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| Mirrors > Home > MPE Home > Th. List > mrcsscl | Structured version Visualization version GIF version | ||
| Description: The closure is the minimal closed set; any closed set which contains the generators is a superset of the closure. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
| Ref | Expression |
|---|---|
| mrcfval.f | β’ πΉ = (mrClsβπΆ) |
| Ref | Expression |
|---|---|
| mrcsscl | β’ ((πΆ β (Mooreβπ) β§ π β π β§ π β πΆ) β (πΉβπ) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mress 17541 | . . . 4 β’ ((πΆ β (Mooreβπ) β§ π β πΆ) β π β π) | |
| 2 | 1 | 3adant2 1131 | . . 3 β’ ((πΆ β (Mooreβπ) β§ π β π β§ π β πΆ) β π β π) |
| 3 | mrcfval.f | . . . 4 β’ πΉ = (mrClsβπΆ) | |
| 4 | 3 | mrcss 17564 | . . 3 β’ ((πΆ β (Mooreβπ) β§ π β π β§ π β π) β (πΉβπ) β (πΉβπ)) |
| 5 | 2, 4 | syld3an3 1409 | . 2 β’ ((πΆ β (Mooreβπ) β§ π β π β§ π β πΆ) β (πΉβπ) β (πΉβπ)) |
| 6 | 3 | mrcid 17561 | . . 3 β’ ((πΆ β (Mooreβπ) β§ π β πΆ) β (πΉβπ) = π) |
| 7 | 6 | 3adant2 1131 | . 2 β’ ((πΆ β (Mooreβπ) β§ π β π β§ π β πΆ) β (πΉβπ) = π) |
| 8 | 5, 7 | sseqtrd 4022 | 1 β’ ((πΆ β (Mooreβπ) β§ π β π β§ π β πΆ) β (πΉβπ) β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1087 = wceq 1541 β wcel 2106 β wss 3948 βcfv 6543 Moorecmre 17530 mrClscmrc 17531 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-mre 17534 df-mrc 17535 |
| This theorem is referenced by: submrc 17576 isacs2 17601 isacs3lem 18499 mrelatlub 18519 mndind 18745 gsumwspan 18763 symggen 19379 cntzspan 19753 dprdspan 19938 subgdmdprd 19945 subgdprd 19946 dprdsn 19947 dprd2dlem1 19952 dprd2da 19953 dmdprdsplit2lem 19956 ablfac1b 19981 pgpfac1lem1 19985 pgpfac1lem5 19990 mrccss 21466 evlseu 21865 ismrcd2 41739 mrefg3 41748 isnacs3 41750 |
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