝒫 cpw 4302   cuni 4588  cfv 6049  (class class class)co 6814  t crest 16303  Topctop 20920  TopOnctopon 20937  Compccmp 21411  𝑘Genckgen 21558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-oadd 7734  df-er 7913  df-en 8124  df-fin 8127  df-fi 8484  df-rest 16305  df-topgen 16326  df-top 20921  df-topon 20938  df-bases 20972  df-cmp 21412  df-kgen 21559
This theorem is referenced by: (None)
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