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Mirrors > Home > MPE Home > Th. List > cfilfil | Structured version Visualization version GIF version |
Description: A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
cfilfil | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscfil 24162 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))) | |
2 | 1 | simprbda 502 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 ⊆ wss 3866 × cxp 5549 “ cima 5554 ‘cfv 6380 (class class class)co 7213 0cc0 10729 ℝ+crp 12586 [,)cico 12937 ∞Metcxmet 20348 Filcfil 22742 CauFilccfil 24149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-map 8510 df-xr 10871 df-xmet 20356 df-cfil 24152 |
This theorem is referenced by: cfil3i 24166 iscfil3 24170 cfilfcls 24171 iscmet3 24190 cfilresi 24192 cmetss 24213 relcmpcmet 24215 cfilucfil4 24218 fmcncfil 31595 |
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