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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 | ⊢ (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ≈ (ClWWalks‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6678 | . . 3 ⊢ (ClWalks‘𝐺) ∈ V | |
2 | 1 | rabex 5228 | . 2 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ∈ V |
3 | fvex 6678 | . 2 ⊢ (ClWWalks‘𝐺) ∈ V | |
4 | 2fveq3 6670 | . . . . 5 ⊢ (𝑤 = 𝑢 → (♯‘(1st ‘𝑤)) = (♯‘(1st ‘𝑢))) | |
5 | 4 | breq2d 5071 | . . . 4 ⊢ (𝑤 = 𝑢 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ (♯‘(1st ‘𝑢)))) |
6 | 5 | cbvrabv 3492 | . . 3 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} = {𝑢 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑢))} |
7 | fveq2 6665 | . . . . 5 ⊢ (𝑑 = 𝑐 → (2nd ‘𝑑) = (2nd ‘𝑐)) | |
8 | 2fveq3 6670 | . . . . . 6 ⊢ (𝑑 = 𝑐 → (♯‘(2nd ‘𝑑)) = (♯‘(2nd ‘𝑐))) | |
9 | 8 | oveq1d 7165 | . . . . 5 ⊢ (𝑑 = 𝑐 → ((♯‘(2nd ‘𝑑)) − 1) = ((♯‘(2nd ‘𝑐)) − 1)) |
10 | 7, 9 | oveq12d 7168 | . . . 4 ⊢ (𝑑 = 𝑐 → ((2nd ‘𝑑) prefix ((♯‘(2nd ‘𝑑)) − 1)) = ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) |
11 | 10 | cbvmptv 5162 | . . 3 ⊢ (𝑑 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ ((2nd ‘𝑑) prefix ((♯‘(2nd ‘𝑑)) − 1))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) |
12 | 6, 11 | clwlkclwwlkf1o 27783 | . 2 ⊢ (𝐺 ∈ USPGraph → (𝑑 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ ((2nd ‘𝑑) prefix ((♯‘(2nd ‘𝑑)) − 1))):{𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}–1-1-onto→(ClWWalks‘𝐺)) |
13 | f1oen2g 8520 | . 2 ⊢ (({𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ∈ V ∧ (ClWWalks‘𝐺) ∈ V ∧ (𝑑 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ ((2nd ‘𝑑) prefix ((♯‘(2nd ‘𝑑)) − 1))):{𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}–1-1-onto→(ClWWalks‘𝐺)) → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ≈ (ClWWalks‘𝐺)) | |
14 | 2, 3, 12, 13 | mp3an12i 1461 | 1 ⊢ (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ≈ (ClWWalks‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 {crab 3142 Vcvv 3495 class class class wbr 5059 ↦ cmpt 5139 –1-1-onto→wf1o 6349 ‘cfv 6350 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 ≈ cen 8500 1c1 10532 ≤ cle 10670 − cmin 10864 ♯chash 13684 prefix cpfx 14026 USPGraphcuspgr 26927 ClWalkscclwlks 27545 ClWWalkscclwwlk 27753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-lsw 13909 df-concat 13917 df-s1 13944 df-substr 13997 df-pfx 14027 df-edg 26827 df-uhgr 26837 df-upgr 26861 df-uspgr 26929 df-wlks 27375 df-clwlks 27546 df-clwwlk 27754 |
This theorem is referenced by: (None) |
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