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Theorem clwlkclwwlken 29809

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 6904	. . 3 ⊢ (ClWalks‘𝐺) ∈ V
2	1	rabex 5328	. 2 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∈ V
3		fvex 6904	. 2 ⊢ (ClWWalks‘𝐺) ∈ V
4		2fveq3 6896	. . . . 5 ⊢ (𝑤 = 𝑢 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑢)))
5	4	breq2d 5154	. . . 4 ⊢ (𝑤 = 𝑢 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑢))))
6	5	cbvrabv 3437	. . 3 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} = {𝑢 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑢))}
7		fveq2 6891	. . . . 5 ⊢ (𝑑 = 𝑐 → (2^nd ‘𝑑) = (2^nd ‘𝑐))
8		2fveq3 6896	. . . . . 6 ⊢ (𝑑 = 𝑐 → (♯‘(2^nd ‘𝑑)) = (♯‘(2^nd ‘𝑐)))
9	8	oveq1d 7429	. . . . 5 ⊢ (𝑑 = 𝑐 → ((♯‘(2^nd ‘𝑑)) − 1) = ((♯‘(2^nd ‘𝑐)) − 1))
10	7, 9	oveq12d 7432	. . . 4 ⊢ (𝑑 = 𝑐 → ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1)) = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
11	10	cbvmptv 5255	. . 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 29808	. 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 8980	. 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 1462	1 ⊢ (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ≈ (ClWWalks‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2099 {crab 3427 Vcvv 3469 class class class wbr 5142 ↦ cmpt 5225 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7414 1^st c1st 7985 2^nd c2nd 7986 ≈ cen 8952 1c1 11131 ≤ cle 11271 − cmin 11466 ♯chash 14313 prefix cpfx 14644 USPGraphcuspgr 28948 ClWalkscclwlks 29571 ClWWalkscclwwlk 29778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8718 df-map 8838 df-pm 8839 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-hash 14314 df-word 14489 df-lsw 14537 df-concat 14545 df-s1 14570 df-substr 14615 df-pfx 14645 df-edg 28848 df-uhgr 28858 df-upgr 28882 df-uspgr 28950 df-wlks 29400 df-clwlks 29572 df-clwwlk 29779

This theorem is referenced by: (None)

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