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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 17537 | . . . 4 β’ ((πΆ β (Mooreβπ) β§ π β πΆ) β π β π) | |
| 2 | 1 | 3adant2 1132 | . . 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 17560 | . . 3 β’ ((πΆ β (Mooreβπ) β§ π β π β§ π β π) β (πΉβπ) β (πΉβπ)) |
| 5 | 2, 4 | syld3an3 1410 | . 2 β’ ((πΆ β (Mooreβπ) β§ π β π β§ π β πΆ) β (πΉβπ) β (πΉβπ)) |
| 6 | 3 | mrcid 17557 | . . 3 β’ ((πΆ β (Mooreβπ) β§ π β πΆ) β (πΉβπ) = π) |
| 7 | 6 | 3adant2 1132 | . 2 β’ ((πΆ β (Mooreβπ) β§ π β π β§ π β πΆ) β (πΉβπ) = π) |
| 8 | 5, 7 | sseqtrd 4023 | 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 1088 = wceq 1542 β wcel 2107 β wss 3949 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-sbc 3779 df-csb 3895 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: submrc 17572 isacs2 17597 isacs3lem 18495 mrelatlub 18515 mndind 18709 gsumwspan 18727 symggen 19338 cntzspan 19712 dprdspan 19897 subgdmdprd 19904 subgdprd 19905 dprdsn 19906 dprd2dlem1 19911 dprd2da 19912 dmdprdsplit2lem 19915 ablfac1b 19940 pgpfac1lem1 19944 pgpfac1lem5 19949 mrccss 21247 evlseu 21646 ismrcd2 41437 mrefg3 41446 isnacs3 41448 |
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