Theorem kelac2lem 40805

Description: Lemma for kelac2 40806 and dfac21 40807: 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 5350	. . . . 5 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ V
2		vex 3426	. . . . . . . 8 ⊢ 𝑥 ∈ V
3	2	elpr 4581	. . . . . . 7 ⊢ (𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}} ↔ (𝑥 = 𝑆 ∨ 𝑥 = {𝒫 ∪ 𝑆}))
4		vex 3426	. . . . . . . 8 ⊢ 𝑦 ∈ V
5	4	elpr 4581	. . . . . . 7 ⊢ (𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}} ↔ (𝑦 = 𝑆 ∨ 𝑦 = {𝒫 ∪ 𝑆}))
6		eqtr3 2764	. . . . . . . . 9 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑆) → 𝑥 = 𝑦)
7	6	orcd 869	. . . . . . . 8 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
8		ineq12 4138	. . . . . . . . . 10 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 ∩ 𝑦) = ({𝒫 ∪ 𝑆} ∩ 𝑆))
9		incom 4131	. . . . . . . . . . 11 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 ∪ 𝑆})
10		pwuninel 8062	. . . . . . . . . . . 12 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
11		disjsn 4644	. . . . . . . . . . . 12 ⊢ ((𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅ ↔ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
12	10, 11	mpbir 230	. . . . . . . . . . 11 ⊢ (𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅
13	9, 12	eqtri 2766	. . . . . . . . . 10 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = ∅
14	8, 13	eqtrdi 2795	. . . . . . . . 9 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 ∩ 𝑦) = ∅)
15	14	olcd 870	. . . . . . . 8 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
16		ineq12 4138	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 ∩ 𝑦) = (𝑆 ∩ {𝒫 ∪ 𝑆}))
17	16, 12	eqtrdi 2795	. . . . . . . . 9 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 ∩ 𝑦) = ∅)
18	17	olcd 870	. . . . . . . 8 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
19		eqtr3 2764	. . . . . . . . 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 1034	. . . . . . 7 ⊢ (((𝑥 = 𝑆 ∨ 𝑥 = {𝒫 ∪ 𝑆}) ∧ (𝑦 = 𝑆 ∨ 𝑦 = {𝒫 ∪ 𝑆})) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
22	3, 5, 21	syl2anb 597	. . . . . 6 ⊢ ((𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}} ∧ 𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
23	22	rgen2 3126	. . . . 5 ⊢ ∀𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}} (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)
24		baspartn 22012	. . . . 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 688	. . . 4 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ TopBases
26		tgcl 22027	. . . 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 9019	. . . . . 6 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin
29		pwfi 8923	. . . . . 6 ⊢ ({𝑆, {𝒫 ∪ 𝑆}} ∈ Fin ↔ 𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin)
30	28, 29	mpbi 229	. . . . 5 ⊢ 𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin
31		tgdom 22036	. . . . . 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 8935	. . . . 5 ⊢ ((𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin ∧ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 ∪ 𝑆}}) → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin)
34	30, 32, 33	mp2an 688	. . . 4 ⊢ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin
35	34	a1i 11	. . 3 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin)
36	27, 35	elind 4124	. 2 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ (Top ∩ Fin))
37		fincmp 22452	. 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 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∩ cin 3882  ∅c0 4253  𝒫 cpw 4530  {csn 4558  {cpr 4560  ∪ cuni 4836   class class class wbr 5070  ‘cfv 6418   ≼ cdom 8689  Fincfn 8691  topGenctg 17065  Topctop 21950  TopBasesctb 22003  Compccmp 22445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-en 8692  df-dom 8693  df-fin 8695  df-topgen 17071  df-top 21951  df-bases 22004  df-cmp 22446

This theorem is referenced by:  kelac2  40806  dfac21  40807