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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 22999 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
2 | fbelss 22984 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) | |
3 | 1, 2 | sylan 580 | 1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ⊆ wss 3887 ‘cfv 6433 fBascfbas 20585 Filcfil 22996 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fv 6441 df-fbas 20594 df-fil 22997 |
This theorem is referenced by: filin 23005 filtop 23006 filuni 23036 trfil2 23038 trfil3 23039 fgtr 23041 trfg 23042 ufilmax 23058 isufil2 23059 ufileu 23070 filufint 23071 cfinufil 23079 ufilen 23081 rnelfm 23104 fmfnfmlem4 23108 fmid 23111 flimclsi 23129 flimrest 23134 txflf 23157 fclsopn 23165 fclsrest 23175 flimfnfcls 23179 fclscmpi 23180 iscfil2 24430 cfil3i 24433 iscmet3lem2 24456 iscmet3 24457 cfilresi 24459 cfilres 24460 filnetlem3 34569 |
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