Theorem cfilfval 25317

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 6953	. . . 4 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
2	1	sseli 4004	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
3		dmeq 5928	. . . . . . 7 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
4	3	dmeqd 5930	. . . . . 6 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
5	4	fveq2d 6924	. . . . 5 ⊢ (𝑑 = 𝐷 → (Fil‘dom dom 𝑑) = (Fil‘dom dom 𝐷))
6		imaeq1 6084	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑑 “ (𝑦 × 𝑦)) = (𝐷 “ (𝑦 × 𝑦)))
7	6	sseq1d 4040	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
8	7	rexbidv 3185	. . . . . 6 ⊢ (𝑑 = 𝐷 → (∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
9	8	ralbidv 3184	. . . . 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 3462	. . . 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 25308	. . . 4 ⊢ CauFil = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
12		fvex 6933	. . . . 5 ⊢ (Fil‘dom dom 𝐷) ∈ V
13	12	rabex 5357	. . . 4 ⊢ {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} ∈ V
14	10, 11, 13	fvmpt 7029	. . 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 24366	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷)
17	16	fveq2d 6924	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (Fil‘𝑋) = (Fil‘dom dom 𝐷))
18	17	rabeqdv 3459	. 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 2783	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443   ⊆ wss 3976  ∪ cuni 4931   × cxp 5698  dom cdm 5700  ran crn 5701   “ cima 5703  ‘cfv 6573  (class class class)co 7448  0cc0 11184  ℝ⁺crp 13057  [,)cico 13409  ∞Metcxmet 21372  Filcfil 23874  CauFilccfil 25305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-xr 11328  df-xmet 21380  df-cfil 25308

This theorem is referenced by:  iscfil  25318