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Theorem cmetcvg 24672
Description: The convergence of a Cauchy filter in a complete metric space. (Contributed by Mario Carneiro, 14-Oct-2015.)
Hypothesis
Ref Expression
iscmet.1 𝐽 = (MetOpenβ€˜π·)
Assertion
Ref Expression
cmetcvg ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐹 ∈ (CauFilβ€˜π·)) β†’ (𝐽 fLim 𝐹) β‰  βˆ…)

Proof of Theorem cmetcvg
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 iscmet.1 . . . 4 𝐽 = (MetOpenβ€˜π·)
21iscmet 24671 . . 3 (𝐷 ∈ (CMetβ€˜π‘‹) ↔ (𝐷 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘“ ∈ (CauFilβ€˜π·)(𝐽 fLim 𝑓) β‰  βˆ…))
32simprbi 498 . 2 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ βˆ€π‘“ ∈ (CauFilβ€˜π·)(𝐽 fLim 𝑓) β‰  βˆ…)
4 oveq2 7369 . . . 4 (𝑓 = 𝐹 β†’ (𝐽 fLim 𝑓) = (𝐽 fLim 𝐹))
54neeq1d 3000 . . 3 (𝑓 = 𝐹 β†’ ((𝐽 fLim 𝑓) β‰  βˆ… ↔ (𝐽 fLim 𝐹) β‰  βˆ…))
65rspccva 3582 . 2 ((βˆ€π‘“ ∈ (CauFilβ€˜π·)(𝐽 fLim 𝑓) β‰  βˆ… ∧ 𝐹 ∈ (CauFilβ€˜π·)) β†’ (𝐽 fLim 𝐹) β‰  βˆ…)
73, 6sylan 581 1 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐹 ∈ (CauFilβ€˜π·)) β†’ (𝐽 fLim 𝐹) β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  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:  cmetcaulem  24675  metsscmetcld  24702  cmetss  24703  cmetcusp  24741  minveclem4a  24817  fmcncfil  32576
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