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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 23672 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
2 | fbsspw 23656 | . 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 2105 ⊆ wss 3948 𝒫 cpw 4602 ‘cfv 6543 fBascfbas 21221 Filcfil 23669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-fbas 21230 df-fil 23670 |
This theorem is referenced by: isfil2 23680 infil 23687 filunibas 23705 trfg 23715 isufil2 23732 filssufilg 23735 ssufl 23742 ufileu 23743 filufint 23744 uffixfr 23747 elflim 23795 fclsfnflim 23851 flimfnfcls 23852 metust 24387 cfilresi 25143 cmetss 25164 |
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