Theorem kelac2lem 43053

Description: Lemma for kelac2 43054 and dfac21 43055: 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 5443	. . . . 5 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ V
2		vex 3482	. . . . . . . 8 ⊢ 𝑥 ∈ V
3	2	elpr 4655	. . . . . . 7 ⊢ (𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}} ↔ (𝑥 = 𝑆 ∨ 𝑥 = {𝒫 ∪ 𝑆}))
4		vex 3482	. . . . . . . 8 ⊢ 𝑦 ∈ V
5	4	elpr 4655	. . . . . . 7 ⊢ (𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}} ↔ (𝑦 = 𝑆 ∨ 𝑦 = {𝒫 ∪ 𝑆}))
6		eqtr3 2761	. . . . . . . . 9 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑆) → 𝑥 = 𝑦)
7	6	orcd 873	. . . . . . . 8 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
8		ineq12 4223	. . . . . . . . . 10 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 ∩ 𝑦) = ({𝒫 ∪ 𝑆} ∩ 𝑆))
9		incom 4217	. . . . . . . . . . 11 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 ∪ 𝑆})
10		pwuninel 8299	. . . . . . . . . . . 12 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
11		disjsn 4716	. . . . . . . . . . . 12 ⊢ ((𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅ ↔ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
12	10, 11	mpbir 231	. . . . . . . . . . 11 ⊢ (𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅
13	9, 12	eqtri 2763	. . . . . . . . . 10 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = ∅
14	8, 13	eqtrdi 2791	. . . . . . . . 9 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 ∩ 𝑦) = ∅)
15	14	olcd 874	. . . . . . . 8 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
16		ineq12 4223	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 ∩ 𝑦) = (𝑆 ∩ {𝒫 ∪ 𝑆}))
17	16, 12	eqtrdi 2791	. . . . . . . . 9 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 ∩ 𝑦) = ∅)
18	17	olcd 874	. . . . . . . 8 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
19		eqtr3 2761	. . . . . . . . 9 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = {𝒫 ∪ 𝑆}) → 𝑥 = 𝑦)
20	19	orcd 873	. . . . . . . 8 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
21	7, 15, 18, 20	ccase 1037	. . . . . . 7 ⊢ (((𝑥 = 𝑆 ∨ 𝑥 = {𝒫 ∪ 𝑆}) ∧ (𝑦 = 𝑆 ∨ 𝑦 = {𝒫 ∪ 𝑆})) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
22	3, 5, 21	syl2anb 598	. . . . . 6 ⊢ ((𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}} ∧ 𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
23	22	rgen2 3197	. . . . 5 ⊢ ∀𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}} (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)
24		baspartn 22977	. . . . 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 692	. . . 4 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ TopBases
26		tgcl 22992	. . . 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 9361	. . . . . 6 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin
29		pwfi 9355	. . . . . 6 ⊢ ({𝑆, {𝒫 ∪ 𝑆}} ∈ Fin ↔ 𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin)
30	28, 29	mpbi 230	. . . . 5 ⊢ 𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin
31		tgdom 23001	. . . . . 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 9227	. . . . 5 ⊢ ((𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin ∧ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 ∪ 𝑆}}) → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin)
34	30, 32, 33	mp2an 692	. . . 4 ⊢ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin
35	34	a1i 11	. . 3 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin)
36	27, 35	elind 4210	. 2 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ (Top ∩ Fin))
37		fincmp 23417	. 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 847   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ∩ cin 3962  ∅c0 4339  𝒫 cpw 4605  {csn 4631  {cpr 4633  ∪ cuni 4912   class class class wbr 5148  ‘cfv 6563   ≼ cdom 8982  Fincfn 8984  topGenctg 17484  Topctop 22915  TopBasesctb 22968  Compccmp 23410

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-2o 8506  df-en 8985  df-dom 8986  df-fin 8988  df-topgen 17490  df-top 22916  df-bases 22969  df-cmp 23411

This theorem is referenced by:  kelac2  43054  dfac21  43055