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Theorem cfilfval 22985
Description: The set of Cauchy filters on a metric space. (Contributed by Mario Carneiro, 13-Oct-2015.)
Assertion
Ref Expression
cfilfval (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
Distinct variable groups:   𝑥,𝑦,𝑓,𝑋   𝐷,𝑓,𝑥,𝑦

Proof of Theorem cfilfval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 fvssunirn 6179 . . . 4 (∞Met‘𝑋) ⊆ ran ∞Met
21sseli 3583 . . 3 (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ran ∞Met)
3 dmeq 5289 . . . . . . 7 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
43dmeqd 5291 . . . . . 6 (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
54fveq2d 6157 . . . . 5 (𝑑 = 𝐷 → (Fil‘dom dom 𝑑) = (Fil‘dom dom 𝐷))
6 imaeq1 5425 . . . . . . . 8 (𝑑 = 𝐷 → (𝑑 “ (𝑦 × 𝑦)) = (𝐷 “ (𝑦 × 𝑦)))
76sseq1d 3616 . . . . . . 7 (𝑑 = 𝐷 → ((𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
87rexbidv 3046 . . . . . 6 (𝑑 = 𝐷 → (∃𝑦𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∃𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
98ralbidv 2981 . . . . 5 (𝑑 = 𝐷 → (∀𝑥 ∈ ℝ+𝑦𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
105, 9rabeqbidv 3184 . . . 4 (𝑑 = 𝐷 → {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
11 df-cfil 22976 . . . 4 CauFil = (𝑑 ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
12 fvex 6163 . . . . 5 (Fil‘dom dom 𝐷) ∈ V
1312rabex 4778 . . . 4 {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} ∈ V
1410, 11, 13fvmpt 6244 . . 3 (𝐷 ran ∞Met → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
152, 14syl 17 . 2 (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
16 xmetdmdm 22063 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷)
1716fveq2d 6157 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (Fil‘𝑋) = (Fil‘dom dom 𝐷))
18 rabeq 3182 . . 3 ((Fil‘𝑋) = (Fil‘dom dom 𝐷) → {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
1917, 18syl 17 . 2 (𝐷 ∈ (∞Met‘𝑋) → {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
2015, 19eqtr4d 2658 1 (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  wss 3559   cuni 4407   × cxp 5077  dom cdm 5079  ran crn 5080  cima 5082  cfv 5852  (class class class)co 6610  0cc0 9888  +crp 11784  [,)cico 12127  ∞Metcxmt 19663  Filcfil 21572  CauFilccfil 22973
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  ax-cnex 9944  ax-resscn 9945
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 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  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-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-xr 10030  df-xmet 19671  df-cfil 22976
This theorem is referenced by:  iscfil  22986
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