Theorem cfilfval 25388

Description: The set of Cauchy filters on a metric space. (Contributed by Mario Carneiro, 13-Oct-2015.)

Assertion

Ref	Expression
cfilfval	⊢ (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})

Distinct variable groups:   𝑥,𝑦,𝑓,𝑋   𝐷,𝑓,𝑥,𝑦

Proof of Theorem cfilfval

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvssunirn 6910	. . . 4 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
2	1	sseli 3941	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
3		dmeq 5891	. . . . . . 7 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
4	3	dmeqd 5893	. . . . . 6 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
5	4	fveq2d 6883	. . . . 5 ⊢ (𝑑 = 𝐷 → (Fil‘dom dom 𝑑) = (Fil‘dom dom 𝐷))
6		imaeq1 6055	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑑 “ (𝑦 × 𝑦)) = (𝐷 “ (𝑦 × 𝑦)))
7	6	sseq1d 3976	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
8	7	rexbidv 3195	. . . . . 6 ⊢ (𝑑 = 𝐷 → (∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
9	8	ralbidv 3194	. . . . 5 ⊢ (𝑑 = 𝐷 → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
10	5, 9	rabeqbidv 3441	. . . 4 ⊢ (𝑑 = 𝐷 → {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
11		df-cfil 25379	. . . 4 ⊢ CauFil = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
12		fvex 6892	. . . . 5 ⊢ (Fil‘dom dom 𝐷) ∈ V
13	12	rabex 5307	. . . 4 ⊢ {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} ∈ V
14	10, 11, 13	fvmpt 6987	. . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
15	2, 14	syl 18	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
16		xmetdmdm 24457	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷)
17	16	fveq2d 6883	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (Fil‘𝑋) = (Fil‘dom dom 𝐷))
18	17	rabeqdv 3438	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
19	15, 18	eqtr4d 2807	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3423   ⊆ wss 3913  ∪ cuni 4873   × cxp 5657  dom cdm 5659  ran crn 5660   “ cima 5662  ‘cfv 6533  (class class class)co 7408  0cc0 11096  ℝ⁺crp 13012  [,)cico 13370  ∞Metcxmet 21472  Filcfil 23967  CauFilccfil 25376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8822  df-xr 11243  df-xmet 21480  df-cfil 25379

This theorem is referenced by:  iscfil  25389