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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 6781 | . . 3 ⊢ (ClWalks‘𝐺) ∈ V | |
2 | 1 | rabex 5259 | . 2 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ∈ V |
3 | fvex 6781 | . 2 ⊢ (ClWWalks‘𝐺) ∈ V | |
4 | 2fveq3 6773 | . . . . 5 ⊢ (𝑤 = 𝑢 → (♯‘(1st ‘𝑤)) = (♯‘(1st ‘𝑢))) | |
5 | 4 | breq2d 5090 | . . . 4 ⊢ (𝑤 = 𝑢 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ (♯‘(1st ‘𝑢)))) |
6 | 5 | cbvrabv 3424 | . . 3 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} = {𝑢 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑢))} |
7 | fveq2 6768 | . . . . 5 ⊢ (𝑑 = 𝑐 → (2nd ‘𝑑) = (2nd ‘𝑐)) | |
8 | 2fveq3 6773 | . . . . . 6 ⊢ (𝑑 = 𝑐 → (♯‘(2nd ‘𝑑)) = (♯‘(2nd ‘𝑐))) | |
9 | 8 | oveq1d 7283 | . . . . 5 ⊢ (𝑑 = 𝑐 → ((♯‘(2nd ‘𝑑)) − 1) = ((♯‘(2nd ‘𝑐)) − 1)) |
10 | 7, 9 | oveq12d 7286 | . . . 4 ⊢ (𝑑 = 𝑐 → ((2nd ‘𝑑) prefix ((♯‘(2nd ‘𝑑)) − 1)) = ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) |
11 | 10 | cbvmptv 5191 | . . 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 28354 | . 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 8727 | . 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 1463 | 1 ⊢ (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ≈ (ClWWalks‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2109 {crab 3069 Vcvv 3430 class class class wbr 5078 ↦ cmpt 5161 –1-1-onto→wf1o 6429 ‘cfv 6430 (class class class)co 7268 1st c1st 7815 2nd c2nd 7816 ≈ cen 8704 1c1 10856 ≤ cle 10994 − cmin 11188 ♯chash 14025 prefix cpfx 14364 USPGraphcuspgr 27499 ClWalkscclwlks 28117 ClWWalkscclwwlk 28324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-oadd 8285 df-er 8472 df-map 8591 df-pm 8592 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-dju 9643 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-n0 12217 df-xnn0 12289 df-z 12303 df-uz 12565 df-rp 12713 df-fz 13222 df-fzo 13365 df-hash 14026 df-word 14199 df-lsw 14247 df-concat 14255 df-s1 14282 df-substr 14335 df-pfx 14365 df-edg 27399 df-uhgr 27409 df-upgr 27433 df-uspgr 27501 df-wlks 27947 df-clwlks 28118 df-clwwlk 28325 |
This theorem is referenced by: (None) |
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