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Theorem cfilucfil3 25253

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 24269	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))
2		cfilucfil2 24482	. . . . 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 23768	. . . . . . 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 25198	. . 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 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ⊆ wss 3911 ∅c0 4292 × cxp 5629 “ cima 5634 ‘cfv 6499 (class class class)co 7369 0cc0 11044 ℝ⁺crp 12927 [,)cico 13284 PsMetcpsmet 21280 ∞Metcxmet 21281 fBascfbas 21284 metUnifcmetu 21287 Filcfil 23765 CauFil_uccfilu 24206 CauFilccfil 25185

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ico 13288 df-psmet 21288 df-xmet 21289 df-fbas 21293 df-fg 21294 df-metu 21295 df-fil 23766 df-ust 24121 df-cfilu 24207 df-cfil 25188

This theorem is referenced by: cfilucfil4 25254

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