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Theorem kgen2ss 21168
 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 1054 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽 ∈ (TopOn‘𝑋))
2 elpwi 4117 . . . . . . . . 9 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
3 resttopon 20775 . . . . . . . . 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 1055 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ (TopOn‘𝑋))
6 resttopon 20775 . . . . . . . . . . 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 20542 . . . . . . . . . 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 6107 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (TopOn‘𝑘) = (TopOn‘ (𝐾t 𝑘)))
114, 10eleqtrd 2690 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)))
12 simpl2 1058 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑋))
13 topontop 20541 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
1412, 13syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ Top)
15 simpl3 1059 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐽𝐾)
16 ssrest 20790 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝐽𝐾) → (𝐽t 𝑘) ⊆ (𝐾t 𝑘))
1714, 15, 16syl2anc 691 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽t 𝑘) ⊆ (𝐾t 𝑘))
18 eqid 2610 . . . . . . . . . 10 (𝐾t 𝑘) = (𝐾t 𝑘)
1918sscmp 21018 . . . . . . . . 9 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐾t 𝑘) ∈ Comp ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘)) → (𝐽t 𝑘) ∈ Comp)
20193com23 1263 . . . . . . . 8 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘) ∧ (𝐾t 𝑘) ∈ Comp) → (𝐽t 𝑘) ∈ Comp)
21203expia 1259 . . . . . . 7 (((𝐽t 𝑘) ∈ (TopOn‘ (𝐾t 𝑘)) ∧ (𝐽t 𝑘) ⊆ (𝐾t 𝑘)) → ((𝐾t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Comp))
2211, 17, 21syl2anc 691 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐾t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Comp))
2317sseld 3567 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝑥𝑘) ∈ (𝐽t 𝑘) → (𝑥𝑘) ∈ (𝐾t 𝑘)))
2422, 23imim12d 79 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)) → ((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘))))
2524ralimdva 2945 . . . 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 587 . . 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 21149 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)))))
28273ad2ant1 1075 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)))))
29 elkgen 21149 . . . 4 (𝐾 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘)))))
30293ad2ant2 1076 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐾t 𝑘)))))
3126, 28, 303imtr4d 282 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐾)))
3231ssrdv 3574 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  TopOnctopon 20518  Compccmp 20999  𝑘Genckgen 21146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-kgen 21147 This theorem is referenced by: (None)
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