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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 21651 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))) | |
2 | 1 | simplbi 476 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∩ cin 3573 ∅c0 3915 𝒫 cpw 4158 ‘cfv 5888 fBascfbas 19734 Filcfil 21649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-fil 21650 |
This theorem is referenced by: 0nelfil 21653 filsspw 21655 filelss 21656 filin 21658 filtop 21659 snfbas 21670 fgfil 21679 elfilss 21680 filfinnfr 21681 fgabs 21683 filconn 21687 fgtr 21694 trfg 21695 ufilb 21710 ufilmax 21711 isufil2 21712 ssufl 21722 ufileu 21723 filufint 21724 ufilen 21734 fmfg 21753 fmufil 21763 fmid 21764 fmco 21765 ufldom 21766 hausflim 21785 flimrest 21787 flimclslem 21788 flfnei 21795 isflf 21797 flfcnp 21808 fclsrest 21828 fclsfnflim 21831 flimfnfcls 21832 isfcf 21838 cnpfcfi 21844 cnpfcf 21845 cnextcn 21871 cfilufg 22097 neipcfilu 22100 cnextucn 22107 ucnextcn 22108 cfilresi 23093 cfilres 23094 cmetss 23113 relcmpcmet 23115 cfilucfil3 23117 minveclem4a 23201 filnetlem4 32376 |
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