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Theorem iscfil 22971
 Description: The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
Assertion
Ref Expression
iscfil (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝑋,𝑦   𝑥,𝐷,𝑦

Proof of Theorem iscfil
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 cfilfval 22970 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
21eleq2d 2684 . 2 (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ 𝐹 ∈ {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)}))
3 rexeq 3128 . . . 4 (𝑓 = 𝐹 → (∃𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∃𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
43ralbidv 2980 . . 3 (𝑓 = 𝐹 → (∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∀𝑥 ∈ ℝ+𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
54elrab 3346 . 2 (𝐹 ∈ {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
62, 5syl6bb 276 1 (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  {crab 2911   ⊆ wss 3555   × cxp 5072   “ cima 5077  ‘cfv 5847  (class class class)co 6604  0cc0 9880  ℝ+crp 11776  [,)cico 12119  ∞Metcxmt 19650  Filcfil 21559  CauFilccfil 22958 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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-xr 10022  df-xmet 19658  df-cfil 22961 This theorem is referenced by:  iscfil2  22972  cfilfil  22973  cfilss  22976  cfilucfil3  23025  cmetcusp  23058
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