Theorem alexsub 23989

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 23995 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 23988	. . . . . . 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 2939	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅)
16	15	ralrimiva 3128	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
17		fibas 22921	. . . . . 6 ⊢ (fi‘𝐵) ∈ TopBases
18		tgtopon 22915	. . . . . 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 2844	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ (fi‘𝐵)))
21	1	elexd 3464	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V)
22	3, 21	eqeltrrd 2837	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ∈ V)
23		uniexb 7709	. . . . . . . 8 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
24	22, 23	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
25		fiuni 9331	. . . . . . 7 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵))
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝐵 = ∪ (fi‘𝐵))
27	3, 26	eqtrd 2771	. . . . 5 ⊢ (𝜑 → 𝑋 = ∪ (fi‘𝐵))
28	27	fveq2d 6838	. . . 4 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ (fi‘𝐵)))
29	20, 28	eleqtrrd 2839	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
30		ufilcmp 23976	. . 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 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  ∪ cuni 4863  ‘cfv 6492  (class class class)co 7358  Fincfn 8883  ficfi 9313  topGenctg 17357  TopOnctopon 22854  TopBasesctb 22889  Compccmp 23330  UFilcufil 23843  UFLcufl 23844   fLim cflim 23878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-1o 8397  df-2o 8398  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9314  df-topgen 17363  df-fbas 21306  df-fg 21307  df-top 22838  df-topon 22855  df-bases 22890  df-cld 22963  df-ntr 22964  df-cls 22965  df-nei 23042  df-cmp 23331  df-fil 23790  df-ufil 23845  df-ufl 23846  df-flim 23883  df-fcls 23885

This theorem is referenced by:  alexsubb  23990  ptcmplem5  24000