Theorem iscmet 25252

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 = (MetOpen‘𝐷)

Assertion

Ref	Expression
iscmet	⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))

Distinct variable groups:   𝐷,𝑓   𝑓,𝐽   𝑓,𝑋

Proof of Theorem iscmet

Dummy variables 𝑑 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6877	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ V)
2		elfvex 6877	. . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ V)
3	2	adantr 480	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅) → 𝑋 ∈ V)
4		fveq2 6842	. . . . . 6 ⊢ (𝑥 = 𝑋 → (Met‘𝑥) = (Met‘𝑋))
5	4	rabeqdv 3416	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} = {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
6		df-cmet 25225	. . . . 5 ⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
7		fvex 6855	. . . . . 6 ⊢ (Met‘𝑋) ∈ V
8	7	rabex 5286	. . . . 5 ⊢ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} ∈ V
9	5, 6, 8	fvmpt 6949	. . . 4 ⊢ (𝑋 ∈ V → (CMet‘𝑋) = {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
10	9	eleq2d 2823	. . 3 ⊢ (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}))
11		fveq2 6842	. . . . 5 ⊢ (𝑑 = 𝐷 → (CauFil‘𝑑) = (CauFil‘𝐷))
12		fveq2 6842	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (MetOpen‘𝑑) = (MetOpen‘𝐷))
13		iscmet.1	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘𝐷)
14	12, 13	eqtr4di 2790	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (MetOpen‘𝑑) = 𝐽)
15	14	oveq1d 7383	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((MetOpen‘𝑑) fLim 𝑓) = (𝐽 fLim 𝑓))
16	15	neeq1d 2992	. . . . 5 ⊢ (𝑑 = 𝐷 → (((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
17	11, 16	raleqbidv 3318	. . . 4 ⊢ (𝑑 = 𝐷 → (∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
18	17	elrab 3648	. . 3 ⊢ (𝐷 ∈ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
19	10, 18	bitrdi 287	. 2 ⊢ (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)))
20	1, 3, 19	pm5.21nii 378	1 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3401  Vcvv 3442  ∅c0 4287  ‘cfv 6500  (class class class)co 7368  Metcmet 21307  MetOpencmopn 21311   fLim cflim 23890  CauFilccfil 25220  CMetccmet 25222

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-cmet 25225

This theorem is referenced by:  cmetcvg  25253  cmetmet  25254  iscmet3  25261  cmetss  25284  equivcmet  25285  relcmpcmet  25286  cmetcusp1  25321