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Mirrors > Home > MPE Home > Th. List > clwlkclwwlkf1o | Structured version Visualization version GIF version |
Description: 𝐹 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 | ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} |
clwlkclwwlkf.f | ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) |
Ref | Expression |
---|---|
clwlkclwwlkf1o | ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1-onto→(ClWWalks‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwlkclwwlkf.c | . . 3 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} | |
2 | clwlkclwwlkf.f | . . 3 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) | |
3 | 1, 2 | clwlkclwwlkf1 29937 | . 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺)) |
4 | 1, 2 | clwlkclwwlkfo 29936 | . 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–onto→(ClWWalks‘𝐺)) |
5 | df-f1o 6550 | . 2 ⊢ (𝐹:𝐶–1-1-onto→(ClWWalks‘𝐺) ↔ (𝐹:𝐶–1-1→(ClWWalks‘𝐺) ∧ 𝐹:𝐶–onto→(ClWWalks‘𝐺))) | |
6 | 3, 4, 5 | sylanbrc 581 | 1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1-onto→(ClWWalks‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {crab 3419 class class class wbr 5143 ↦ cmpt 5226 –1-1→wf1 6540 –onto→wfo 6541 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7413 1st c1st 7990 2nd c2nd 7991 1c1 11147 ≤ cle 11287 − cmin 11482 ♯chash 14339 prefix cpfx 14670 USPGraphcuspgr 29078 ClWalkscclwlks 29701 ClWWalkscclwwlk 29908 |
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 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8723 df-map 8846 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-dju 9934 df-card 9972 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-2 12318 df-n0 12516 df-xnn0 12588 df-z 12602 df-uz 12866 df-rp 13020 df-fz 13530 df-fzo 13673 df-hash 14340 df-word 14515 df-lsw 14563 df-concat 14571 df-s1 14596 df-substr 14641 df-pfx 14671 df-edg 28978 df-uhgr 28988 df-upgr 29012 df-uspgr 29080 df-wlks 29530 df-clwlks 29702 df-clwwlk 29909 |
This theorem is referenced by: clwlkclwwlken 29939 clwlknf1oclwwlkn 30011 |
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