Theorem alexsub 24020

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 24026 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 716	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
9		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑓 ∈ (UFil‘𝑋))
10		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → (𝐽 fLim 𝑓) = ∅)
11	2, 4, 6, 8, 9, 10	alexsublem 24019	. . . . . . 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 2940	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅)
16	15	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
17		fibas 22952	. . . . . 6 ⊢ (fi‘𝐵) ∈ TopBases
18		tgtopon 22946	. . . . . 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 2845	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ (fi‘𝐵)))
21	1	elexd 3454	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V)
22	3, 21	eqeltrrd 2838	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ∈ V)
23		uniexb 7711	. . . . . . . 8 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
24	22, 23	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
25		fiuni 9334	. . . . . . 7 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵))
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝐵 = ∪ (fi‘𝐵))
27	3, 26	eqtrd 2772	. . . . 5 ⊢ (𝜑 → 𝑋 = ∪ (fi‘𝐵))
28	27	fveq2d 6838	. . . 4 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ (fi‘𝐵)))
29	20, 28	eleqtrrd 2840	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
30		ufilcmp 24007	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
31	1, 29, 30	syl2anc 585	. 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 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851  ‘cfv 6492  (class class class)co 7360  Fincfn 8886  ficfi 9316  topGenctg 17391  TopOnctopon 22885  TopBasesctb 22920  Compccmp 23361  UFilcufil 23874  UFLcufl 23875   fLim cflim 23909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  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 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-1o 8398  df-2o 8399  df-en 8887  df-dom 8888  df-fin 8890  df-fi 9317  df-topgen 17397  df-fbas 21341  df-fg 21342  df-top 22869  df-topon 22886  df-bases 22921  df-cld 22994  df-ntr 22995  df-cls 22996  df-nei 23073  df-cmp 23362  df-fil 23821  df-ufil 23876  df-ufl 23877  df-flim 23914  df-fcls 23916

This theorem is referenced by:  alexsubb  24021  ptcmplem5  24031