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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 23765 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
2 | fbelss 23750 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) | |
3 | 1, 2 | sylan 579 | 1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ⊆ wss 3947 ‘cfv 6548 fBascfbas 21267 Filcfil 23762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fv 6556 df-fbas 21276 df-fil 23763 |
This theorem is referenced by: filin 23771 filtop 23772 filuni 23802 trfil2 23804 trfil3 23805 fgtr 23807 trfg 23808 ufilmax 23824 isufil2 23825 ufileu 23836 filufint 23837 cfinufil 23845 ufilen 23847 rnelfm 23870 fmfnfmlem4 23874 fmid 23877 flimclsi 23895 flimrest 23900 txflf 23923 fclsopn 23931 fclsrest 23941 flimfnfcls 23945 fclscmpi 23946 iscfil2 25207 cfil3i 25210 iscmet3lem2 25233 iscmet3 25234 cfilresi 25236 cfilres 25237 filnetlem3 35864 |
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