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Theorem cmetcvg 23023
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 = (MetOpen‘𝐷)
Assertion
Ref Expression
cmetcvg ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝐹) ≠ ∅)

Proof of Theorem cmetcvg
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 iscmet.1 . . . 4 𝐽 = (MetOpen‘𝐷)
21iscmet 23022 . . 3 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
32simprbi 480 . 2 (𝐷 ∈ (CMet‘𝑋) → ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)
4 oveq2 6623 . . . 4 (𝑓 = 𝐹 → (𝐽 fLim 𝑓) = (𝐽 fLim 𝐹))
54neeq1d 2849 . . 3 (𝑓 = 𝐹 → ((𝐽 fLim 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝐹) ≠ ∅))
65rspccva 3298 . 2 ((∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅ ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝐹) ≠ ∅)
73, 6sylan 488 1 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝐹) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2908  c0 3897  cfv 5857  (class class class)co 6615  Metcme 19672  MetOpencmopn 19676   fLim cflim 21678  CauFilccfil 22990  CMetcms 22992
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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-cmet 22995
This theorem is referenced by:  cmetcaulem  23026  cmetss  23053  cmetcusp  23090  minveclem4a  23141  fmcncfil  29801
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