Theorem kelac2lem 39657

Description: Lemma for kelac2 39658 and dfac21 39659: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
kelac2lem	⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp)

Proof of Theorem kelac2lem

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prex 5324	. . . . 5 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ V
2		vex 3497	. . . . . . . 8 ⊢ 𝑥 ∈ V
3	2	elpr 4583	. . . . . . 7 ⊢ (𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}} ↔ (𝑥 = 𝑆 ∨ 𝑥 = {𝒫 ∪ 𝑆}))
4		vex 3497	. . . . . . . 8 ⊢ 𝑦 ∈ V
5	4	elpr 4583	. . . . . . 7 ⊢ (𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}} ↔ (𝑦 = 𝑆 ∨ 𝑦 = {𝒫 ∪ 𝑆}))
6		eqtr3 2843	. . . . . . . . 9 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑆) → 𝑥 = 𝑦)
7	6	orcd 869	. . . . . . . 8 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
8		ineq12 4183	. . . . . . . . . 10 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 ∩ 𝑦) = ({𝒫 ∪ 𝑆} ∩ 𝑆))
9		incom 4177	. . . . . . . . . . 11 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 ∪ 𝑆})
10		pwuninel 7935	. . . . . . . . . . . 12 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
11		disjsn 4640	. . . . . . . . . . . 12 ⊢ ((𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅ ↔ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
12	10, 11	mpbir 233	. . . . . . . . . . 11 ⊢ (𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅
13	9, 12	eqtri 2844	. . . . . . . . . 10 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = ∅
14	8, 13	syl6eq 2872	. . . . . . . . 9 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 ∩ 𝑦) = ∅)
15	14	olcd 870	. . . . . . . 8 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
16		ineq12 4183	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 ∩ 𝑦) = (𝑆 ∩ {𝒫 ∪ 𝑆}))
17	16, 12	syl6eq 2872	. . . . . . . . 9 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 ∩ 𝑦) = ∅)
18	17	olcd 870	. . . . . . . 8 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
19		eqtr3 2843	. . . . . . . . 9 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = {𝒫 ∪ 𝑆}) → 𝑥 = 𝑦)
20	19	orcd 869	. . . . . . . 8 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
21	7, 15, 18, 20	ccase 1032	. . . . . . 7 ⊢ (((𝑥 = 𝑆 ∨ 𝑥 = {𝒫 ∪ 𝑆}) ∧ (𝑦 = 𝑆 ∨ 𝑦 = {𝒫 ∪ 𝑆})) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
22	3, 5, 21	syl2anb 599	. . . . . 6 ⊢ ((𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}} ∧ 𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
23	22	rgen2 3203	. . . . 5 ⊢ ∀𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}} (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)
24		baspartn 21556	. . . . 5 ⊢ (({𝑆, {𝒫 ∪ 𝑆}} ∈ V ∧ ∀𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}} (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → {𝑆, {𝒫 ∪ 𝑆}} ∈ TopBases)
25	1, 23, 24	mp2an 690	. . . 4 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ TopBases
26		tgcl 21571	. . . 4 ⊢ ({𝑆, {𝒫 ∪ 𝑆}} ∈ TopBases → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Top)
27	25, 26	mp1i 13	. . 3 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Top)
28		prfi 8787	. . . . . 6 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin
29		pwfi 8813	. . . . . 6 ⊢ ({𝑆, {𝒫 ∪ 𝑆}} ∈ Fin ↔ 𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin)
30	28, 29	mpbi 232	. . . . 5 ⊢ 𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin
31		tgdom 21580	. . . . . 6 ⊢ ({𝑆, {𝒫 ∪ 𝑆}} ∈ V → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 ∪ 𝑆}})
32	1, 31	ax-mp 5	. . . . 5 ⊢ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 ∪ 𝑆}}
33		domfi 8733	. . . . 5 ⊢ ((𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin ∧ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 ∪ 𝑆}}) → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin)
34	30, 32, 33	mp2an 690	. . . 4 ⊢ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin
35	34	a1i 11	. . 3 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin)
36	27, 35	elind 4170	. 2 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ (Top ∩ Fin))
37		fincmp 21995	. 2 ⊢ ((topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ (Top ∩ Fin) → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp)
38	36, 37	syl 17	1 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494   ∩ cin 3934  ∅c0 4290  𝒫 cpw 4538  {csn 4560  {cpr 4562  ∪ cuni 4831   class class class wbr 5058  ‘cfv 6349   ≼ cdom 8501  Fincfn 8503  topGenctg 16705  Topctop 21495  TopBasesctb 21547  Compccmp 21988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-topgen 16711  df-top 21496  df-bases 21548  df-cmp 21989

This theorem is referenced by:  kelac2  39658  dfac21  39659