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Mirrors > Home > MPE Home > Th. List > clwlkclwwlkf1o | Structured version Visualization version GIF version |
Description: πΉ is a bijection between the nonempty closed walks and the closed walks as words in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 29-Oct-2022.) |
Ref | Expression |
---|---|
clwlkclwwlkf.c | β’ πΆ = {π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))} |
clwlkclwwlkf.f | β’ πΉ = (π β πΆ β¦ ((2nd βπ) prefix ((β―β(2nd βπ)) β 1))) |
Ref | Expression |
---|---|
clwlkclwwlkf1o | β’ (πΊ β USPGraph β πΉ:πΆβ1-1-ontoβ(ClWWalksβπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwlkclwwlkf.c | . . 3 β’ πΆ = {π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))} | |
2 | clwlkclwwlkf.f | . . 3 β’ πΉ = (π β πΆ β¦ ((2nd βπ) prefix ((β―β(2nd βπ)) β 1))) | |
3 | 1, 2 | clwlkclwwlkf1 28662 | . 2 β’ (πΊ β USPGraph β πΉ:πΆβ1-1β(ClWWalksβπΊ)) |
4 | 1, 2 | clwlkclwwlkfo 28661 | . 2 β’ (πΊ β USPGraph β πΉ:πΆβontoβ(ClWWalksβπΊ)) |
5 | df-f1o 6486 | . 2 β’ (πΉ:πΆβ1-1-ontoβ(ClWWalksβπΊ) β (πΉ:πΆβ1-1β(ClWWalksβπΊ) β§ πΉ:πΆβontoβ(ClWWalksβπΊ))) | |
6 | 3, 4, 5 | sylanbrc 583 | 1 β’ (πΊ β USPGraph β πΉ:πΆβ1-1-ontoβ(ClWWalksβπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 {crab 3403 class class class wbr 5092 β¦ cmpt 5175 β1-1βwf1 6476 βontoβwfo 6477 β1-1-ontoβwf1o 6478 βcfv 6479 (class class class)co 7337 1st c1st 7897 2nd c2nd 7898 1c1 10973 β€ cle 11111 β cmin 11306 β―chash 14145 prefix cpfx 14481 USPGraphcuspgr 27807 ClWalkscclwlks 28426 ClWWalkscclwwlk 28633 |
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 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-2o 8368 df-oadd 8371 df-er 8569 df-map 8688 df-pm 8689 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-dju 9758 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-n0 12335 df-xnn0 12407 df-z 12421 df-uz 12684 df-rp 12832 df-fz 13341 df-fzo 13484 df-hash 14146 df-word 14318 df-lsw 14366 df-concat 14374 df-s1 14400 df-substr 14452 df-pfx 14482 df-edg 27707 df-uhgr 27717 df-upgr 27741 df-uspgr 27809 df-wlks 28255 df-clwlks 28427 df-clwwlk 28634 |
This theorem is referenced by: clwlkclwwlken 28664 clwlknf1oclwwlkn 28736 |
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