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Theorem cfinfil 21616
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 3705 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
21eleq1d 2683 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑥) ∈ Fin ↔ (𝐴𝑦) ∈ Fin))
32elrab 3350 . . . 4 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦) ∈ Fin))
4 selpw 4142 . . . . 5 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
54anbi1i 730 . . . 4 ((𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦) ∈ Fin) ↔ (𝑦𝑋 ∧ (𝐴𝑦) ∈ Fin))
63, 5bitri 264 . . 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 3201 . . 3 (𝑋𝑉𝑋 ∈ V)
983ad2ant1 1080 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → 𝑋 ∈ V)
10 ssdif0 3921 . . . . 5 (𝐴𝑋 ↔ (𝐴𝑋) = ∅)
11 0fin 8139 . . . . . 6 ∅ ∈ Fin
12 eleq1 2686 . . . . . 6 ((𝐴𝑋) = ∅ → ((𝐴𝑋) ∈ Fin ↔ ∅ ∈ Fin))
1311, 12mpbiri 248 . . . . 5 ((𝐴𝑋) = ∅ → (𝐴𝑋) ∈ Fin)
1410, 13sylbi 207 . . . 4 (𝐴𝑋 → (𝐴𝑋) ∈ Fin)
15 difeq2 3705 . . . . . . 7 (𝑦 = 𝑋 → (𝐴𝑦) = (𝐴𝑋))
1615eleq1d 2683 . . . . . 6 (𝑦 = 𝑋 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
1716sbcieg 3454 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
1817biimpar 502 . . . 4 ((𝑋𝑉 ∧ (𝐴𝑋) ∈ Fin) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
1914, 18sylan2 491 . . 3 ((𝑋𝑉𝐴𝑋) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
20193adant3 1079 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
21 0ex 4755 . . . . . 6 ∅ ∈ V
22 difeq2 3705 . . . . . . 7 (𝑦 = ∅ → (𝐴𝑦) = (𝐴 ∖ ∅))
2322eleq1d 2683 . . . . . 6 (𝑦 = ∅ → ((𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin))
2421, 23sbcie 3456 . . . . 5 ([∅ / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin)
25 dif0 3929 . . . . . 6 (𝐴 ∖ ∅) = 𝐴
2625eleq1i 2689 . . . . 5 ((𝐴 ∖ ∅) ∈ Fin ↔ 𝐴 ∈ Fin)
2724, 26sylbb 209 . . . 4 ([∅ / 𝑦](𝐴𝑦) ∈ Fin → 𝐴 ∈ Fin)
2827con3i 150 . . 3 𝐴 ∈ Fin → ¬ [∅ / 𝑦](𝐴𝑦) ∈ Fin)
29283ad2ant3 1082 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → ¬ [∅ / 𝑦](𝐴𝑦) ∈ Fin)
30 sscon 3727 . . . . 5 (𝑤𝑧 → (𝐴𝑧) ⊆ (𝐴𝑤))
31 ssfi 8131 . . . . . 6 (((𝐴𝑤) ∈ Fin ∧ (𝐴𝑧) ⊆ (𝐴𝑤)) → (𝐴𝑧) ∈ Fin)
3231expcom 451 . . . . 5 ((𝐴𝑧) ⊆ (𝐴𝑤) → ((𝐴𝑤) ∈ Fin → (𝐴𝑧) ∈ Fin))
3330, 32syl 17 . . . 4 (𝑤𝑧 → ((𝐴𝑤) ∈ Fin → (𝐴𝑧) ∈ Fin))
34 vex 3192 . . . . 5 𝑤 ∈ V
35 difeq2 3705 . . . . . 6 (𝑦 = 𝑤 → (𝐴𝑦) = (𝐴𝑤))
3635eleq1d 2683 . . . . 5 (𝑦 = 𝑤 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑤) ∈ Fin))
3734, 36sbcie 3456 . . . 4 ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑤) ∈ Fin)
38 vex 3192 . . . . 5 𝑧 ∈ V
39 difeq2 3705 . . . . . 6 (𝑦 = 𝑧 → (𝐴𝑦) = (𝐴𝑧))
4039eleq1d 2683 . . . . 5 (𝑦 = 𝑧 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑧) ∈ Fin))
4138, 40sbcie 3456 . . . 4 ([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑧) ∈ Fin)
4233, 37, 413imtr4g 285 . . 3 (𝑤𝑧 → ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴𝑦) ∈ Fin))
43423ad2ant3 1082 . 2 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑧) → ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴𝑦) ∈ Fin))
44 difindi 3862 . . . . 5 (𝐴 ∖ (𝑧𝑤)) = ((𝐴𝑧) ∪ (𝐴𝑤))
45 unfi 8178 . . . . 5 (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → ((𝐴𝑧) ∪ (𝐴𝑤)) ∈ Fin)
4644, 45syl5eqel 2702 . . . 4 (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → (𝐴 ∖ (𝑧𝑤)) ∈ Fin)
4746a1i 11 . . 3 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑋) → (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → (𝐴 ∖ (𝑧𝑤)) ∈ Fin))
4841, 37anbi12i 732 . . 3 (([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴𝑦) ∈ Fin) ↔ ((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin))
4938inex1 4764 . . . 4 (𝑧𝑤) ∈ V
50 difeq2 3705 . . . . 5 (𝑦 = (𝑧𝑤) → (𝐴𝑦) = (𝐴 ∖ (𝑧𝑤)))
5150eleq1d 2683 . . . 4 (𝑦 = (𝑧𝑤) → ((𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧𝑤)) ∈ Fin))
5249, 51sbcie 3456 . . 3 ([(𝑧𝑤) / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧𝑤)) ∈ Fin)
5347, 48, 523imtr4g 285 . 2 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑋) → (([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴𝑦) ∈ Fin) → [(𝑧𝑤) / 𝑦](𝐴𝑦) ∈ Fin))
547, 9, 20, 29, 43, 53isfild 21581 1 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3189  [wsbc 3421  cdif 3556  cun 3557  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  cfv 5852  Fincfn 7906  Filcfil 21568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-oadd 7516  df-er 7694  df-en 7907  df-fin 7910  df-fbas 19671  df-fil 21569
This theorem is referenced by:  ufinffr  21652
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