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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 28759 | . 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺)) |
4 | 1, 2 | clwlkclwwlkfo 28758 | . 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–onto→(ClWWalks‘𝐺)) |
5 | df-f1o 6499 | . 2 ⊢ (𝐹:𝐶–1-1-onto→(ClWWalks‘𝐺) ↔ (𝐹:𝐶–1-1→(ClWWalks‘𝐺) ∧ 𝐹:𝐶–onto→(ClWWalks‘𝐺))) | |
6 | 3, 4, 5 | sylanbrc 584 | 1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1-onto→(ClWWalks‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3406 class class class wbr 5104 ↦ cmpt 5187 –1-1→wf1 6489 –onto→wfo 6490 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7350 1st c1st 7910 2nd c2nd 7911 1c1 10986 ≤ cle 11124 − cmin 11319 ♯chash 14159 prefix cpfx 14491 USPGraphcuspgr 27904 ClWalkscclwlks 28523 ClWWalkscclwwlk 28730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-er 8582 df-map 8701 df-pm 8702 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-dju 9771 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-n0 12348 df-xnn0 12420 df-z 12434 df-uz 12698 df-rp 12846 df-fz 13355 df-fzo 13498 df-hash 14160 df-word 14332 df-lsw 14380 df-concat 14388 df-s1 14413 df-substr 14462 df-pfx 14492 df-edg 27804 df-uhgr 27814 df-upgr 27838 df-uspgr 27906 df-wlks 28352 df-clwlks 28524 df-clwwlk 28731 |
This theorem is referenced by: clwlkclwwlken 28761 clwlknf1oclwwlkn 28833 |
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