Theorem cfilfval 25253

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 6920	. . . 4 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
2	1	sseli 3961	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
3		dmeq 5896	. . . . . . 7 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
4	3	dmeqd 5898	. . . . . 6 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
5	4	fveq2d 6891	. . . . 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 3997	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
8	7	rexbidv 3166	. . . . . 6 ⊢ (𝑑 = 𝐷 → (∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
9	8	ralbidv 3165	. . . . 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 3439	. . . 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 25244	. . . 4 ⊢ CauFil = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
12		fvex 6900	. . . . 5 ⊢ (Fil‘dom dom 𝐷) ∈ V
13	12	rabex 5321	. . . 4 ⊢ {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} ∈ V
14	10, 11, 13	fvmpt 6997	. . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
15	2, 14	syl 17	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
16		xmetdmdm 24309	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷)
17	16	fveq2d 6891	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (Fil‘𝑋) = (Fil‘dom dom 𝐷))
18	17	rabeqdv 3436	. 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 2772	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059  {crab 3420   ⊆ wss 3933  ∪ cuni 4889   × cxp 5665  dom cdm 5667  ran crn 5668   “ cima 5670  ‘cfv 6542  (class class class)co 7414  0cc0 11138  ℝ⁺crp 13017  [,)cico 13372  ∞Metcxmet 21316  Filcfil 23818  CauFilccfil 25241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738  ax-cnex 11194  ax-resscn 11195

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-sbc 3773  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8851  df-xr 11282  df-xmet 21324  df-cfil 25244

This theorem is referenced by:  iscfil  25254