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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 6683 | . . 3 ⊢ (ClWalks‘𝐺) ∈ V | |
2 | 1 | rabex 5235 | . 2 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ∈ V |
3 | fvex 6683 | . 2 ⊢ (ClWWalks‘𝐺) ∈ V | |
4 | 2fveq3 6675 | . . . . 5 ⊢ (𝑤 = 𝑢 → (♯‘(1st ‘𝑤)) = (♯‘(1st ‘𝑢))) | |
5 | 4 | breq2d 5078 | . . . 4 ⊢ (𝑤 = 𝑢 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ (♯‘(1st ‘𝑢)))) |
6 | 5 | cbvrabv 3491 | . . 3 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} = {𝑢 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑢))} |
7 | fveq2 6670 | . . . . 5 ⊢ (𝑑 = 𝑐 → (2nd ‘𝑑) = (2nd ‘𝑐)) | |
8 | 2fveq3 6675 | . . . . . 6 ⊢ (𝑑 = 𝑐 → (♯‘(2nd ‘𝑑)) = (♯‘(2nd ‘𝑐))) | |
9 | 8 | oveq1d 7171 | . . . . 5 ⊢ (𝑑 = 𝑐 → ((♯‘(2nd ‘𝑑)) − 1) = ((♯‘(2nd ‘𝑐)) − 1)) |
10 | 7, 9 | oveq12d 7174 | . . . 4 ⊢ (𝑑 = 𝑐 → ((2nd ‘𝑑) prefix ((♯‘(2nd ‘𝑑)) − 1)) = ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) |
11 | 10 | cbvmptv 5169 | . . 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 27789 | . 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 8526 | . 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 2114 {crab 3142 Vcvv 3494 class class class wbr 5066 ↦ cmpt 5146 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 2nd c2nd 7688 ≈ cen 8506 1c1 10538 ≤ cle 10676 − cmin 10870 ♯chash 13691 prefix cpfx 14032 USPGraphcuspgr 26933 ClWalkscclwlks 27551 ClWWalkscclwwlk 27759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-lsw 13915 df-concat 13923 df-s1 13950 df-substr 14003 df-pfx 14033 df-edg 26833 df-uhgr 26843 df-upgr 26867 df-uspgr 26935 df-wlks 27381 df-clwlks 27552 df-clwwlk 27760 |
This theorem is referenced by: (None) |
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