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Mirrors > Home > MPE Home > Th. List > clwwlkf1o | Structured version Visualization version GIF version |
Description: F is a 1-1 onto function, that means that there is a bijection between the set of closed walks of a fixed length represented by walks (as words) and the set of closed walks (as words) of the fixed length. The difference between these two representations is that in the first case the starting vertex is repeated at the end of the word, and in the second case it is not. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.) |
Ref | Expression |
---|---|
clwwlkf1o.d | β’ π· = {π€ β (π WWalksN πΊ) β£ (lastSβπ€) = (π€β0)} |
clwwlkf1o.f | β’ πΉ = (π‘ β π· β¦ (π‘ prefix π)) |
Ref | Expression |
---|---|
clwwlkf1o | β’ (π β β β πΉ:π·β1-1-ontoβ(π ClWWalksN πΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwwlkf1o.d | . . 3 β’ π· = {π€ β (π WWalksN πΊ) β£ (lastSβπ€) = (π€β0)} | |
2 | clwwlkf1o.f | . . 3 β’ πΉ = (π‘ β π· β¦ (π‘ prefix π)) | |
3 | 1, 2 | clwwlkf1 29811 | . 2 β’ (π β β β πΉ:π·β1-1β(π ClWWalksN πΊ)) |
4 | 1, 2 | clwwlkfo 29812 | . 2 β’ (π β β β πΉ:π·βontoβ(π ClWWalksN πΊ)) |
5 | df-f1o 6544 | . 2 β’ (πΉ:π·β1-1-ontoβ(π ClWWalksN πΊ) β (πΉ:π·β1-1β(π ClWWalksN πΊ) β§ πΉ:π·βontoβ(π ClWWalksN πΊ))) | |
6 | 3, 4, 5 | sylanbrc 582 | 1 β’ (π β β β πΉ:π·β1-1-ontoβ(π ClWWalksN πΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {crab 3426 β¦ cmpt 5224 β1-1βwf1 6534 βontoβwfo 6535 β1-1-ontoβwf1o 6536 βcfv 6537 (class class class)co 7405 0cc0 11112 βcn 12216 lastSclsw 14518 prefix cpfx 14626 WWalksN cwwlksn 29589 ClWWalksN cclwwlkn 29786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-lsw 14519 df-concat 14527 df-s1 14552 df-substr 14597 df-pfx 14627 df-wwlks 29593 df-wwlksn 29594 df-clwwlk 29744 df-clwwlkn 29787 |
This theorem is referenced by: clwwlken 29814 clwwlkvbij 29875 |
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