Theorem cfinfil 24033

Description: Relative complements of the finite parts of an infinite set is a filter. When 𝐴 = ℕ the set of the relative complements is called Frechet's filter and is used to define the concept of limit of a sequence. (Contributed by FL, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
cfinfil	⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ∈ (Fil‘𝑋))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem cfinfil

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difeq2 4083	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
2	1	eleq1d 2855	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin))
3	2	elrab 3658	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin))
4		velpw 4572	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
5	4	anbi1i 635	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin) ↔ (𝑦 ⊆ 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin))
6	3, 5	bitri 278	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ↔ (𝑦 ⊆ 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin))
7	6	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ↔ (𝑦 ⊆ 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin)))
8		simp1 1152	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → 𝑋 ∈ 𝑉)
9		ssdif0 4329	. . . . 5 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∖ 𝑋) = ∅)
10		0fi 9042	. . . . . 6 ⊢ ∅ ∈ Fin
11		eleq1 2858	. . . . . 6 ⊢ ((𝐴 ∖ 𝑋) = ∅ → ((𝐴 ∖ 𝑋) ∈ Fin ↔ ∅ ∈ Fin))
12	10, 11	mpbiri 261	. . . . 5 ⊢ ((𝐴 ∖ 𝑋) = ∅ → (𝐴 ∖ 𝑋) ∈ Fin)
13	9, 12	sylbi 220	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → (𝐴 ∖ 𝑋) ∈ Fin)
14		difeq2 4083	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑋))
15	14	eleq1d 2855	. . . . . 6 ⊢ (𝑦 = 𝑋 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑋) ∈ Fin))
16	15	sbcieg 3791	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑋) ∈ Fin))
17	16	biimpar 482	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ∖ 𝑋) ∈ Fin) → [𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
18	13, 17	sylan2 604	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → [𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
19	18	3adant3 1148	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → [𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
20		0ex 5275	. . . . . 6 ⊢ ∅ ∈ V
21		difeq2 4083	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝐴 ∖ 𝑦) = (𝐴 ∖ ∅))
22	21	eleq1d 2855	. . . . . 6 ⊢ (𝑦 = ∅ → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin))
23	20, 22	sbcie 3793	. . . . 5 ⊢ ([∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin)
24		dif0 4341	. . . . . 6 ⊢ (𝐴 ∖ ∅) = 𝐴
25	24	eleq1i 2861	. . . . 5 ⊢ ((𝐴 ∖ ∅) ∈ Fin ↔ 𝐴 ∈ Fin)
26	23, 25	sylbb 222	. . . 4 ⊢ ([∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin → 𝐴 ∈ Fin)
27	26	con3i 155	. . 3 ⊢ (¬ 𝐴 ∈ Fin → ¬ [∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
28	27	3ad2ant3 1151	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → ¬ [∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
29		sscon 4105	. . . . 5 ⊢ (𝑤 ⊆ 𝑧 → (𝐴 ∖ 𝑧) ⊆ (𝐴 ∖ 𝑤))
30		ssfi 9160	. . . . . 6 ⊢ (((𝐴 ∖ 𝑤) ∈ Fin ∧ (𝐴 ∖ 𝑧) ⊆ (𝐴 ∖ 𝑤)) → (𝐴 ∖ 𝑧) ∈ Fin)
31	30	expcom 418	. . . . 5 ⊢ ((𝐴 ∖ 𝑧) ⊆ (𝐴 ∖ 𝑤) → ((𝐴 ∖ 𝑤) ∈ Fin → (𝐴 ∖ 𝑧) ∈ Fin))
32	29, 31	syl 18	. . . 4 ⊢ (𝑤 ⊆ 𝑧 → ((𝐴 ∖ 𝑤) ∈ Fin → (𝐴 ∖ 𝑧) ∈ Fin))
33		vex 3466	. . . . 5 ⊢ 𝑤 ∈ V
34		difeq2 4083	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑤))
35	34	eleq1d 2855	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑤) ∈ Fin))
36	33, 35	sbcie 3793	. . . 4 ⊢ ([𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑤) ∈ Fin)
37		vex 3466	. . . . 5 ⊢ 𝑧 ∈ V
38		difeq2 4083	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑧))
39	38	eleq1d 2855	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑧) ∈ Fin))
40	37, 39	sbcie 3793	. . . 4 ⊢ ([𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑧) ∈ Fin)
41	32, 36, 40	3imtr4g 299	. . 3 ⊢ (𝑤 ⊆ 𝑧 → ([𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin))
42	41	3ad2ant3 1151	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → ([𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin))
43		difindi 4253	. . . . 5 ⊢ (𝐴 ∖ (𝑧 ∩ 𝑤)) = ((𝐴 ∖ 𝑧) ∪ (𝐴 ∖ 𝑤))
44		unfi 9158	. . . . 5 ⊢ (((𝐴 ∖ 𝑧) ∈ Fin ∧ (𝐴 ∖ 𝑤) ∈ Fin) → ((𝐴 ∖ 𝑧) ∪ (𝐴 ∖ 𝑤)) ∈ Fin)
45	43, 44	eqeltrid 2874	. . . 4 ⊢ (((𝐴 ∖ 𝑧) ∈ Fin ∧ (𝐴 ∖ 𝑤) ∈ Fin) → (𝐴 ∖ (𝑧 ∩ 𝑤)) ∈ Fin)
46	45	a1i 11	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑋) → (((𝐴 ∖ 𝑧) ∈ Fin ∧ (𝐴 ∖ 𝑤) ∈ Fin) → (𝐴 ∖ (𝑧 ∩ 𝑤)) ∈ Fin))
47	40, 36	anbi12i 639	. . 3 ⊢ (([𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin) ↔ ((𝐴 ∖ 𝑧) ∈ Fin ∧ (𝐴 ∖ 𝑤) ∈ Fin))
48	37	inex1 5291	. . . 4 ⊢ (𝑧 ∩ 𝑤) ∈ V
49		difeq2 4083	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝐴 ∖ 𝑦) = (𝐴 ∖ (𝑧 ∩ 𝑤)))
50	49	eleq1d 2855	. . . 4 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧 ∩ 𝑤)) ∈ Fin))
51	48, 50	sbcie 3793	. . 3 ⊢ ([(𝑧 ∩ 𝑤) / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧 ∩ 𝑤)) ∈ Fin)
52	46, 47, 51	3imtr4g 299	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑋) → (([𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin) → [(𝑧 ∩ 𝑤) / 𝑦](𝐴 ∖ 𝑦) ∈ Fin))
53	7, 8, 19, 28, 42, 52	isfild 23998	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ∈ (Fil‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  {crab 3423  [wsbc 3752   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  ‘cfv 6540  Fincfn 8946  Filcfil 23985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-nel 3072  df-ral 3087  df-rex 3097  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5560  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7866  df-1o 8456  df-en 8947  df-fin 8950  df-fbas 21502  df-fil 23986

This theorem is referenced by:  ufinffr  24069