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| Mirrors > Home > MPE Home > Th. List > Mathboxes > clsf2 | Structured version Visualization version GIF version | ||
| Description: The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 22871. (Contributed by RP, 22-Apr-2021.) |
| Ref | Expression |
|---|---|
| clselmap.x | β’ π = βͺ π½ |
| clselmap.k | β’ πΎ = (clsβπ½) |
| Ref | Expression |
|---|---|
| clsf2 | β’ (π½ β Top β πΎ:π« πβΆπ« π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | clselmap.x | . . . 4 β’ π = βͺ π½ | |
| 2 | 1 | clsf 22871 | . . 3 β’ (π½ β Top β (clsβπ½):π« πβΆ(Clsdβπ½)) |
| 3 | clselmap.k | . . . . 5 β’ πΎ = (clsβπ½) | |
| 4 | 3 | feq1i 6708 | . . . 4 β’ (πΎ:π« πβΆ(Clsdβπ½) β (clsβπ½):π« πβΆ(Clsdβπ½)) |
| 5 | df-f 6547 | . . . 4 β’ (πΎ:π« πβΆ(Clsdβπ½) β (πΎ Fn π« π β§ ran πΎ β (Clsdβπ½))) | |
| 6 | 4, 5 | sylbb1 236 | . . 3 β’ ((clsβπ½):π« πβΆ(Clsdβπ½) β (πΎ Fn π« π β§ ran πΎ β (Clsdβπ½))) |
| 7 | 1 | cldss2 22853 | . . . . 5 β’ (Clsdβπ½) β π« π |
| 8 | sstr2 3989 | . . . . 5 β’ (ran πΎ β (Clsdβπ½) β ((Clsdβπ½) β π« π β ran πΎ β π« π)) | |
| 9 | 7, 8 | mpi 20 | . . . 4 β’ (ran πΎ β (Clsdβπ½) β ran πΎ β π« π) |
| 10 | 9 | anim2i 616 | . . 3 β’ ((πΎ Fn π« π β§ ran πΎ β (Clsdβπ½)) β (πΎ Fn π« π β§ ran πΎ β π« π)) |
| 11 | 2, 6, 10 | 3syl 18 | . 2 β’ (π½ β Top β (πΎ Fn π« π β§ ran πΎ β π« π)) |
| 12 | df-f 6547 | . 2 β’ (πΎ:π« πβΆπ« π β (πΎ Fn π« π β§ ran πΎ β π« π)) | |
| 13 | 11, 12 | sylibr 233 | 1 β’ (π½ β Top β πΎ:π« πβΆπ« π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 β wss 3948 π« cpw 4602 βͺ cuni 4908 ran crn 5677 Fn wfn 6538 βΆwf 6539 βcfv 6543 Topctop 22714 Clsdccld 22839 clsccl 22841 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 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-iun 4999 df-iin 5000 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-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-top 22715 df-cld 22842 df-cls 22844 |
| This theorem is referenced by: clselmap 43340 |
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