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Theorem wlknwwlksnen 29979

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 2741	. . 3 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁}
2		eqid 2741	. . 3 ⊢ (𝑁 WWalksN 𝐺) = (𝑁 WWalksN 𝐺)
3		eqid 2741	. . 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 29978	. 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 6844	. . . 4 ⊢ (Walks‘𝐺) ∈ V
6	5	rabex 5270	. . 3 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑝)) = 𝑁} ∈ V
7	6	f1oen 8913	. 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 397 = wceq 1548 ∈ wcel 2121 {crab 3393 class class class wbr 5075 ↦ cmpt 5156 –1-1-onto→wf1o 6488 ‘cfv 6489 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 ≈ cen 8884 ℕ₀cn0 12432 ♯chash 14287 USPGraphcuspgr 29239 Walkscwlks 29687 WWalksN cwwlksn 29916

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-ifp 1070 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-dju 9820 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-n0 12433 df-xnn0 12506 df-z 12520 df-uz 12784 df-fz 13457 df-fzo 13604 df-hash 14288 df-word 14471 df-edg 29139 df-uhgr 29149 df-upgr 29173 df-uspgr 29241 df-wlks 29690 df-wwlks 29920 df-wwlksn 29921

This theorem is referenced by: wlknwwlksneqs 29980 wlksnfi 29997

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