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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 23862 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))) | |
2 | 1 | simprbda 501 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 × cxp 5548 “ cima 5553 ‘cfv 6350 (class class class)co 7150 0cc0 10531 ℝ+crp 12383 [,)cico 12734 ∞Metcxmet 20524 Filcfil 22447 CauFilccfil 23849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-xr 10673 df-xmet 20532 df-cfil 23852 |
This theorem is referenced by: cfil3i 23866 iscfil3 23870 cfilfcls 23871 iscmet3 23890 cfilresi 23892 cmetss 23913 relcmpcmet 23915 cfilucfil4 23918 fmcncfil 31169 |
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