Theorem cfinfil 24221

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 4056	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
2	1	eleq1d 2845	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin))
3	2	elrab 3644	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin))
4		velpw 4561	. . . . 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 4313	. . . . 5 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∖ 𝑋) = ∅)
10		0fi 9077	. . . . . 6 ⊢ ∅ ∈ Fin
11		eleq1 2848	. . . . . 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 4056	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑋))
15	14	eleq1d 2845	. . . . . 6 ⊢ (𝑦 = 𝑋 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑋) ∈ Fin))
16	15	sbcieg 3777	. . . . 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 5260	. . . . . 6 ⊢ ∅ ∈ V
21		difeq2 4056	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝐴 ∖ 𝑦) = (𝐴 ∖ ∅))
22	21	eleq1d 2845	. . . . . 6 ⊢ (𝑦 = ∅ → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin))
23	20, 22	sbcie 3779	. . . . 5 ⊢ ([∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin)
24		dif0 4326	. . . . . 6 ⊢ (𝐴 ∖ ∅) = 𝐴
25	24	eleq1i 2851	. . . . 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 4078	. . . . 5 ⊢ (𝑤 ⊆ 𝑧 → (𝐴 ∖ 𝑧) ⊆ (𝐴 ∖ 𝑤))
30		ssfi 9195	. . . . . 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 3454	. . . . 5 ⊢ 𝑤 ∈ V
34		difeq2 4056	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑤))
35	34	eleq1d 2845	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑤) ∈ Fin))
36	33, 35	sbcie 3779	. . . 4 ⊢ ([𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑤) ∈ Fin)
37		vex 3454	. . . . 5 ⊢ 𝑧 ∈ V
38		difeq2 4056	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑧))
39	38	eleq1d 2845	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑧) ∈ Fin))
40	37, 39	sbcie 3779	. . . 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 4236	. . . . 5 ⊢ (𝐴 ∖ (𝑧 ∩ 𝑤)) = ((𝐴 ∖ 𝑧) ∪ (𝐴 ∖ 𝑤))
44		unfi 9193	. . . . 5 ⊢ (((𝐴 ∖ 𝑧) ∈ Fin ∧ (𝐴 ∖ 𝑤) ∈ Fin) → ((𝐴 ∖ 𝑧) ∪ (𝐴 ∖ 𝑤)) ∈ Fin)
45	43, 44	eqeltrid 2864	. . . 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 5277	. . . 4 ⊢ (𝑧 ∩ 𝑤) ∈ V
49		difeq2 4056	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝐴 ∖ 𝑦) = (𝐴 ∖ (𝑧 ∩ 𝑤)))
50	49	eleq1d 2845	. . . 4 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧 ∩ 𝑤)) ∈ Fin))
51	48, 50	sbcie 3779	. . 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 24186	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 1569   ∈ wcel 2144  {crab 3412  [wsbc 3738   ⊆ wss 3896   ∖ cdif 4044   ∩ cin 4088   ∪ cun 4148  ∅c0 4277  𝒫 cpw 4556  ‘cfv 6543  Fincfn 8980  Filcfil 24173

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7882  df-1o 8483  df-en 8981  df-fin 8984  df-fbas 21684  df-fil 24174

This theorem is used by:  ufinffr  24257