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Mirrors > Home > MPE Home > Th. List > fbelss | Structured version Visualization version GIF version |
Description: An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
Ref | Expression |
---|---|
fbelss | ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝑋 ∈ 𝐹) → 𝑋 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fbsspw 23855 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐹 ⊆ 𝒫 𝐵) | |
2 | 1 | sselda 3994 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝑋 ∈ 𝐹) → 𝑋 ∈ 𝒫 𝐵) |
3 | 2 | elpwid 4613 | 1 ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝑋 ∈ 𝐹) → 𝑋 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ⊆ wss 3962 𝒫 cpw 4604 ‘cfv 6562 fBascfbas 21369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fv 6570 df-fbas 21378 |
This theorem is referenced by: fbdmn0 23857 filelss 23875 ssfg 23895 fgcl 23901 fbasrn 23907 fmfnfmlem4 23980 fmfnfm 23981 fmucnd 24316 cfilucfil 24587 fmcfil 25319 |
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