Theorem alexsub 23942

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 23948 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 714	. . . . . . . 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 23941	. . . . . . 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 2943	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅)
16	15	ralrimiva 3142	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
17		fibas 22873	. . . . . 6 ⊢ (fi‘𝐵) ∈ TopBases
18		tgtopon 22867	. . . . . 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 2837	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ (fi‘𝐵)))
21	1	elexd 3491	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V)
22	3, 21	eqeltrrd 2830	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ∈ V)
23		uniexb 7760	. . . . . . . 8 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
24	22, 23	sylibr 233	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
25		fiuni 9445	. . . . . . 7 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵))
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝐵 = ∪ (fi‘𝐵))
27	3, 26	eqtrd 2768	. . . . 5 ⊢ (𝜑 → 𝑋 = ∪ (fi‘𝐵))
28	27	fveq2d 6895	. . . 4 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ (fi‘𝐵)))
29	20, 28	eleqtrrd 2832	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
30		ufilcmp 23929	. . 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 257	1 ⊢ (𝜑 → 𝐽 ∈ Comp)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ≠ wne 2936  ∀wral 3057  ∃wrex 3066  Vcvv 3470   ∩ cin 3944   ⊆ wss 3945  ∅c0 4318  𝒫 cpw 4598  ∪ cuni 4903  ‘cfv 6542  (class class class)co 7414  Fincfn 8957  ficfi 9427  topGenctg 17412  TopOnctopon 22805  TopBasesctb 22841  Compccmp 23283  UFilcufil 23796  UFLcufl 23797   fLim cflim 23831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-1o 8480  df-er 8718  df-en 8958  df-dom 8959  df-fin 8961  df-fi 9428  df-topgen 17418  df-fbas 21269  df-fg 21270  df-top 22789  df-topon 22806  df-bases 22842  df-cld 22916  df-ntr 22917  df-cls 22918  df-nei 22995  df-cmp 23284  df-fil 23743  df-ufil 23798  df-ufl 23799  df-flim 23836  df-fcls 23838

This theorem is referenced by:  alexsubb  23943  ptcmplem5  23953