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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 22599 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
2 | fbelss 22584 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) | |
3 | 1, 2 | sylan 583 | 1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2114 ⊆ wss 3843 ‘cfv 6339 fBascfbas 20205 Filcfil 22596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fv 6347 df-fbas 20214 df-fil 22597 |
This theorem is referenced by: filin 22605 filtop 22606 filuni 22636 trfil2 22638 trfil3 22639 fgtr 22641 trfg 22642 ufilmax 22658 isufil2 22659 ufileu 22670 filufint 22671 cfinufil 22679 ufilen 22681 rnelfm 22704 fmfnfmlem4 22708 fmid 22711 flimclsi 22729 flimrest 22734 txflf 22757 fclsopn 22765 fclsrest 22775 flimfnfcls 22779 fclscmpi 22780 iscfil2 24018 cfil3i 24021 iscmet3lem2 24044 iscmet3 24045 cfilresi 24047 cfilres 24048 filnetlem3 34212 |
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