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Theorem fgcfil 24788

Description: The Cauchy filter condition for a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)

**Assertion**
Ref	Expression
fgcfil	⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐵) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))

Distinct variable groups: 𝑥,𝑤,𝑦,𝑧,𝐵 𝑤,𝑋,𝑥,𝑦,𝑧 𝑤,𝐷,𝑥,𝑦,𝑧

Proof of Theorem fgcfil Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfili 24785	. . . . . 6 ⊢ (((𝑋filGen𝐵) ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ ℝ⁺) → ∃𝑢 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥)
2	1	adantll 713	. . . . 5 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ (𝑋filGen𝐵) ∈ (CauFil‘𝐷)) ∧ 𝑥 ∈ ℝ⁺) → ∃𝑢 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥)
3		elfg 23375	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑢 ∈ (𝑋filGen𝐵) ↔ (𝑢 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑢)))
4	3	ad3antlr 730	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ (𝑋filGen𝐵) ∈ (CauFil‘𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑢 ∈ (𝑋filGen𝐵) ↔ (𝑢 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑢)))
5		ssralv 4051	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ 𝑢 → (∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
6	5	ralimdv 3170	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑢 → (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
7		ssralv 4051	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑢 → (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥 → ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
8	6, 7	syldc 48	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → (𝑦 ⊆ 𝑢 → ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
9	8	reximdv 3171	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → (∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑢 → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
10	9	com12 32	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑢 → (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
11	10	adantl 483	. . . . . . 7 ⊢ ((𝑢 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑢) → (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
12	4, 11	syl6bi 253	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ (𝑋filGen𝐵) ∈ (CauFil‘𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑢 ∈ (𝑋filGen𝐵) → (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)))
13	12	rexlimdv 3154	. . . . 5 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ (𝑋filGen𝐵) ∈ (CauFil‘𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (∃𝑢 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
14	2, 13	mpd 15	. . . 4 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ (𝑋filGen𝐵) ∈ (CauFil‘𝐷)) ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)
15	14	ralrimiva 3147	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ (𝑋filGen𝐵) ∈ (CauFil‘𝐷)) → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)
16	15	ex 414	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐵) ∈ (CauFil‘𝐷) → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
17		ssfg 23376	. . . . . 6 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
18	17	adantl 483	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → 𝐵 ⊆ (𝑋filGen𝐵))
19		ssrexv 4052	. . . . . 6 ⊢ (𝐵 ⊆ (𝑋filGen𝐵) → (∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥 → ∃𝑦 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
20	19	ralimdv 3170	. . . . 5 ⊢ (𝐵 ⊆ (𝑋filGen𝐵) → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥 → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
21	18, 20	syl 17	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥 → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))
22		fgcl 23382	. . . . 5 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑋filGen𝐵) ∈ (Fil‘𝑋))
23	22	adantl 483	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝑋filGen𝐵) ∈ (Fil‘𝑋))
24	21, 23	jctild 527	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥 → ((𝑋filGen𝐵) ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)))
25		iscfil2 24783	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝑋filGen𝐵) ∈ (CauFil‘𝐷) ↔ ((𝑋filGen𝐵) ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)))
26	25	adantr 482	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐵) ∈ (CauFil‘𝐷) ↔ ((𝑋filGen𝐵) ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)))
27	24, 26	sylibrd 259	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥 → (𝑋filGen𝐵) ∈ (CauFil‘𝐷)))
28	16, 27	impbid 211	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐵) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 < clt 11248 ℝ⁺crp 12974 ∞Metcxmet 20929 fBascfbas 20932 filGencfg 20933 Filcfil 23349 CauFilccfil 24769

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-2 12275 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ico 13330 df-xmet 20937 df-fbas 20941 df-fg 20942 df-fil 23350 df-cfil 24772

This theorem is referenced by: fmcfil 24789 cfilresi 24812 minveclem3 24946

W3C validator