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Mirrors > Home > MPE Home > Th. List > filfbas | Structured version Visualization version GIF version |
Description: A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.) |
Ref | Expression |
---|---|
filfbas | ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfil 22059 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))) | |
2 | 1 | simplbi 493 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2968 ∀wral 3089 ∩ cin 3790 ∅c0 4140 𝒫 cpw 4378 ‘cfv 6135 fBascfbas 20130 Filcfil 22057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fv 6143 df-fil 22058 |
This theorem is referenced by: 0nelfil 22061 filsspw 22063 filelss 22064 filin 22066 filtop 22067 snfbas 22078 fgfil 22087 elfilss 22088 filfinnfr 22089 fgabs 22091 filconn 22095 fgtr 22102 trfg 22103 ufilb 22118 ufilmax 22119 isufil2 22120 ssufl 22130 ufileu 22131 filufint 22132 ufilen 22142 fmfg 22161 fmufil 22171 fmid 22172 fmco 22173 ufldom 22174 hausflim 22193 flimrest 22195 flimclslem 22196 flfnei 22203 isflf 22205 flfcnp 22216 fclsrest 22236 fclsfnflim 22239 flimfnfcls 22240 isfcf 22246 cnpfcfi 22252 cnpfcf 22253 cnextcn 22279 cfilufg 22505 neipcfilu 22508 cnextucn 22515 ucnextcn 22516 cfilresi 23501 cfilres 23502 cmetss 23522 relcmpcmet 23524 cfilucfil3 23526 minveclem4a 23636 filnetlem4 32964 |
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