Theorem cfinfil 23296

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 4096	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
2	1	eleq1d 2817	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin))
3	2	elrab 3663	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin))
4		velpw 4585	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
5	4	anbi1i 624	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin) ↔ (𝑦 ⊆ 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin))
6	3, 5	bitri 274	. . 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 1136	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → 𝑋 ∈ 𝑉)
9		ssdif0 4343	. . . . 5 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∖ 𝑋) = ∅)
10		0fin 9137	. . . . . 6 ⊢ ∅ ∈ Fin
11		eleq1 2820	. . . . . 6 ⊢ ((𝐴 ∖ 𝑋) = ∅ → ((𝐴 ∖ 𝑋) ∈ Fin ↔ ∅ ∈ Fin))
12	10, 11	mpbiri 257	. . . . 5 ⊢ ((𝐴 ∖ 𝑋) = ∅ → (𝐴 ∖ 𝑋) ∈ Fin)
13	9, 12	sylbi 216	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → (𝐴 ∖ 𝑋) ∈ Fin)
14		difeq2 4096	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑋))
15	14	eleq1d 2817	. . . . . 6 ⊢ (𝑦 = 𝑋 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑋) ∈ Fin))
16	15	sbcieg 3797	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑋) ∈ Fin))
17	16	biimpar 478	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ∖ 𝑋) ∈ Fin) → [𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
18	13, 17	sylan2 593	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → [𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
19	18	3adant3 1132	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → [𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
20		0ex 5284	. . . . . 6 ⊢ ∅ ∈ V
21		difeq2 4096	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝐴 ∖ 𝑦) = (𝐴 ∖ ∅))
22	21	eleq1d 2817	. . . . . 6 ⊢ (𝑦 = ∅ → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin))
23	20, 22	sbcie 3800	. . . . 5 ⊢ ([∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin)
24		dif0 4352	. . . . . 6 ⊢ (𝐴 ∖ ∅) = 𝐴
25	24	eleq1i 2823	. . . . 5 ⊢ ((𝐴 ∖ ∅) ∈ Fin ↔ 𝐴 ∈ Fin)
26	23, 25	sylbb 218	. . . 4 ⊢ ([∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin → 𝐴 ∈ Fin)
27	26	con3i 154	. . 3 ⊢ (¬ 𝐴 ∈ Fin → ¬ [∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
28	27	3ad2ant3 1135	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → ¬ [∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
29		sscon 4118	. . . . 5 ⊢ (𝑤 ⊆ 𝑧 → (𝐴 ∖ 𝑧) ⊆ (𝐴 ∖ 𝑤))
30		ssfi 9139	. . . . . 6 ⊢ (((𝐴 ∖ 𝑤) ∈ Fin ∧ (𝐴 ∖ 𝑧) ⊆ (𝐴 ∖ 𝑤)) → (𝐴 ∖ 𝑧) ∈ Fin)
31	30	expcom 414	. . . . 5 ⊢ ((𝐴 ∖ 𝑧) ⊆ (𝐴 ∖ 𝑤) → ((𝐴 ∖ 𝑤) ∈ Fin → (𝐴 ∖ 𝑧) ∈ Fin))
32	29, 31	syl 17	. . . 4 ⊢ (𝑤 ⊆ 𝑧 → ((𝐴 ∖ 𝑤) ∈ Fin → (𝐴 ∖ 𝑧) ∈ Fin))
33		vex 3463	. . . . 5 ⊢ 𝑤 ∈ V
34		difeq2 4096	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑤))
35	34	eleq1d 2817	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑤) ∈ Fin))
36	33, 35	sbcie 3800	. . . 4 ⊢ ([𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑤) ∈ Fin)
37		vex 3463	. . . . 5 ⊢ 𝑧 ∈ V
38		difeq2 4096	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑧))
39	38	eleq1d 2817	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑧) ∈ Fin))
40	37, 39	sbcie 3800	. . . 4 ⊢ ([𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑧) ∈ Fin)
41	32, 36, 40	3imtr4g 295	. . 3 ⊢ (𝑤 ⊆ 𝑧 → ([𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin))
42	41	3ad2ant3 1135	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → ([𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin))
43		difindi 4261	. . . . 5 ⊢ (𝐴 ∖ (𝑧 ∩ 𝑤)) = ((𝐴 ∖ 𝑧) ∪ (𝐴 ∖ 𝑤))
44		unfi 9138	. . . . 5 ⊢ (((𝐴 ∖ 𝑧) ∈ Fin ∧ (𝐴 ∖ 𝑤) ∈ Fin) → ((𝐴 ∖ 𝑧) ∪ (𝐴 ∖ 𝑤)) ∈ Fin)
45	43, 44	eqeltrid 2836	. . . 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 627	. . 3 ⊢ (([𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin) ↔ ((𝐴 ∖ 𝑧) ∈ Fin ∧ (𝐴 ∖ 𝑤) ∈ Fin))
48	37	inex1 5294	. . . 4 ⊢ (𝑧 ∩ 𝑤) ∈ V
49		difeq2 4096	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝐴 ∖ 𝑦) = (𝐴 ∖ (𝑧 ∩ 𝑤)))
50	49	eleq1d 2817	. . . 4 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧 ∩ 𝑤)) ∈ Fin))
51	48, 50	sbcie 3800	. . 3 ⊢ ([(𝑧 ∩ 𝑤) / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧 ∩ 𝑤)) ∈ Fin)
52	46, 47, 51	3imtr4g 295	. 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 23261	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ∈ (Fil‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3418  [wsbc 3757   ∖ cdif 3925   ∪ cun 3926   ∩ cin 3927   ⊆ wss 3928  ∅c0 4302  𝒫 cpw 4580  ‘cfv 6516  Fincfn 8905  Filcfil 23248

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-om 7823  df-1o 8432  df-en 8906  df-fin 8909  df-fbas 20845  df-fil 23249

This theorem is referenced by:  ufinffr  23332