![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 23744 | . 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 2099 ≠ wne 2935 ∀wral 3056 ∩ cin 3943 ∅c0 4318 𝒫 cpw 4598 ‘cfv 6542 fBascfbas 21260 Filcfil 23742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-fil 23743 |
This theorem is referenced by: 0nelfil 23746 filsspw 23748 filelss 23749 filin 23751 filtop 23752 snfbas 23763 fgfil 23772 elfilss 23773 filfinnfr 23774 fgabs 23776 filconn 23780 fgtr 23787 trfg 23788 ufilb 23803 ufilmax 23804 isufil2 23805 ssufl 23815 ufileu 23816 filufint 23817 ufilen 23827 fmfg 23846 fmufil 23856 fmid 23857 fmco 23858 ufldom 23859 hausflim 23878 flimrest 23880 flimclslem 23881 flfnei 23888 isflf 23890 flfcnp 23901 fclsrest 23921 fclsfnflim 23924 flimfnfcls 23925 isfcf 23931 cnpfcfi 23937 cnpfcf 23938 cnextcn 23964 cfilufg 24191 neipcfilu 24194 cnextucn 24201 ucnextcn 24202 cfilresi 25216 cfilres 25217 cmetss 25237 relcmpcmet 25239 cfilucfil3 25241 minveclem4a 25351 filnetlem4 35855 |
Copyright terms: Public domain | W3C validator |