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| Mirrors > Home > MPE Home > Th. List > filsspw | Structured version Visualization version GIF version | ||
| Description: A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.) | 
| Ref | Expression | 
|---|---|
| filsspw | ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | filfbas 23857 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
| 2 | fbsspw 23841 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ 𝒫 𝑋) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ⊆ wss 3950 𝒫 cpw 4599 ‘cfv 6560 fBascfbas 21353 Filcfil 23854 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-fbas 21362 df-fil 23855 | 
| This theorem is referenced by: isfil2 23865 infil 23872 filunibas 23890 trfg 23900 isufil2 23917 filssufilg 23920 ssufl 23927 ufileu 23928 filufint 23929 uffixfr 23932 elflim 23980 fclsfnflim 24036 flimfnfcls 24037 metust 24572 cfilresi 25330 cmetss 25351 | 
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