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Theorem cfilucfil3 25355

Description: Given a metric 𝐷 and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the Cauchy filters for the metric. (Contributed by Thierry Arnoux, 15-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

**Assertion**
Ref	Expression
cfilucfil3	⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFil_u‘(metUnif‘𝐷))) ↔ 𝐶 ∈ (CauFil‘𝐷)))

Proof of Theorem cfilucfil3 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmetpsmet 24359	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))
2		cfilucfil2 24575	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐶 ∈ (CauFil_u‘(metUnif‘𝐷)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
3	2	anbi2d 630	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFil_u‘(metUnif‘𝐷))) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))))
4		filfbas 23857	. . . . . . 7 ⊢ (𝐶 ∈ (Fil‘𝑋) → 𝐶 ∈ (fBas‘𝑋))
5	4	pm4.71i 559	. . . . . 6 ⊢ (𝐶 ∈ (Fil‘𝑋) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (fBas‘𝑋)))
6	5	anbi1i 624	. . . . 5 ⊢ ((𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)) ↔ ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
7		anass 468	. . . . 5 ⊢ (((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
8	6, 7	bitr2i 276	. . . 4 ⊢ ((𝐶 ∈ (Fil‘𝑋) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
9	3, 8	bitrdi 287	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFil_u‘(metUnif‘𝐷))) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
10	1, 9	sylan2 593	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFil_u‘(metUnif‘𝐷))) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
11		iscfil 25300	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐶 ∈ (CauFil‘𝐷) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
12	11	adantl 481	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐶 ∈ (CauFil‘𝐷) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
13	10, 12	bitr4d 282	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFil_u‘(metUnif‘𝐷))) ↔ 𝐶 ∈ (CauFil‘𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ⊆ wss 3950 ∅c0 4332 × cxp 5682 “ cima 5687 ‘cfv 6560 (class class class)co 7432 0cc0 11156 ℝ⁺crp 13035 [,)cico 13390 PsMetcpsmet 21349 ∞Metcxmet 21350 fBascfbas 21353 metUnifcmetu 21356 Filcfil 23854 CauFil_uccfilu 24296 CauFilccfil 25287

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 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ico 13394 df-psmet 21357 df-xmet 21358 df-fbas 21362 df-fg 21363 df-metu 21364 df-fil 23855 df-ust 24210 df-cfilu 24297 df-cfil 25290

This theorem is referenced by: cfilucfil4 25356

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