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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 23855 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∩ cin 3950 ∅c0 4333 𝒫 cpw 4600 ‘cfv 6561 fBascfbas 21352 Filcfil 23853 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-fil 23854 |
| This theorem is referenced by: 0nelfil 23857 filsspw 23859 filelss 23860 filin 23862 filtop 23863 snfbas 23874 fgfil 23883 elfilss 23884 filfinnfr 23885 fgabs 23887 filconn 23891 fgtr 23898 trfg 23899 ufilb 23914 ufilmax 23915 isufil2 23916 ssufl 23926 ufileu 23927 filufint 23928 ufilen 23938 fmfg 23957 fmufil 23967 fmid 23968 fmco 23969 ufldom 23970 hausflim 23989 flimrest 23991 flimclslem 23992 flfnei 23999 isflf 24001 flfcnp 24012 fclsrest 24032 fclsfnflim 24035 flimfnfcls 24036 isfcf 24042 cnpfcfi 24048 cnpfcf 24049 cnextcn 24075 cfilufg 24302 neipcfilu 24305 cnextucn 24312 ucnextcn 24313 cfilresi 25329 cfilres 25330 cmetss 25350 relcmpcmet 25352 cfilucfil3 25354 minveclem4a 25464 filnetlem4 36382 |
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