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Theorem iscmet 24671
Description: The property "𝐷 is a complete metric." meaning all Cauchy filters converge to a point in the space. (Contributed by Mario Carneiro, 1-May-2014.) (Revised by Mario Carneiro, 13-Oct-2015.)
Hypothesis
Ref Expression
iscmet.1 𝐽 = (MetOpenβ€˜π·)
Assertion
Ref Expression
iscmet (𝐷 ∈ (CMetβ€˜π‘‹) ↔ (𝐷 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘“ ∈ (CauFilβ€˜π·)(𝐽 fLim 𝑓) β‰  βˆ…))
Distinct variable groups:   𝐷,𝑓   𝑓,𝐽   𝑓,𝑋

Proof of Theorem iscmet
Dummy variables 𝑑 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6884 . 2 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝑋 ∈ V)
2 elfvex 6884 . . 3 (𝐷 ∈ (Metβ€˜π‘‹) β†’ 𝑋 ∈ V)
32adantr 482 . 2 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘“ ∈ (CauFilβ€˜π·)(𝐽 fLim 𝑓) β‰  βˆ…) β†’ 𝑋 ∈ V)
4 fveq2 6846 . . . . . 6 (π‘₯ = 𝑋 β†’ (Metβ€˜π‘₯) = (Metβ€˜π‘‹))
54rabeqdv 3421 . . . . 5 (π‘₯ = 𝑋 β†’ {𝑑 ∈ (Metβ€˜π‘₯) ∣ βˆ€π‘“ ∈ (CauFilβ€˜π‘‘)((MetOpenβ€˜π‘‘) fLim 𝑓) β‰  βˆ…} = {𝑑 ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘“ ∈ (CauFilβ€˜π‘‘)((MetOpenβ€˜π‘‘) fLim 𝑓) β‰  βˆ…})
6 df-cmet 24644 . . . . 5 CMet = (π‘₯ ∈ V ↦ {𝑑 ∈ (Metβ€˜π‘₯) ∣ βˆ€π‘“ ∈ (CauFilβ€˜π‘‘)((MetOpenβ€˜π‘‘) fLim 𝑓) β‰  βˆ…})
7 fvex 6859 . . . . . 6 (Metβ€˜π‘‹) ∈ V
87rabex 5293 . . . . 5 {𝑑 ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘“ ∈ (CauFilβ€˜π‘‘)((MetOpenβ€˜π‘‘) fLim 𝑓) β‰  βˆ…} ∈ V
95, 6, 8fvmpt 6952 . . . 4 (𝑋 ∈ V β†’ (CMetβ€˜π‘‹) = {𝑑 ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘“ ∈ (CauFilβ€˜π‘‘)((MetOpenβ€˜π‘‘) fLim 𝑓) β‰  βˆ…})
109eleq2d 2820 . . 3 (𝑋 ∈ V β†’ (𝐷 ∈ (CMetβ€˜π‘‹) ↔ 𝐷 ∈ {𝑑 ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘“ ∈ (CauFilβ€˜π‘‘)((MetOpenβ€˜π‘‘) fLim 𝑓) β‰  βˆ…}))
11 fveq2 6846 . . . . 5 (𝑑 = 𝐷 β†’ (CauFilβ€˜π‘‘) = (CauFilβ€˜π·))
12 fveq2 6846 . . . . . . . 8 (𝑑 = 𝐷 β†’ (MetOpenβ€˜π‘‘) = (MetOpenβ€˜π·))
13 iscmet.1 . . . . . . . 8 𝐽 = (MetOpenβ€˜π·)
1412, 13eqtr4di 2791 . . . . . . 7 (𝑑 = 𝐷 β†’ (MetOpenβ€˜π‘‘) = 𝐽)
1514oveq1d 7376 . . . . . 6 (𝑑 = 𝐷 β†’ ((MetOpenβ€˜π‘‘) fLim 𝑓) = (𝐽 fLim 𝑓))
1615neeq1d 3000 . . . . 5 (𝑑 = 𝐷 β†’ (((MetOpenβ€˜π‘‘) fLim 𝑓) β‰  βˆ… ↔ (𝐽 fLim 𝑓) β‰  βˆ…))
1711, 16raleqbidv 3318 . . . 4 (𝑑 = 𝐷 β†’ (βˆ€π‘“ ∈ (CauFilβ€˜π‘‘)((MetOpenβ€˜π‘‘) fLim 𝑓) β‰  βˆ… ↔ βˆ€π‘“ ∈ (CauFilβ€˜π·)(𝐽 fLim 𝑓) β‰  βˆ…))
1817elrab 3649 . . 3 (𝐷 ∈ {𝑑 ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘“ ∈ (CauFilβ€˜π‘‘)((MetOpenβ€˜π‘‘) fLim 𝑓) β‰  βˆ…} ↔ (𝐷 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘“ ∈ (CauFilβ€˜π·)(𝐽 fLim 𝑓) β‰  βˆ…))
1910, 18bitrdi 287 . 2 (𝑋 ∈ V β†’ (𝐷 ∈ (CMetβ€˜π‘‹) ↔ (𝐷 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘“ ∈ (CauFilβ€˜π·)(𝐽 fLim 𝑓) β‰  βˆ…)))
201, 3, 19pm5.21nii 380 1 (𝐷 ∈ (CMetβ€˜π‘‹) ↔ (𝐷 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘“ ∈ (CauFilβ€˜π·)(𝐽 fLim 𝑓) β‰  βˆ…))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  {crab 3406  Vcvv 3447  βˆ…c0 4286  β€˜cfv 6500  (class class class)co 7361  Metcmet 20805  MetOpencmopn 20809   fLim cflim 23308  CauFilccfil 24639  CMetccmet 24641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-cmet 24644
This theorem is referenced by:  cmetcvg  24672  cmetmet  24673  iscmet3  24680  cmetss  24703  equivcmet  24704  relcmpcmet  24705  cmetcusp1  24740
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