Theorem cfinfil 24101

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 4075	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
2	1	eleq1d 2850	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin))
3	2	elrab 3652	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin))
4		velpw 4569	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
5	4	anbi1i 636	. . . 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 1154	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → 𝑋 ∈ 𝑉)
9		ssdif0 4321	. . . . 5 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∖ 𝑋) = ∅)
10		0fi 9046	. . . . . 6 ⊢ ∅ ∈ Fin
11		eleq1 2853	. . . . . 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 4075	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑋))
15	14	eleq1d 2850	. . . . . 6 ⊢ (𝑦 = 𝑋 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑋) ∈ Fin))
16	15	sbcieg 3785	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑋) ∈ Fin))
17	16	biimpar 483	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ∖ 𝑋) ∈ Fin) → [𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
18	13, 17	sylan2 605	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → [𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
19	18	3adant3 1150	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → [𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
20		0ex 5272	. . . . . 6 ⊢ ∅ ∈ V
21		difeq2 4075	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝐴 ∖ 𝑦) = (𝐴 ∖ ∅))
22	21	eleq1d 2850	. . . . . 6 ⊢ (𝑦 = ∅ → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin))
23	20, 22	sbcie 3787	. . . . 5 ⊢ ([∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin)
24		dif0 4334	. . . . . 6 ⊢ (𝐴 ∖ ∅) = 𝐴
25	24	eleq1i 2856	. . . . 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 1153	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → ¬ [∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
29		sscon 4097	. . . . 5 ⊢ (𝑤 ⊆ 𝑧 → (𝐴 ∖ 𝑧) ⊆ (𝐴 ∖ 𝑤))
30		ssfi 9164	. . . . . 6 ⊢ (((𝐴 ∖ 𝑤) ∈ Fin ∧ (𝐴 ∖ 𝑧) ⊆ (𝐴 ∖ 𝑤)) → (𝐴 ∖ 𝑧) ∈ Fin)
31	30	expcom 419	. . . . 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 3461	. . . . 5 ⊢ 𝑤 ∈ V
34		difeq2 4075	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑤))
35	34	eleq1d 2850	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑤) ∈ Fin))
36	33, 35	sbcie 3787	. . . 4 ⊢ ([𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑤) ∈ Fin)
37		vex 3461	. . . . 5 ⊢ 𝑧 ∈ V
38		difeq2 4075	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑧))
39	38	eleq1d 2850	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑧) ∈ Fin))
40	37, 39	sbcie 3787	. . . 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 1153	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → ([𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin))
43		difindi 4245	. . . . 5 ⊢ (𝐴 ∖ (𝑧 ∩ 𝑤)) = ((𝐴 ∖ 𝑧) ∪ (𝐴 ∖ 𝑤))
44		unfi 9162	. . . . 5 ⊢ (((𝐴 ∖ 𝑧) ∈ Fin ∧ (𝐴 ∖ 𝑤) ∈ Fin) → ((𝐴 ∖ 𝑧) ∪ (𝐴 ∖ 𝑤)) ∈ Fin)
45	43, 44	eqeltrid 2869	. . . 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 640	. . 3 ⊢ (([𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin) ↔ ((𝐴 ∖ 𝑧) ∈ Fin ∧ (𝐴 ∖ 𝑤) ∈ Fin))
48	37	inex1 5288	. . . 4 ⊢ (𝑧 ∩ 𝑤) ∈ V
49		difeq2 4075	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝐴 ∖ 𝑦) = (𝐴 ∖ (𝑧 ∩ 𝑤)))
50	49	eleq1d 2850	. . . 4 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧 ∩ 𝑤)) ∈ Fin))
51	48, 50	sbcie 3787	. . 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 24066	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ∈ (Fil‘𝑋))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {crab 3418  [wsbc 3746   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  ‘cfv 6540  Fincfn 8949  Filcfil 24053

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7869  df-1o 8459  df-en 8950  df-fin 8953  df-fbas 21569  df-fil 24054

This theorem is used by:  ufinffr  24137