Theorem iscmet 25276

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 6869	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ V)
2		elfvex 6869	. . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ V)
3	2	adantr 481	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅) → 𝑋 ∈ V)
4		fveq2 6834	. . . . . 6 ⊢ (𝑥 = 𝑋 → (Met‘𝑥) = (Met‘𝑋))
5	4	rabeqdv 3407	. . . . 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 25249	. . . . 5 ⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
7		fvex 6847	. . . . . 6 ⊢ (Met‘𝑋) ∈ V
8	7	rabex 5274	. . . . 5 ⊢ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} ∈ V
9	5, 6, 8	fvmpt 6942	. . . 4 ⊢ (𝑋 ∈ V → (CMet‘𝑋) = {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
10	9	eleq2d 2826	. . 3 ⊢ (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}))
11		fveq2 6834	. . . . 5 ⊢ (𝑑 = 𝐷 → (CauFil‘𝑑) = (CauFil‘𝐷))
12		fveq2 6834	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (MetOpen‘𝑑) = (MetOpen‘𝐷))
13		iscmet.1	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘𝐷)
14	12, 13	eqtr4di 2793	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (MetOpen‘𝑑) = 𝐽)
15	14	oveq1d 7378	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((MetOpen‘𝑑) fLim 𝑓) = (𝐽 fLim 𝑓))
16	15	neeq1d 2994	. . . . 5 ⊢ (𝑑 = 𝐷 → (((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
17	11, 16	raleqbidv 3314	. . . 4 ⊢ (𝑑 = 𝐷 → (∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
18	17	elrab 3636	. . 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 288	. 2 ⊢ (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)))
20	1, 3, 19	pm5.21nii 379	1 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  {crab 3392  Vcvv 3432  ∅c0 4268  ‘cfv 6492  (class class class)co 7363  Metcmet 21340  MetOpencmopn 21344   fLim cflim 23924  CauFilccfil 25244  CMetccmet 25246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-cmet 25249

This theorem is referenced by:  cmetcvg  25277  cmetmet  25278  iscmet3  25285  cmetss  25308  equivcmet  25309  relcmpcmet  25310  cmetcusp1  25345