Theorem cmetcvg 24801

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.

Step	Hyp	Ref	Expression
1		iscmet.1	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐷)
2	1	iscmet 24800	. . 3 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
3	2	simprbi 497	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)
4		oveq2 7416	. . . 4 ⊢ (𝑓 = 𝐹 → (𝐽 fLim 𝑓) = (𝐽 fLim 𝐹))
5	4	neeq1d 3000	. . 3 ⊢ (𝑓 = 𝐹 → ((𝐽 fLim 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝐹) ≠ ∅))
6	5	rspccva 3611	. 2 ⊢ ((∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅ ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝐹) ≠ ∅)
7	3, 6	sylan 580	1 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝐹) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∅c0 4322  ‘cfv 6543  (class class class)co 7408  Metcmet 20929  MetOpencmopn 20933   fLim cflim 23437  CauFilccfil 24768  CMetccmet 24770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-cmet 24773

This theorem is referenced by:  cmetcaulem  24804  metsscmetcld  24831  cmetss  24832  cmetcusp  24870  minveclem4a  24946  fmcncfil  32906