Theorem cfinfil 23876

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 4051	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
2	1	eleq1d 2824	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin))
3	2	elrab 3629	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin))
4		velpw 4534	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
5	4	anbi1i 630	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin) ↔ (𝑦 ⊆ 𝑋 ∧ (𝐴 ∖ 𝑦) ∈ Fin))
6	3, 5	bitri 276	. . 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 1142	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → 𝑋 ∈ 𝑉)
9		ssdif0 4294	. . . . 5 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∖ 𝑋) = ∅)
10		0fi 8979	. . . . . 6 ⊢ ∅ ∈ Fin
11		eleq1 2827	. . . . . 6 ⊢ ((𝐴 ∖ 𝑋) = ∅ → ((𝐴 ∖ 𝑋) ∈ Fin ↔ ∅ ∈ Fin))
12	10, 11	mpbiri 259	. . . . 5 ⊢ ((𝐴 ∖ 𝑋) = ∅ → (𝐴 ∖ 𝑋) ∈ Fin)
13	9, 12	sylbi 218	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → (𝐴 ∖ 𝑋) ∈ Fin)
14		difeq2 4051	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑋))
15	14	eleq1d 2824	. . . . . 6 ⊢ (𝑦 = 𝑋 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑋) ∈ Fin))
16	15	sbcieg 3762	. . . . 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 599	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → [𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
19	18	3adant3 1138	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → [𝑋 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
20		0ex 5229	. . . . . 6 ⊢ ∅ ∈ V
21		difeq2 4051	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝐴 ∖ 𝑦) = (𝐴 ∖ ∅))
22	21	eleq1d 2824	. . . . . 6 ⊢ (𝑦 = ∅ → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin))
23	20, 22	sbcie 3764	. . . . 5 ⊢ ([∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin)
24		dif0 4306	. . . . . 6 ⊢ (𝐴 ∖ ∅) = 𝐴
25	24	eleq1i 2830	. . . . 5 ⊢ ((𝐴 ∖ ∅) ∈ Fin ↔ 𝐴 ∈ Fin)
26	23, 25	sylbb 220	. . . 4 ⊢ ([∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin → 𝐴 ∈ Fin)
27	26	con3i 154	. . 3 ⊢ (¬ 𝐴 ∈ Fin → ¬ [∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
28	27	3ad2ant3 1141	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → ¬ [∅ / 𝑦](𝐴 ∖ 𝑦) ∈ Fin)
29		sscon 4073	. . . . 5 ⊢ (𝑤 ⊆ 𝑧 → (𝐴 ∖ 𝑧) ⊆ (𝐴 ∖ 𝑤))
30		ssfi 9097	. . . . . 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 3435	. . . . 5 ⊢ 𝑤 ∈ V
34		difeq2 4051	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑤))
35	34	eleq1d 2824	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑤) ∈ Fin))
36	33, 35	sbcie 3764	. . . 4 ⊢ ([𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑤) ∈ Fin)
37		vex 3435	. . . . 5 ⊢ 𝑧 ∈ V
38		difeq2 4051	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝑧))
39	38	eleq1d 2824	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑧) ∈ Fin))
40	37, 39	sbcie 3764	. . . 4 ⊢ ([𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑧) ∈ Fin)
41	32, 36, 40	3imtr4g 297	. . 3 ⊢ (𝑤 ⊆ 𝑧 → ([𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin))
42	41	3ad2ant3 1141	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → ([𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin))
43		difindi 4220	. . . . 5 ⊢ (𝐴 ∖ (𝑧 ∩ 𝑤)) = ((𝐴 ∖ 𝑧) ∪ (𝐴 ∖ 𝑤))
44		unfi 9095	. . . . 5 ⊢ (((𝐴 ∖ 𝑧) ∈ Fin ∧ (𝐴 ∖ 𝑤) ∈ Fin) → ((𝐴 ∖ 𝑧) ∪ (𝐴 ∖ 𝑤)) ∈ Fin)
45	43, 44	eqeltrid 2843	. . . 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 634	. . 3 ⊢ (([𝑧 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴 ∖ 𝑦) ∈ Fin) ↔ ((𝐴 ∖ 𝑧) ∈ Fin ∧ (𝐴 ∖ 𝑤) ∈ Fin))
48	37	inex1 5245	. . . 4 ⊢ (𝑧 ∩ 𝑤) ∈ V
49		difeq2 4051	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝐴 ∖ 𝑦) = (𝐴 ∖ (𝑧 ∩ 𝑤)))
50	49	eleq1d 2824	. . . 4 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → ((𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧 ∩ 𝑤)) ∈ Fin))
51	48, 50	sbcie 3764	. . 3 ⊢ ([(𝑧 ∩ 𝑤) / 𝑦](𝐴 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧 ∩ 𝑤)) ∈ Fin)
52	46, 47, 51	3imtr4g 297	. 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 23841	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ∈ (Fil‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  {crab 3391  [wsbc 3723   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  ‘cfv 6485  Fincfn 8883  Filcfil 23828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-om 7807  df-1o 8395  df-en 8884  df-fin 8887  df-fbas 21344  df-fil 23829

This theorem is referenced by:  ufinffr  23912