Theorem iscmet 25424

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 6918	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ V)
2		elfvex 6918	. . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ V)
3	2	adantr 485	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅) → 𝑋 ∈ V)
4		fveq2 6883	. . . . . 6 ⊢ (𝑥 = 𝑋 → (Met‘𝑥) = (Met‘𝑋))
5	4	rabeqdv 3431	. . . . 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 25397	. . . . 5 ⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
7		fvex 6896	. . . . . 6 ⊢ (Met‘𝑋) ∈ V
8	7	rabex 5311	. . . . 5 ⊢ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} ∈ V
9	5, 6, 8	fvmpt 6991	. . . 4 ⊢ (𝑋 ∈ V → (CMet‘𝑋) = {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
10	9	eleq2d 2849	. . 3 ⊢ (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}))
11		fveq2 6883	. . . . 5 ⊢ (𝑑 = 𝐷 → (CauFil‘𝑑) = (CauFil‘𝐷))
12		fveq2 6883	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (MetOpen‘𝑑) = (MetOpen‘𝐷))
13		iscmet.1	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘𝐷)
14	12, 13	eqtr4di 2816	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (MetOpen‘𝑑) = 𝐽)
15	14	oveq1d 7427	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((MetOpen‘𝑑) fLim 𝑓) = (𝐽 fLim 𝑓))
16	15	neeq1d 3017	. . . . 5 ⊢ (𝑑 = 𝐷 → (((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
17	11, 16	raleqbidv 3338	. . . 4 ⊢ (𝑑 = 𝐷 → (∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
18	17	elrab 3651	. . 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 290	. 2 ⊢ (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)))
20	1, 3, 19	pm5.21nii 381	1 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  {crab 3416  Vcvv 3455  ∅c0 4287  ‘cfv 6538  (class class class)co 7412  Metcmet 21489  MetOpencmopn 21493   fLim cflim 24072  CauFilccfil 25392  CMetccmet 25394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-cmet 25397

This theorem is referenced by:  cmetcvg  25425  cmetmet  25426  iscmet3  25433  cmetss  25456  equivcmet  25457  relcmpcmet  25458  cmetcusp1  25493