Theorem alexsub 23104

Description: The Alexander Subbase Theorem: If 𝐵 is a subbase for the topology 𝐽, and any cover taken from 𝐵 has a finite subcover, then the generated topology is compact. This proof uses the ultrafilter lemma; see alexsubALT 23110 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.)

Hypotheses

Ref	Expression
alexsub.1	⊢ (𝜑 → 𝑋 ∈ UFL)
alexsub.2	⊢ (𝜑 → 𝑋 = ∪ 𝐵)
alexsub.3	⊢ (𝜑 → 𝐽 = (topGen‘(fi‘𝐵)))
alexsub.4	⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)

Assertion

Ref	Expression
alexsub	⊢ (𝜑 → 𝐽 ∈ Comp)

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐽,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦

Proof of Theorem alexsub

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alexsub.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ UFL)
2	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 ∈ UFL)
3		alexsub.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ∪ 𝐵)
4	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 = ∪ 𝐵)
5		alexsub.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 = (topGen‘(fi‘𝐵)))
6	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝐽 = (topGen‘(fi‘𝐵)))
7		alexsub.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
8	7	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
9		simprl 767	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑓 ∈ (UFil‘𝑋))
10		simprr 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → (𝐽 fLim 𝑓) = ∅)
11	2, 4, 6, 8, 9, 10	alexsublem 23103	. . . . . . 7 ⊢ ¬ (𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅))
12	11	pm2.21i 119	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → ¬ (𝐽 fLim 𝑓) = ∅)
13	12	expr 456	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ((𝐽 fLim 𝑓) = ∅ → ¬ (𝐽 fLim 𝑓) = ∅))
14	13	pm2.01d 189	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ¬ (𝐽 fLim 𝑓) = ∅)
15	14	neqned 2949	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅)
16	15	ralrimiva 3107	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
17		fibas 22035	. . . . . 6 ⊢ (fi‘𝐵) ∈ TopBases
18		tgtopon 22029	. . . . . 6 ⊢ ((fi‘𝐵) ∈ TopBases → (topGen‘(fi‘𝐵)) ∈ (TopOn‘∪ (fi‘𝐵)))
19	17, 18	ax-mp 5	. . . . 5 ⊢ (topGen‘(fi‘𝐵)) ∈ (TopOn‘∪ (fi‘𝐵))
20	5, 19	eqeltrdi 2847	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ (fi‘𝐵)))
21	1	elexd 3442	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V)
22	3, 21	eqeltrrd 2840	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ∈ V)
23		uniexb 7592	. . . . . . . 8 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
24	22, 23	sylibr 233	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
25		fiuni 9117	. . . . . . 7 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵))
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝐵 = ∪ (fi‘𝐵))
27	3, 26	eqtrd 2778	. . . . 5 ⊢ (𝜑 → 𝑋 = ∪ (fi‘𝐵))
28	27	fveq2d 6760	. . . 4 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ (fi‘𝐵)))
29	20, 28	eleqtrrd 2842	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
30		ufilcmp 23091	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
31	1, 29, 30	syl2anc 583	. 2 ⊢ (𝜑 → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
32	16, 31	mpbird 256	1 ⊢ (𝜑 → 𝐽 ∈ Comp)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  ficfi 9099  topGenctg 17065  TopOnctopon 21967  TopBasesctb 22003  Compccmp 22445  UFilcufil 22958  UFLcufl 22959   fLim cflim 22993

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-nel 3049  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-int 4877  df-iun 4923  df-iin 4924  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-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-fin 8695  df-fi 9100  df-topgen 17071  df-fbas 20507  df-fg 20508  df-top 21951  df-topon 21968  df-bases 22004  df-cld 22078  df-ntr 22079  df-cls 22080  df-nei 22157  df-cmp 22446  df-fil 22905  df-ufil 22960  df-ufl 22961  df-flim 22998  df-fcls 23000

This theorem is referenced by:  alexsubb  23105  ptcmplem5  23115