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Theorem wlknwwlksnen 30096

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 2763	. . 3 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}
2		eqid 2763	. . 3 ⊢ (𝑁 WWalksN 𝐺) = (𝑁 WWalksN 𝐺)
3		eqid 2763	. . 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 30095	. 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 6880	. . . 4 ⊢ (Walks‘𝐺) ∈ V
6	5	rabex 5296	. . 3 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ∈ V
7	6	f1oen 8953	. 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 399 = wceq 1561 ∈ wcel 2143 {crab 3415 class class class wbr 5101 ↦ cmpt 5182 –1-1-onto→wf1o 6520 ‘cfv 6521 (class class class)co 7396 1^st c1st 7968 2^nd c2nd 7969 ≈ cen 8924 ℕ₀cn0 12491 ♯chash 14353 USPGraphcuspgr 29356 Walkscwlks 29804 WWalksN cwwlksn 30033

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-ifp 1075 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8678 df-map 8810 df-pm 8811 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-dju 9871 df-card 9909 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-nn 12221 df-2 12290 df-n0 12492 df-xnn0 12565 df-z 12579 df-uz 12850 df-fz 13523 df-fzo 13670 df-hash 14354 df-word 14537 df-edg 29256 df-uhgr 29266 df-upgr 29290 df-uspgr 29358 df-wlks 29807 df-wwlks 30037 df-wwlksn 30038

This theorem is referenced by: wlknwwlksneqs 30097 wlksnfi 30114

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