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| Mirrors > Home > MPE Home > Th. List > mrccl | Structured version Visualization version GIF version | ||
| Description: The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
| Ref | Expression |
|---|---|
| mrcfval.f | β’ πΉ = (mrClsβπΆ) |
| Ref | Expression |
|---|---|
| mrccl | β’ ((πΆ β (Mooreβπ) β§ π β π) β (πΉβπ) β πΆ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mrcfval.f | . . . 4 β’ πΉ = (mrClsβπΆ) | |
| 2 | 1 | mrcf 17553 | . . 3 β’ (πΆ β (Mooreβπ) β πΉ:π« πβΆπΆ) |
| 3 | 2 | adantr 482 | . 2 β’ ((πΆ β (Mooreβπ) β§ π β π) β πΉ:π« πβΆπΆ) |
| 4 | mre1cl 17538 | . . . 4 β’ (πΆ β (Mooreβπ) β π β πΆ) | |
| 5 | elpw2g 5345 | . . . 4 β’ (π β πΆ β (π β π« π β π β π)) | |
| 6 | 4, 5 | syl 17 | . . 3 β’ (πΆ β (Mooreβπ) β (π β π« π β π β π)) |
| 7 | 6 | biimpar 479 | . 2 β’ ((πΆ β (Mooreβπ) β§ π β π) β π β π« π) |
| 8 | 3, 7 | ffvelcdmd 7088 | 1 β’ ((πΆ β (Mooreβπ) β§ π β π) β (πΉβπ) β πΆ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3949 π« cpw 4603 βΆwf 6540 βcfv 6544 Moorecmre 17526 mrClscmrc 17527 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-mre 17530 df-mrc 17531 |
| This theorem is referenced by: mrcsncl 17556 mrcidb 17559 mrcidm 17563 submrc 17572 isacs2 17597 mrelatlub 18515 mreclatBAD 18516 gsumwspan 18727 cycsubg2cl 19088 symggen 19338 odf1o1 19440 cntzspan 19712 gsumzsplit 19795 gsumzoppg 19812 gsumpt 19830 dmdprdd 19869 dprdfeq0 19892 dprdspan 19897 dprdres 19898 dprdz 19900 subgdmdprd 19904 subgdprd 19905 dprd2dlem1 19911 dprd2da 19912 dmdprdsplit2lem 19915 mrccss 21247 ismrcd2 41437 proot1mul 41941 mrelatlubALT 47620 mreclat 47622 |
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