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Theorem cfinfil 22417
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 4096 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
21eleq1d 2901 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑥) ∈ Fin ↔ (𝐴𝑦) ∈ Fin))
32elrab 3683 . . . 4 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦) ∈ Fin))
4 velpw 4549 . . . . 5 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
54anbi1i 623 . . . 4 ((𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦) ∈ Fin) ↔ (𝑦𝑋 ∧ (𝐴𝑦) ∈ Fin))
63, 5bitri 276 . . 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 elex 3517 . . 3 (𝑋𝑉𝑋 ∈ V)
983ad2ant1 1127 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → 𝑋 ∈ V)
10 ssdif0 4326 . . . . 5 (𝐴𝑋 ↔ (𝐴𝑋) = ∅)
11 0fin 8738 . . . . . 6 ∅ ∈ Fin
12 eleq1 2904 . . . . . 6 ((𝐴𝑋) = ∅ → ((𝐴𝑋) ∈ Fin ↔ ∅ ∈ Fin))
1311, 12mpbiri 259 . . . . 5 ((𝐴𝑋) = ∅ → (𝐴𝑋) ∈ Fin)
1410, 13sylbi 218 . . . 4 (𝐴𝑋 → (𝐴𝑋) ∈ Fin)
15 difeq2 4096 . . . . . . 7 (𝑦 = 𝑋 → (𝐴𝑦) = (𝐴𝑋))
1615eleq1d 2901 . . . . . 6 (𝑦 = 𝑋 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
1716sbcieg 3813 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
1817biimpar 478 . . . 4 ((𝑋𝑉 ∧ (𝐴𝑋) ∈ Fin) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
1914, 18sylan2 592 . . 3 ((𝑋𝑉𝐴𝑋) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
20193adant3 1126 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
21 0ex 5207 . . . . . 6 ∅ ∈ V
22 difeq2 4096 . . . . . . 7 (𝑦 = ∅ → (𝐴𝑦) = (𝐴 ∖ ∅))
2322eleq1d 2901 . . . . . 6 (𝑦 = ∅ → ((𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin))
2421, 23sbcie 3815 . . . . 5 ([∅ / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin)
25 dif0 4335 . . . . . 6 (𝐴 ∖ ∅) = 𝐴
2625eleq1i 2907 . . . . 5 ((𝐴 ∖ ∅) ∈ Fin ↔ 𝐴 ∈ Fin)
2724, 26sylbb 220 . . . 4 ([∅ / 𝑦](𝐴𝑦) ∈ Fin → 𝐴 ∈ Fin)
2827con3i 157 . . 3 𝐴 ∈ Fin → ¬ [∅ / 𝑦](𝐴𝑦) ∈ Fin)
29283ad2ant3 1129 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → ¬ [∅ / 𝑦](𝐴𝑦) ∈ Fin)
30 sscon 4118 . . . . 5 (𝑤𝑧 → (𝐴𝑧) ⊆ (𝐴𝑤))
31 ssfi 8730 . . . . . 6 (((𝐴𝑤) ∈ Fin ∧ (𝐴𝑧) ⊆ (𝐴𝑤)) → (𝐴𝑧) ∈ Fin)
3231expcom 414 . . . . 5 ((𝐴𝑧) ⊆ (𝐴𝑤) → ((𝐴𝑤) ∈ Fin → (𝐴𝑧) ∈ Fin))
3330, 32syl 17 . . . 4 (𝑤𝑧 → ((𝐴𝑤) ∈ Fin → (𝐴𝑧) ∈ Fin))
34 vex 3502 . . . . 5 𝑤 ∈ V
35 difeq2 4096 . . . . . 6 (𝑦 = 𝑤 → (𝐴𝑦) = (𝐴𝑤))
3635eleq1d 2901 . . . . 5 (𝑦 = 𝑤 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑤) ∈ Fin))
3734, 36sbcie 3815 . . . 4 ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑤) ∈ Fin)
38 vex 3502 . . . . 5 𝑧 ∈ V
39 difeq2 4096 . . . . . 6 (𝑦 = 𝑧 → (𝐴𝑦) = (𝐴𝑧))
4039eleq1d 2901 . . . . 5 (𝑦 = 𝑧 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑧) ∈ Fin))
4138, 40sbcie 3815 . . . 4 ([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑧) ∈ Fin)
4233, 37, 413imtr4g 297 . . 3 (𝑤𝑧 → ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴𝑦) ∈ Fin))
43423ad2ant3 1129 . 2 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑧) → ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴𝑦) ∈ Fin))
44 difindi 4261 . . . . 5 (𝐴 ∖ (𝑧𝑤)) = ((𝐴𝑧) ∪ (𝐴𝑤))
45 unfi 8777 . . . . 5 (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → ((𝐴𝑧) ∪ (𝐴𝑤)) ∈ Fin)
4644, 45eqeltrid 2921 . . . 4 (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → (𝐴 ∖ (𝑧𝑤)) ∈ Fin)
4746a1i 11 . . 3 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑋) → (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → (𝐴 ∖ (𝑧𝑤)) ∈ Fin))
4841, 37anbi12i 626 . . 3 (([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴𝑦) ∈ Fin) ↔ ((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin))
4938inex1 5217 . . . 4 (𝑧𝑤) ∈ V
50 difeq2 4096 . . . . 5 (𝑦 = (𝑧𝑤) → (𝐴𝑦) = (𝐴 ∖ (𝑧𝑤)))
5150eleq1d 2901 . . . 4 (𝑦 = (𝑧𝑤) → ((𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧𝑤)) ∈ Fin))
5249, 51sbcie 3815 . . 3 ([(𝑧𝑤) / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧𝑤)) ∈ Fin)
5347, 48, 523imtr4g 297 . 2 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑋) → (([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴𝑦) ∈ Fin) → [(𝑧𝑤) / 𝑦](𝐴𝑦) ∈ Fin))
547, 9, 20, 29, 43, 53isfild 22382 1 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  {crab 3146  Vcvv 3499  [wsbc 3775  cdif 3936  cun 3937  cin 3938  wss 3939  c0 4294  𝒫 cpw 4541  cfv 6351  Fincfn 8501  Filcfil 22369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-oadd 8100  df-er 8282  df-en 8502  df-fin 8505  df-fbas 20458  df-fil 22370
This theorem is referenced by:  ufinffr  22453
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