Theorem alexsub 23932

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 23938 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 715	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
9		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑓 ∈ (UFil‘𝑋))
10		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → (𝐽 fLim 𝑓) = ∅)
11	2, 4, 6, 8, 9, 10	alexsublem 23931	. . . . . . 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 190	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ¬ (𝐽 fLim 𝑓) = ∅)
15	14	neqned 2932	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅)
16	15	ralrimiva 3125	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
17		fibas 22864	. . . . . 6 ⊢ (fi‘𝐵) ∈ TopBases
18		tgtopon 22858	. . . . . 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 2836	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ (fi‘𝐵)))
21	1	elexd 3471	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V)
22	3, 21	eqeltrrd 2829	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ∈ V)
23		uniexb 7740	. . . . . . . 8 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
24	22, 23	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
25		fiuni 9379	. . . . . . 7 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵))
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝐵 = ∪ (fi‘𝐵))
27	3, 26	eqtrd 2764	. . . . 5 ⊢ (𝜑 → 𝑋 = ∪ (fi‘𝐵))
28	27	fveq2d 6862	. . . 4 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ (fi‘𝐵)))
29	20, 28	eleqtrrd 2831	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
30		ufilcmp 23919	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
31	1, 29, 30	syl2anc 584	. 2 ⊢ (𝜑 → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
32	16, 31	mpbird 257	1 ⊢ (𝜑 → 𝐽 ∈ Comp)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  ∪ cuni 4871  ‘cfv 6511  (class class class)co 7387  Fincfn 8918  ficfi 9361  topGenctg 17400  TopOnctopon 22797  TopBasesctb 22832  Compccmp 23273  UFilcufil 23786  UFLcufl 23787   fLim cflim 23821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-1o 8434  df-2o 8435  df-en 8919  df-dom 8920  df-fin 8922  df-fi 9362  df-topgen 17406  df-fbas 21261  df-fg 21262  df-top 22781  df-topon 22798  df-bases 22833  df-cld 22906  df-ntr 22907  df-cls 22908  df-nei 22985  df-cmp 23274  df-fil 23733  df-ufil 23788  df-ufl 23789  df-flim 23826  df-fcls 23828

This theorem is referenced by:  alexsubb  23933  ptcmplem5  23943