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Theorem wlknwwlksnen 28241

Description: In a simple pseudograph, the set of walks of a fixed length and the set of walks represented by words are equinumerous. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 5-Aug-2022.)

**Assertion**
Ref	Expression
wlknwwlksnen	⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺))

Distinct variable groups: 𝐺,𝑝 𝑁,𝑝

Proof of Theorem wlknwwlksnen Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}
2		eqid 2738	. . 3 ⊢ (𝑁 WWalksN 𝐺) = (𝑁 WWalksN 𝐺)
3		eqid 2738	. . 3 ⊢ (𝑤 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ↦ (2^nd ‘𝑤)) = (𝑤 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ↦ (2^nd ‘𝑤))
4	1, 2, 3	wlknwwlksnbij 28240	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → (𝑤 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ↦ (2^nd ‘𝑤)):{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}–1-1-onto→(𝑁 WWalksN 𝐺))
5		fvex 6781	. . . 4 ⊢ (Walks‘𝐺) ∈ V
6	5	rabex 5256	. . 3 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ∈ V
7	6	f1oen 8750	. 2 ⊢ ((𝑤 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ↦ (2^nd ‘𝑤)):{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}–1-1-onto→(𝑁 WWalksN 𝐺) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺))
8	4, 7	syl 17	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 class class class wbr 5075 ↦ cmpt 5158 –1-1-onto→wf1o 6427 ‘cfv 6428 (class class class)co 7269 1^st c1st 7820 2^nd c2nd 7821 ≈ cen 8719 ℕ₀cn0 12222 ♯chash 14033 USPGraphcuspgr 27507 Walkscwlks 27952 WWalksN cwwlksn 28178

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-oadd 8290 df-er 8487 df-map 8606 df-pm 8607 df-en 8723 df-dom 8724 df-sdom 8725 df-fin 8726 df-dju 9648 df-card 9686 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-nn 11963 df-2 12025 df-n0 12223 df-xnn0 12295 df-z 12309 df-uz 12572 df-fz 13229 df-fzo 13372 df-hash 14034 df-word 14207 df-edg 27407 df-uhgr 27417 df-upgr 27441 df-uspgr 27509 df-wlks 27955 df-wwlks 28182 df-wwlksn 28183

This theorem is referenced by: wlknwwlksneqs 28242 wlksnfi 28259

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