Theorem cmpfiiin 42713

Description: In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Hypotheses

Ref	Expression
cmpfiiin.x	⊢ 𝑋 = ∪ 𝐽
cmpfiiin.j	⊢ (𝜑 → 𝐽 ∈ Comp)
cmpfiiin.s	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑆 ∈ (Clsd‘𝐽))
cmpfiiin.z	⊢ ((𝜑 ∧ (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin)) → (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆) ≠ ∅)

Assertion

Ref	Expression
cmpfiiin	⊢ (𝜑 → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) ≠ ∅)

Distinct variable groups:   𝜑,𝑘,𝑙   𝑘,𝐼,𝑙   𝑘,𝐽,𝑙   𝑆,𝑙   𝑘,𝑋,𝑙

Allowed substitution hint:   𝑆(𝑘)

Proof of Theorem cmpfiiin

Step	Hyp	Ref	Expression
1		cmpfiiin.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Comp)
2		cmptop 23404	. . . . 5 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
4		cmpfiiin.x	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
5	4	topcld 23044	. . . 4 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Clsd‘𝐽))
7		cmpfiiin.s	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑆 ∈ (Clsd‘𝐽))
8	4	cldss 23038	. . . . 5 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋)
9	7, 8	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑆 ⊆ 𝑋)
10	9	ralrimiva 3145	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋)
11		riinint 5981	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
12	6, 10, 11	syl2anc 584	. 2 ⊢ (𝜑 → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))
13	6	snssd 4808	. . . 4 ⊢ (𝜑 → {𝑋} ⊆ (Clsd‘𝐽))
14	7	fmpttd 7134	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐼 ↦ 𝑆):𝐼⟶(Clsd‘𝐽))
15	14	frnd 6743	. . . 4 ⊢ (𝜑 → ran (𝑘 ∈ 𝐼 ↦ 𝑆) ⊆ (Clsd‘𝐽))
16	13, 15	unssd 4191	. . 3 ⊢ (𝜑 → ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) ⊆ (Clsd‘𝐽))
17		elin 3966	. . . . . . 7 ⊢ (𝑙 ∈ (𝒫 𝐼 ∩ Fin) ↔ (𝑙 ∈ 𝒫 𝐼 ∧ 𝑙 ∈ Fin))
18		elpwi 4606	. . . . . . . 8 ⊢ (𝑙 ∈ 𝒫 𝐼 → 𝑙 ⊆ 𝐼)
19	18	anim1i 615	. . . . . . 7 ⊢ ((𝑙 ∈ 𝒫 𝐼 ∧ 𝑙 ∈ Fin) → (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin))
20	17, 19	sylbi 217	. . . . . 6 ⊢ (𝑙 ∈ (𝒫 𝐼 ∩ Fin) → (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin))
21		cmpfiiin.z	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin)) → (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆) ≠ ∅)
22		nesym 2996	. . . . . . 7 ⊢ ((𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆) ≠ ∅ ↔ ¬ ∅ = (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆))
23	21, 22	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin)) → ¬ ∅ = (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆))
24	20, 23	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝒫 𝐼 ∩ Fin)) → ¬ ∅ = (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆))
25	24	nrexdv 3148	. . . 4 ⊢ (𝜑 → ¬ ∃𝑙 ∈ (𝒫 𝐼 ∩ Fin)∅ = (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆))
26		elrfirn2 42712	. . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (∅ ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) ↔ ∃𝑙 ∈ (𝒫 𝐼 ∩ Fin)∅ = (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆)))
27	6, 10, 26	syl2anc 584	. . . 4 ⊢ (𝜑 → (∅ ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) ↔ ∃𝑙 ∈ (𝒫 𝐼 ∩ Fin)∅ = (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆)))
28	25, 27	mtbird 325	. . 3 ⊢ (𝜑 → ¬ ∅ ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆))))
29		cmpfii 23418	. . 3 ⊢ ((𝐽 ∈ Comp ∧ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)))) → ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) ≠ ∅)
30	1, 16, 28, 29	syl3anc 1372	. 2 ⊢ (𝜑 → ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆)) ≠ ∅)
31	12, 30	eqnetrd 3007	1 ⊢ (𝜑 → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069   ∪ cun 3948   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  𝒫 cpw 4599  {csn 4625  ∪ cuni 4906  ∩ cint 4945  ∩ ciin 4991   ↦ cmpt 5224  ran crn 5685  ‘cfv 6560  Fincfn 8986  ficfi 9451  Topctop 22900  Clsdccld 23025  Compccmp 23395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-om 7889  df-1o 8507  df-en 8987  df-dom 8988  df-fin 8990  df-fi 9452  df-top 22901  df-cld 23028  df-cmp 23396

This theorem is referenced by:  kelac1  43080