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Theorem clwlkclwwlken 29993

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.)

**Assertion**
Ref	Expression
clwlkclwwlken	⊢ (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ≈ (ClWWalks‘𝐺))

Distinct variable group: 𝑤,𝐺

Proof of Theorem clwlkclwwlken Dummy variables 𝑐 𝑑 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6889	. . 3 ⊢ (ClWalks‘𝐺) ∈ V
2	1	rabex 5309	. 2 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∈ V
3		fvex 6889	. 2 ⊢ (ClWWalks‘𝐺) ∈ V
4		2fveq3 6881	. . . . 5 ⊢ (𝑤 = 𝑢 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑢)))
5	4	breq2d 5131	. . . 4 ⊢ (𝑤 = 𝑢 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑢))))
6	5	cbvrabv 3426	. . 3 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} = {𝑢 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑢))}
7		fveq2 6876	. . . . 5 ⊢ (𝑑 = 𝑐 → (2^nd ‘𝑑) = (2^nd ‘𝑐))
8		2fveq3 6881	. . . . . 6 ⊢ (𝑑 = 𝑐 → (♯‘(2^nd ‘𝑑)) = (♯‘(2^nd ‘𝑐)))
9	8	oveq1d 7420	. . . . 5 ⊢ (𝑑 = 𝑐 → ((♯‘(2^nd ‘𝑑)) − 1) = ((♯‘(2^nd ‘𝑐)) − 1))
10	7, 9	oveq12d 7423	. . . 4 ⊢ (𝑑 = 𝑐 → ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1)) = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
11	10	cbvmptv 5225	. . 3 ⊢ (𝑑 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
12	6, 11	clwlkclwwlkf1o 29992	. 2 ⊢ (𝐺 ∈ USPGraph → (𝑑 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1))):{𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}–1-1-onto→(ClWWalks‘𝐺))
13		f1oen2g 8983	. 2 ⊢ (({𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∈ V ∧ (ClWWalks‘𝐺) ∈ V ∧ (𝑑 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1))):{𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}–1-1-onto→(ClWWalks‘𝐺)) → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ≈ (ClWWalks‘𝐺))
14	2, 3, 12, 13	mp3an12i 1467	1 ⊢ (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ≈ (ClWWalks‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2108 {crab 3415 Vcvv 3459 class class class wbr 5119 ↦ cmpt 5201 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7405 1^st c1st 7986 2^nd c2nd 7987 ≈ cen 8956 1c1 11130 ≤ cle 11270 − cmin 11466 ♯chash 14348 prefix cpfx 14688 USPGraphcuspgr 29127 ClWalkscclwlks 29752 ClWWalkscclwwlk 29962

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8719 df-map 8842 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-n0 12502 df-xnn0 12575 df-z 12589 df-uz 12853 df-rp 13009 df-fz 13525 df-fzo 13672 df-hash 14349 df-word 14532 df-lsw 14581 df-concat 14589 df-s1 14614 df-substr 14659 df-pfx 14689 df-edg 29027 df-uhgr 29037 df-upgr 29061 df-uspgr 29129 df-wlks 29579 df-clwlks 29753 df-clwwlk 29963

This theorem is referenced by: (None)

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