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Theorem cfinfil 22508
 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 4044 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
21eleq1d 2874 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑥) ∈ Fin ↔ (𝐴𝑦) ∈ Fin))
32elrab 3628 . . . 4 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦) ∈ Fin))
4 velpw 4502 . . . . 5 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
54anbi1i 626 . . . 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 1133 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → 𝑋𝑉)
9 ssdif0 4277 . . . . 5 (𝐴𝑋 ↔ (𝐴𝑋) = ∅)
10 0fin 8733 . . . . . 6 ∅ ∈ Fin
11 eleq1 2877 . . . . . 6 ((𝐴𝑋) = ∅ → ((𝐴𝑋) ∈ Fin ↔ ∅ ∈ Fin))
1210, 11mpbiri 261 . . . . 5 ((𝐴𝑋) = ∅ → (𝐴𝑋) ∈ Fin)
139, 12sylbi 220 . . . 4 (𝐴𝑋 → (𝐴𝑋) ∈ Fin)
14 difeq2 4044 . . . . . . 7 (𝑦 = 𝑋 → (𝐴𝑦) = (𝐴𝑋))
1514eleq1d 2874 . . . . . 6 (𝑦 = 𝑋 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
1615sbcieg 3758 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
1716biimpar 481 . . . 4 ((𝑋𝑉 ∧ (𝐴𝑋) ∈ Fin) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
1813, 17sylan2 595 . . 3 ((𝑋𝑉𝐴𝑋) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
19183adant3 1129 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
20 0ex 5176 . . . . . 6 ∅ ∈ V
21 difeq2 4044 . . . . . . 7 (𝑦 = ∅ → (𝐴𝑦) = (𝐴 ∖ ∅))
2221eleq1d 2874 . . . . . 6 (𝑦 = ∅ → ((𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin))
2320, 22sbcie 3760 . . . . 5 ([∅ / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin)
24 dif0 4286 . . . . . 6 (𝐴 ∖ ∅) = 𝐴
2524eleq1i 2880 . . . . 5 ((𝐴 ∖ ∅) ∈ Fin ↔ 𝐴 ∈ Fin)
2623, 25sylbb 222 . . . 4 ([∅ / 𝑦](𝐴𝑦) ∈ Fin → 𝐴 ∈ Fin)
2726con3i 157 . . 3 𝐴 ∈ Fin → ¬ [∅ / 𝑦](𝐴𝑦) ∈ Fin)
28273ad2ant3 1132 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → ¬ [∅ / 𝑦](𝐴𝑦) ∈ Fin)
29 sscon 4066 . . . . 5 (𝑤𝑧 → (𝐴𝑧) ⊆ (𝐴𝑤))
30 ssfi 8725 . . . . . 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 3444 . . . . 5 𝑤 ∈ V
34 difeq2 4044 . . . . . 6 (𝑦 = 𝑤 → (𝐴𝑦) = (𝐴𝑤))
3534eleq1d 2874 . . . . 5 (𝑦 = 𝑤 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑤) ∈ Fin))
3633, 35sbcie 3760 . . . 4 ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑤) ∈ Fin)
37 vex 3444 . . . . 5 𝑧 ∈ V
38 difeq2 4044 . . . . . 6 (𝑦 = 𝑧 → (𝐴𝑦) = (𝐴𝑧))
3938eleq1d 2874 . . . . 5 (𝑦 = 𝑧 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑧) ∈ Fin))
4037, 39sbcie 3760 . . . 4 ([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑧) ∈ Fin)
4132, 36, 403imtr4g 299 . . 3 (𝑤𝑧 → ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴𝑦) ∈ Fin))
42413ad2ant3 1132 . 2 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑧) → ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴𝑦) ∈ Fin))
43 difindi 4208 . . . . 5 (𝐴 ∖ (𝑧𝑤)) = ((𝐴𝑧) ∪ (𝐴𝑤))
44 unfi 8772 . . . . 5 (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → ((𝐴𝑧) ∪ (𝐴𝑤)) ∈ Fin)
4543, 44eqeltrid 2894 . . . 4 (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → (𝐴 ∖ (𝑧𝑤)) ∈ Fin)
4645a1i 11 . . 3 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑋) → (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → (𝐴 ∖ (𝑧𝑤)) ∈ Fin))
4740, 36anbi12i 629 . . 3 (([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴𝑦) ∈ Fin) ↔ ((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin))
4837inex1 5186 . . . 4 (𝑧𝑤) ∈ V
49 difeq2 4044 . . . . 5 (𝑦 = (𝑧𝑤) → (𝐴𝑦) = (𝐴 ∖ (𝑧𝑤)))
5049eleq1d 2874 . . . 4 (𝑦 = (𝑧𝑤) → ((𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧𝑤)) ∈ Fin))
5148, 50sbcie 3760 . . 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 22473 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 1084   = wceq 1538   ∈ wcel 2111  {crab 3110  [wsbc 3720   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  ‘cfv 6325  Fincfn 8495  Filcfil 22460 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-oadd 8092  df-er 8275  df-en 8496  df-fin 8499  df-fbas 20092  df-fil 22461 This theorem is referenced by:  ufinffr  22544
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