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Mirrors > Home > MPE Home > Th. List > filin | Structured version Visualization version Unicode version |
Description: A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
Ref | Expression |
---|---|
filin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | filfbas 21652 | . . 3 | |
2 | fbasssin 21640 | . . 3 | |
3 | 1, 2 | syl3an1 1359 | . 2 |
4 | inss1 3833 | . . . . 5 | |
5 | filelss 21656 | . . . . 5 | |
6 | 4, 5 | syl5ss 3614 | . . . 4 |
7 | filss 21657 | . . . . . . . 8 | |
8 | 7 | 3exp2 1285 | . . . . . . 7 |
9 | 8 | com23 86 | . . . . . 6 |
10 | 9 | imp 445 | . . . . 5 |
11 | 10 | rexlimdv 3030 | . . . 4 |
12 | 6, 11 | syldan 487 | . . 3 |
13 | 12 | 3adant3 1081 | . 2 |
14 | 3, 13 | mpd 15 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wcel 1990 wrex 2913 cin 3573 wss 3574 cfv 5888 cfbas 19734 cfil 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-nel 2898 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-fbas 19743 df-fil 21650 |
This theorem is referenced by: isfil2 21660 filfi 21663 filinn0 21664 infil 21667 filconn 21687 filuni 21689 trfil2 21691 trfilss 21693 ufprim 21713 filufint 21724 rnelfmlem 21756 rnelfm 21757 fmfnfmlem2 21759 fmfnfmlem3 21760 fmfnfmlem4 21761 fmfnfm 21762 txflf 21810 fclsrest 21828 metust 22363 filnetlem3 32375 |
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