MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cfilfval Structured version   Visualization version   GIF version

Theorem cfilfval 23564
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 6522 . . . 4 (∞Met‘𝑋) ⊆ ran ∞Met
21sseli 3848 . . 3 (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ran ∞Met)
3 dmeq 5616 . . . . . . 7 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
43dmeqd 5618 . . . . . 6 (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
54fveq2d 6497 . . . . 5 (𝑑 = 𝐷 → (Fil‘dom dom 𝑑) = (Fil‘dom dom 𝐷))
6 imaeq1 5759 . . . . . . . 8 (𝑑 = 𝐷 → (𝑑 “ (𝑦 × 𝑦)) = (𝐷 “ (𝑦 × 𝑦)))
76sseq1d 3882 . . . . . . 7 (𝑑 = 𝐷 → ((𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
87rexbidv 3236 . . . . . 6 (𝑑 = 𝐷 → (∃𝑦𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∃𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
98ralbidv 3141 . . . . 5 (𝑑 = 𝐷 → (∀𝑥 ∈ ℝ+𝑦𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
105, 9rabeqbidv 3402 . . . 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 23555 . . . 4 CauFil = (𝑑 ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
12 fvex 6506 . . . . 5 (Fil‘dom dom 𝐷) ∈ V
1312rabex 5085 . . . 4 {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} ∈ V
1410, 11, 13fvmpt 6589 . . 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 22642 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷)
1716fveq2d 6497 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (Fil‘𝑋) = (Fil‘dom dom 𝐷))
1817rabeqdv 3401 . 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)})
1915, 18eqtr4d 2811 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 1507  wcel 2050  wral 3082  wrex 3083  {crab 3086  wss 3823   cuni 4706   × cxp 5399  dom cdm 5401  ran crn 5402  cima 5404  cfv 6182  (class class class)co 6970  0cc0 10329  +crp 12198  [,)cico 12550  ∞Metcxmet 20226  Filcfil 22151  CauFilccfil 23552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10385  ax-resscn 10386
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-map 8202  df-xr 10472  df-xmet 20234  df-cfil 23555
This theorem is referenced by:  iscfil  23565
  Copyright terms: Public domain W3C validator