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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 6510 | . . 3 ⊢ (ClWalks‘𝐺) ∈ V | |
2 | 1 | rabex 5088 | . 2 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ∈ V |
3 | fvex 6510 | . 2 ⊢ (ClWWalks‘𝐺) ∈ V | |
4 | 2fveq3 6502 | . . . . 5 ⊢ (𝑤 = 𝑢 → (♯‘(1st ‘𝑤)) = (♯‘(1st ‘𝑢))) | |
5 | 4 | breq2d 4938 | . . . 4 ⊢ (𝑤 = 𝑢 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ (♯‘(1st ‘𝑢)))) |
6 | 5 | cbvrabv 3407 | . . 3 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} = {𝑢 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑢))} |
7 | fveq2 6497 | . . . . 5 ⊢ (𝑑 = 𝑐 → (2nd ‘𝑑) = (2nd ‘𝑐)) | |
8 | 2fveq3 6502 | . . . . . 6 ⊢ (𝑑 = 𝑐 → (♯‘(2nd ‘𝑑)) = (♯‘(2nd ‘𝑐))) | |
9 | 8 | oveq1d 6990 | . . . . 5 ⊢ (𝑑 = 𝑐 → ((♯‘(2nd ‘𝑑)) − 1) = ((♯‘(2nd ‘𝑐)) − 1)) |
10 | 7, 9 | oveq12d 6993 | . . . 4 ⊢ (𝑑 = 𝑐 → ((2nd ‘𝑑) prefix ((♯‘(2nd ‘𝑑)) − 1)) = ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) |
11 | 10 | cbvmptv 5025 | . . 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 27541 | . 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 8322 | . 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 1445 | 1 ⊢ (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ≈ (ClWWalks‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2051 {crab 3087 Vcvv 3410 class class class wbr 4926 ↦ cmpt 5005 –1-1-onto→wf1o 6185 ‘cfv 6186 (class class class)co 6975 1st c1st 7498 2nd c2nd 7499 ≈ cen 8302 1c1 10335 ≤ cle 10474 − cmin 10669 ♯chash 13504 prefix cpfx 13851 USPGraphcuspgr 26652 ClWalkscclwlks 27275 ClWWalkscclwwlk 27503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-ifp 1045 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-2o 7905 df-oadd 7908 df-er 8088 df-map 8207 df-pm 8208 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-dju 9123 df-card 9161 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-n0 11707 df-xnn0 11779 df-z 11793 df-uz 12058 df-rp 12204 df-fz 12708 df-fzo 12849 df-hash 13505 df-word 13672 df-lsw 13725 df-concat 13733 df-s1 13758 df-substr 13803 df-pfx 13852 df-edg 26552 df-uhgr 26562 df-upgr 26586 df-uspgr 26654 df-wlks 27100 df-clwlks 27276 df-clwwlk 27504 |
This theorem is referenced by: (None) |
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