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Mirrors > Home > MPE Home > Th. List > filelss | Structured version Visualization version GIF version |
Description: An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
Ref | Expression |
---|---|
filelss | ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | filfbas 22029 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
2 | fbelss 22014 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) | |
3 | 1, 2 | sylan 575 | 1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2164 ⊆ wss 3798 ‘cfv 6127 fBascfbas 20101 Filcfil 22026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fv 6135 df-fbas 20110 df-fil 22027 |
This theorem is referenced by: filin 22035 filtop 22036 filuni 22066 trfil2 22068 trfil3 22069 fgtr 22071 trfg 22072 ufilmax 22088 isufil2 22089 ufileu 22100 filufint 22101 cfinufil 22109 ufilen 22111 rnelfm 22134 fmfnfmlem4 22138 fmid 22141 flimclsi 22159 flimrest 22164 txflf 22187 fclsopn 22195 fclsrest 22205 flimfnfcls 22209 fclscmpi 22210 iscfil2 23441 cfil3i 23444 iscmet3lem2 23467 iscmet3 23468 cfilresi 23470 cfilres 23471 filnetlem3 32908 |
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