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Theorem cfilucfil3 23925

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 22960	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))
2		cfilucfil2 23173	. . . . 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 22458	. . . . . . 7 ⊢ (𝐶 ∈ (Fil‘𝑋) → 𝐶 ∈ (fBas‘𝑋))
5	4	pm4.71i 562	. . . . . 6 ⊢ (𝐶 ∈ (Fil‘𝑋) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (fBas‘𝑋)))
6	5	anbi1i 625	. . . . 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 471	. . . . 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 278	. . . 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	syl6bb 289	. . 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 594	. 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 23870	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐶 ∈ (CauFil‘𝐷) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
12	11	adantl 484	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐶 ∈ (CauFil‘𝐷) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
13	10, 12	bitr4d 284	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 208 ∧ wa 398 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 ∅c0 4293 × cxp 5555 “ cima 5560 ‘cfv 6357 (class class class)co 7158 0cc0 10539 ℝ⁺crp 12392 [,)cico 12743 PsMetcpsmet 20531 ∞Metcxmet 20532 fBascfbas 20535 metUnifcmetu 20538 Filcfil 22455 CauFil_uccfilu 22897 CauFilccfil 23857

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ico 12747 df-psmet 20539 df-xmet 20540 df-fbas 20544 df-fg 20545 df-metu 20546 df-fil 22456 df-ust 22811 df-cfilu 22898 df-cfil 23860

This theorem is referenced by: cfilucfil4 23926

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