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Theorem cfinfil 22790
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 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ 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.
StepHypRef Expression
1 difeq2 4031 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
21eleq1d 2822 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑥) ∈ Fin ↔ (𝐴𝑦) ∈ Fin))
32elrab 3602 . . . 4 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦) ∈ Fin))
4 velpw 4518 . . . . 5 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
54anbi1i 627 . . . 4 ((𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦) ∈ Fin) ↔ (𝑦𝑋 ∧ (𝐴𝑦) ∈ Fin))
63, 5bitri 278 . . 3 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ↔ (𝑦𝑋 ∧ (𝐴𝑦) ∈ Fin))
76a1i 11 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ↔ (𝑦𝑋 ∧ (𝐴𝑦) ∈ Fin)))
8 simp1 1138 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → 𝑋𝑉)
9 ssdif0 4278 . . . . 5 (𝐴𝑋 ↔ (𝐴𝑋) = ∅)
10 0fin 8849 . . . . . 6 ∅ ∈ Fin
11 eleq1 2825 . . . . . 6 ((𝐴𝑋) = ∅ → ((𝐴𝑋) ∈ Fin ↔ ∅ ∈ Fin))
1210, 11mpbiri 261 . . . . 5 ((𝐴𝑋) = ∅ → (𝐴𝑋) ∈ Fin)
139, 12sylbi 220 . . . 4 (𝐴𝑋 → (𝐴𝑋) ∈ Fin)
14 difeq2 4031 . . . . . . 7 (𝑦 = 𝑋 → (𝐴𝑦) = (𝐴𝑋))
1514eleq1d 2822 . . . . . 6 (𝑦 = 𝑋 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
1615sbcieg 3734 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
1716biimpar 481 . . . 4 ((𝑋𝑉 ∧ (𝐴𝑋) ∈ Fin) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
1813, 17sylan2 596 . . 3 ((𝑋𝑉𝐴𝑋) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
19183adant3 1134 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
20 0ex 5200 . . . . . 6 ∅ ∈ V
21 difeq2 4031 . . . . . . 7 (𝑦 = ∅ → (𝐴𝑦) = (𝐴 ∖ ∅))
2221eleq1d 2822 . . . . . 6 (𝑦 = ∅ → ((𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin))
2320, 22sbcie 3737 . . . . 5 ([∅ / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin)
24 dif0 4287 . . . . . 6 (𝐴 ∖ ∅) = 𝐴
2524eleq1i 2828 . . . . 5 ((𝐴 ∖ ∅) ∈ Fin ↔ 𝐴 ∈ Fin)
2623, 25sylbb 222 . . . 4 ([∅ / 𝑦](𝐴𝑦) ∈ Fin → 𝐴 ∈ Fin)
2726con3i 157 . . 3 𝐴 ∈ Fin → ¬ [∅ / 𝑦](𝐴𝑦) ∈ Fin)
28273ad2ant3 1137 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → ¬ [∅ / 𝑦](𝐴𝑦) ∈ Fin)
29 sscon 4053 . . . . 5 (𝑤𝑧 → (𝐴𝑧) ⊆ (𝐴𝑤))
30 ssfi 8851 . . . . . 6 (((𝐴𝑤) ∈ Fin ∧ (𝐴𝑧) ⊆ (𝐴𝑤)) → (𝐴𝑧) ∈ Fin)
3130expcom 417 . . . . 5 ((𝐴𝑧) ⊆ (𝐴𝑤) → ((𝐴𝑤) ∈ Fin → (𝐴𝑧) ∈ Fin))
3229, 31syl 17 . . . 4 (𝑤𝑧 → ((𝐴𝑤) ∈ Fin → (𝐴𝑧) ∈ Fin))
33 vex 3412 . . . . 5 𝑤 ∈ V
34 difeq2 4031 . . . . . 6 (𝑦 = 𝑤 → (𝐴𝑦) = (𝐴𝑤))
3534eleq1d 2822 . . . . 5 (𝑦 = 𝑤 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑤) ∈ Fin))
3633, 35sbcie 3737 . . . 4 ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑤) ∈ Fin)
37 vex 3412 . . . . 5 𝑧 ∈ V
38 difeq2 4031 . . . . . 6 (𝑦 = 𝑧 → (𝐴𝑦) = (𝐴𝑧))
3938eleq1d 2822 . . . . 5 (𝑦 = 𝑧 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑧) ∈ Fin))
4037, 39sbcie 3737 . . . 4 ([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑧) ∈ Fin)
4132, 36, 403imtr4g 299 . . 3 (𝑤𝑧 → ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴𝑦) ∈ Fin))
42413ad2ant3 1137 . 2 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑧) → ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴𝑦) ∈ Fin))
43 difindi 4196 . . . . 5 (𝐴 ∖ (𝑧𝑤)) = ((𝐴𝑧) ∪ (𝐴𝑤))
44 unfi 8850 . . . . 5 (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → ((𝐴𝑧) ∪ (𝐴𝑤)) ∈ Fin)
4543, 44eqeltrid 2842 . . . 4 (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → (𝐴 ∖ (𝑧𝑤)) ∈ Fin)
4645a1i 11 . . 3 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑋) → (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → (𝐴 ∖ (𝑧𝑤)) ∈ Fin))
4740, 36anbi12i 630 . . 3 (([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴𝑦) ∈ Fin) ↔ ((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin))
4837inex1 5210 . . . 4 (𝑧𝑤) ∈ V
49 difeq2 4031 . . . . 5 (𝑦 = (𝑧𝑤) → (𝐴𝑦) = (𝐴 ∖ (𝑧𝑤)))
5049eleq1d 2822 . . . 4 (𝑦 = (𝑧𝑤) → ((𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧𝑤)) ∈ Fin))
5148, 50sbcie 3737 . . 3 ([(𝑧𝑤) / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧𝑤)) ∈ Fin)
5246, 47, 513imtr4g 299 . 2 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑋) → (([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴𝑦) ∈ Fin) → [(𝑧𝑤) / 𝑦](𝐴𝑦) ∈ Fin))
537, 8, 19, 28, 42, 52isfild 22755 1 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  {crab 3065  [wsbc 3694  cdif 3863  cun 3864  cin 3865  wss 3866  c0 4237  𝒫 cpw 4513  cfv 6380  Fincfn 8626  Filcfil 22742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-om 7645  df-1o 8202  df-en 8627  df-fin 8630  df-fbas 20360  df-fil 22743
This theorem is referenced by:  ufinffr  22826
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