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Theorem cfinfil 21976
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 3884 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
21eleq1d 2829 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑥) ∈ Fin ↔ (𝐴𝑦) ∈ Fin))
32elrab 3519 . . . 4 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦) ∈ Fin))
4 selpw 4322 . . . . 5 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
54anbi1i 617 . . . 4 ((𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦) ∈ Fin) ↔ (𝑦𝑋 ∧ (𝐴𝑦) ∈ Fin))
63, 5bitri 266 . . 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 3365 . . 3 (𝑋𝑉𝑋 ∈ V)
983ad2ant1 1163 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → 𝑋 ∈ V)
10 ssdif0 4106 . . . . 5 (𝐴𝑋 ↔ (𝐴𝑋) = ∅)
11 0fin 8395 . . . . . 6 ∅ ∈ Fin
12 eleq1 2832 . . . . . 6 ((𝐴𝑋) = ∅ → ((𝐴𝑋) ∈ Fin ↔ ∅ ∈ Fin))
1311, 12mpbiri 249 . . . . 5 ((𝐴𝑋) = ∅ → (𝐴𝑋) ∈ Fin)
1410, 13sylbi 208 . . . 4 (𝐴𝑋 → (𝐴𝑋) ∈ Fin)
15 difeq2 3884 . . . . . . 7 (𝑦 = 𝑋 → (𝐴𝑦) = (𝐴𝑋))
1615eleq1d 2829 . . . . . 6 (𝑦 = 𝑋 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
1716sbcieg 3629 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑋) ∈ Fin))
1817biimpar 469 . . . 4 ((𝑋𝑉 ∧ (𝐴𝑋) ∈ Fin) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
1914, 18sylan2 586 . . 3 ((𝑋𝑉𝐴𝑋) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
20193adant3 1162 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → [𝑋 / 𝑦](𝐴𝑦) ∈ Fin)
21 0ex 4950 . . . . . 6 ∅ ∈ V
22 difeq2 3884 . . . . . . 7 (𝑦 = ∅ → (𝐴𝑦) = (𝐴 ∖ ∅))
2322eleq1d 2829 . . . . . 6 (𝑦 = ∅ → ((𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin))
2421, 23sbcie 3631 . . . . 5 ([∅ / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ ∅) ∈ Fin)
25 dif0 4115 . . . . . 6 (𝐴 ∖ ∅) = 𝐴
2625eleq1i 2835 . . . . 5 ((𝐴 ∖ ∅) ∈ Fin ↔ 𝐴 ∈ Fin)
2724, 26sylbb 210 . . . 4 ([∅ / 𝑦](𝐴𝑦) ∈ Fin → 𝐴 ∈ Fin)
2827con3i 151 . . 3 𝐴 ∈ Fin → ¬ [∅ / 𝑦](𝐴𝑦) ∈ Fin)
29283ad2ant3 1165 . 2 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → ¬ [∅ / 𝑦](𝐴𝑦) ∈ Fin)
30 sscon 3906 . . . . 5 (𝑤𝑧 → (𝐴𝑧) ⊆ (𝐴𝑤))
31 ssfi 8387 . . . . . 6 (((𝐴𝑤) ∈ Fin ∧ (𝐴𝑧) ⊆ (𝐴𝑤)) → (𝐴𝑧) ∈ Fin)
3231expcom 402 . . . . 5 ((𝐴𝑧) ⊆ (𝐴𝑤) → ((𝐴𝑤) ∈ Fin → (𝐴𝑧) ∈ Fin))
3330, 32syl 17 . . . 4 (𝑤𝑧 → ((𝐴𝑤) ∈ Fin → (𝐴𝑧) ∈ Fin))
34 vex 3353 . . . . 5 𝑤 ∈ V
35 difeq2 3884 . . . . . 6 (𝑦 = 𝑤 → (𝐴𝑦) = (𝐴𝑤))
3635eleq1d 2829 . . . . 5 (𝑦 = 𝑤 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑤) ∈ Fin))
3734, 36sbcie 3631 . . . 4 ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑤) ∈ Fin)
38 vex 3353 . . . . 5 𝑧 ∈ V
39 difeq2 3884 . . . . . 6 (𝑦 = 𝑧 → (𝐴𝑦) = (𝐴𝑧))
4039eleq1d 2829 . . . . 5 (𝑦 = 𝑧 → ((𝐴𝑦) ∈ Fin ↔ (𝐴𝑧) ∈ Fin))
4138, 40sbcie 3631 . . . 4 ([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴𝑧) ∈ Fin)
4233, 37, 413imtr4g 287 . . 3 (𝑤𝑧 → ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴𝑦) ∈ Fin))
43423ad2ant3 1165 . 2 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑧) → ([𝑤 / 𝑦](𝐴𝑦) ∈ Fin → [𝑧 / 𝑦](𝐴𝑦) ∈ Fin))
44 difindi 4046 . . . . 5 (𝐴 ∖ (𝑧𝑤)) = ((𝐴𝑧) ∪ (𝐴𝑤))
45 unfi 8434 . . . . 5 (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → ((𝐴𝑧) ∪ (𝐴𝑤)) ∈ Fin)
4644, 45syl5eqel 2848 . . . 4 (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → (𝐴 ∖ (𝑧𝑤)) ∈ Fin)
4746a1i 11 . . 3 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑋) → (((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin) → (𝐴 ∖ (𝑧𝑤)) ∈ Fin))
4841, 37anbi12i 620 . . 3 (([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴𝑦) ∈ Fin) ↔ ((𝐴𝑧) ∈ Fin ∧ (𝐴𝑤) ∈ Fin))
4938inex1 4960 . . . 4 (𝑧𝑤) ∈ V
50 difeq2 3884 . . . . 5 (𝑦 = (𝑧𝑤) → (𝐴𝑦) = (𝐴 ∖ (𝑧𝑤)))
5150eleq1d 2829 . . . 4 (𝑦 = (𝑧𝑤) → ((𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧𝑤)) ∈ Fin))
5249, 51sbcie 3631 . . 3 ([(𝑧𝑤) / 𝑦](𝐴𝑦) ∈ Fin ↔ (𝐴 ∖ (𝑧𝑤)) ∈ Fin)
5347, 48, 523imtr4g 287 . 2 (((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧𝑋𝑤𝑋) → (([𝑧 / 𝑦](𝐴𝑦) ∈ Fin ∧ [𝑤 / 𝑦](𝐴𝑦) ∈ Fin) → [(𝑧𝑤) / 𝑦](𝐴𝑦) ∈ Fin))
547, 9, 20, 29, 43, 53isfild 21941 1 ((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  {crab 3059  Vcvv 3350  [wsbc 3596  cdif 3729  cun 3730  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315  cfv 6068  Fincfn 8160  Filcfil 21928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-oadd 7768  df-er 7947  df-en 8161  df-fin 8164  df-fbas 20016  df-fil 21929
This theorem is referenced by:  ufinffr  22012
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