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Mirrors > Home > MPE Home > Th. List > clwlkclwwlken | Structured version Visualization version GIF version |
Description: The set of the nonempty closed walks and the set of closed walks as word are equinumerous in a simple pseudograph. (Contributed by AV, 25-May-2022.) (Proof shortened by AV, 4-Nov-2022.) |
Ref | Expression |
---|---|
clwlkclwwlken | β’ (πΊ β USPGraph β {π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))} β (ClWWalksβπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6904 | . . 3 β’ (ClWalksβπΊ) β V | |
2 | 1 | rabex 5328 | . 2 β’ {π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))} β V |
3 | fvex 6904 | . 2 β’ (ClWWalksβπΊ) β V | |
4 | 2fveq3 6896 | . . . . 5 β’ (π€ = π’ β (β―β(1st βπ€)) = (β―β(1st βπ’))) | |
5 | 4 | breq2d 5154 | . . . 4 β’ (π€ = π’ β (1 β€ (β―β(1st βπ€)) β 1 β€ (β―β(1st βπ’)))) |
6 | 5 | cbvrabv 3437 | . . 3 β’ {π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))} = {π’ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ’))} |
7 | fveq2 6891 | . . . . 5 β’ (π = π β (2nd βπ) = (2nd βπ)) | |
8 | 2fveq3 6896 | . . . . . 6 β’ (π = π β (β―β(2nd βπ)) = (β―β(2nd βπ))) | |
9 | 8 | oveq1d 7429 | . . . . 5 β’ (π = π β ((β―β(2nd βπ)) β 1) = ((β―β(2nd βπ)) β 1)) |
10 | 7, 9 | oveq12d 7432 | . . . 4 β’ (π = π β ((2nd βπ) prefix ((β―β(2nd βπ)) β 1)) = ((2nd βπ) prefix ((β―β(2nd βπ)) β 1))) |
11 | 10 | cbvmptv 5255 | . . 3 β’ (π β {π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))} β¦ ((2nd βπ) prefix ((β―β(2nd βπ)) β 1))) = (π β {π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))} β¦ ((2nd βπ) prefix ((β―β(2nd βπ)) β 1))) |
12 | 6, 11 | clwlkclwwlkf1o 29808 | . 2 β’ (πΊ β USPGraph β (π β {π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))} β¦ ((2nd βπ) prefix ((β―β(2nd βπ)) β 1))):{π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))}β1-1-ontoβ(ClWWalksβπΊ)) |
13 | f1oen2g 8980 | . 2 β’ (({π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))} β V β§ (ClWWalksβπΊ) β V β§ (π β {π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))} β¦ ((2nd βπ) prefix ((β―β(2nd βπ)) β 1))):{π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))}β1-1-ontoβ(ClWWalksβπΊ)) β {π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))} β (ClWWalksβπΊ)) | |
14 | 2, 3, 12, 13 | mp3an12i 1462 | 1 β’ (πΊ β USPGraph β {π€ β (ClWalksβπΊ) β£ 1 β€ (β―β(1st βπ€))} β (ClWWalksβπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2099 {crab 3427 Vcvv 3469 class class class wbr 5142 β¦ cmpt 5225 β1-1-ontoβwf1o 6541 βcfv 6542 (class class class)co 7414 1st c1st 7985 2nd c2nd 7986 β cen 8952 1c1 11131 β€ cle 11271 β cmin 11466 β―chash 14313 prefix cpfx 14644 USPGraphcuspgr 28948 ClWalkscclwlks 29571 ClWWalkscclwwlk 29778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8718 df-map 8838 df-pm 8839 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-hash 14314 df-word 14489 df-lsw 14537 df-concat 14545 df-s1 14570 df-substr 14615 df-pfx 14645 df-edg 28848 df-uhgr 28858 df-upgr 28882 df-uspgr 28950 df-wlks 29400 df-clwlks 29572 df-clwwlk 29779 |
This theorem is referenced by: (None) |
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