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Theorem kelac2lem 39938
Description: Lemma for kelac2 39939 and dfac21 39940: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
kelac2lem (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Comp)

Proof of Theorem kelac2lem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prex 5310 . . . . 5 {𝑆, {𝒫 𝑆}} ∈ V
2 vex 3472 . . . . . . . 8 𝑥 ∈ V
32elpr 4562 . . . . . . 7 (𝑥 ∈ {𝑆, {𝒫 𝑆}} ↔ (𝑥 = 𝑆𝑥 = {𝒫 𝑆}))
4 vex 3472 . . . . . . . 8 𝑦 ∈ V
54elpr 4562 . . . . . . 7 (𝑦 ∈ {𝑆, {𝒫 𝑆}} ↔ (𝑦 = 𝑆𝑦 = {𝒫 𝑆}))
6 eqtr3 2844 . . . . . . . . 9 ((𝑥 = 𝑆𝑦 = 𝑆) → 𝑥 = 𝑦)
76orcd 870 . . . . . . . 8 ((𝑥 = 𝑆𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
8 ineq12 4158 . . . . . . . . . 10 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = 𝑆) → (𝑥𝑦) = ({𝒫 𝑆} ∩ 𝑆))
9 incom 4152 . . . . . . . . . . 11 ({𝒫 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 𝑆})
10 pwuninel 7928 . . . . . . . . . . . 12 ¬ 𝒫 𝑆𝑆
11 disjsn 4621 . . . . . . . . . . . 12 ((𝑆 ∩ {𝒫 𝑆}) = ∅ ↔ ¬ 𝒫 𝑆𝑆)
1210, 11mpbir 234 . . . . . . . . . . 11 (𝑆 ∩ {𝒫 𝑆}) = ∅
139, 12eqtri 2845 . . . . . . . . . 10 ({𝒫 𝑆} ∩ 𝑆) = ∅
148, 13syl6eq 2873 . . . . . . . . 9 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = 𝑆) → (𝑥𝑦) = ∅)
1514olcd 871 . . . . . . . 8 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
16 ineq12 4158 . . . . . . . . . 10 ((𝑥 = 𝑆𝑦 = {𝒫 𝑆}) → (𝑥𝑦) = (𝑆 ∩ {𝒫 𝑆}))
1716, 12syl6eq 2873 . . . . . . . . 9 ((𝑥 = 𝑆𝑦 = {𝒫 𝑆}) → (𝑥𝑦) = ∅)
1817olcd 871 . . . . . . . 8 ((𝑥 = 𝑆𝑦 = {𝒫 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
19 eqtr3 2844 . . . . . . . . 9 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = {𝒫 𝑆}) → 𝑥 = 𝑦)
2019orcd 870 . . . . . . . 8 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = {𝒫 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
217, 15, 18, 20ccase 1033 . . . . . . 7 (((𝑥 = 𝑆𝑥 = {𝒫 𝑆}) ∧ (𝑦 = 𝑆𝑦 = {𝒫 𝑆})) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
223, 5, 21syl2anb 600 . . . . . 6 ((𝑥 ∈ {𝑆, {𝒫 𝑆}} ∧ 𝑦 ∈ {𝑆, {𝒫 𝑆}}) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
2322rgen2 3193 . . . . 5 𝑥 ∈ {𝑆, {𝒫 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 𝑆}} (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)
24 baspartn 21557 . . . . 5 (({𝑆, {𝒫 𝑆}} ∈ V ∧ ∀𝑥 ∈ {𝑆, {𝒫 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 𝑆}} (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → {𝑆, {𝒫 𝑆}} ∈ TopBases)
251, 23, 24mp2an 691 . . . 4 {𝑆, {𝒫 𝑆}} ∈ TopBases
26 tgcl 21572 . . . 4 ({𝑆, {𝒫 𝑆}} ∈ TopBases → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Top)
2725, 26mp1i 13 . . 3 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Top)
28 prfi 8781 . . . . . 6 {𝑆, {𝒫 𝑆}} ∈ Fin
29 pwfi 8807 . . . . . 6 ({𝑆, {𝒫 𝑆}} ∈ Fin ↔ 𝒫 {𝑆, {𝒫 𝑆}} ∈ Fin)
3028, 29mpbi 233 . . . . 5 𝒫 {𝑆, {𝒫 𝑆}} ∈ Fin
31 tgdom 21581 . . . . . 6 ({𝑆, {𝒫 𝑆}} ∈ V → (topGen‘{𝑆, {𝒫 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 𝑆}})
321, 31ax-mp 5 . . . . 5 (topGen‘{𝑆, {𝒫 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 𝑆}}
33 domfi 8727 . . . . 5 ((𝒫 {𝑆, {𝒫 𝑆}} ∈ Fin ∧ (topGen‘{𝑆, {𝒫 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 𝑆}}) → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Fin)
3430, 32, 33mp2an 691 . . . 4 (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Fin
3534a1i 11 . . 3 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Fin)
3627, 35elind 4145 . 2 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ (Top ∩ Fin))
37 fincmp 21996 . 2 ((topGen‘{𝑆, {𝒫 𝑆}}) ∈ (Top ∩ Fin) → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Comp)
3836, 37syl 17 1 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844   = wceq 1538  wcel 2114  wral 3130  Vcvv 3469  cin 3907  c0 4265  𝒫 cpw 4511  {csn 4539  {cpr 4541   cuni 4813   class class class wbr 5042  cfv 6334  cdom 8494  Fincfn 8496  topGenctg 16702  Topctop 21496  TopBasesctb 21548  Compccmp 21989
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-topgen 16708  df-top 21497  df-bases 21549  df-cmp 21990
This theorem is referenced by:  kelac2  39939  dfac21  39940
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