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Theorem clwlkssizeeq 29602

Description: The size of the set of closed walks as words of length 𝑁 corresponds to the size of the set of closed walks of length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 4-May-2021.) (Revised by AV, 26-May-2022.) (Proof shortened by AV, 3-Nov-2022.)

**Assertion**
Ref	Expression
clwlkssizeeq	⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘(𝑁 ClWWalksN 𝐺)) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}))

Distinct variable groups: 𝑤,𝐺 𝑤,𝑁

Proof of Theorem clwlkssizeeq Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6905	. . . . 5 ⊢ (ClWalks‘𝐺) ∈ V
2	1	rabex 5333	. . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ∈ V
3	2	a1i 11	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ∈ V)
4		eqid 2731	. . . 4 ⊢ (1^st ‘𝑐) = (1^st ‘𝑐)
5		eqid 2731	. . . 4 ⊢ (2^nd ‘𝑐) = (2^nd ‘𝑐)
6		eqid 2731	. . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}
7		eqid 2731	. . . 4 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))))
8	4, 5, 6, 7	clwlknf1oclwwlkn 29601	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺))
9	3, 8	hasheqf1od 14318	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}) = (♯‘(𝑁 ClWWalksN 𝐺)))
10	9	eqcomd 2737	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘(𝑁 ClWWalksN 𝐺)) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {crab 3431 Vcvv 3473 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7412 1^st c1st 7976 2^nd c2nd 7977 ℕcn 12217 ♯chash 14295 prefix cpfx 14625 USPGraphcuspgr 28672 ClWalkscclwlks 29291 ClWWalksN cclwwlkn 29541

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-oadd 8473 df-er 8706 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-dju 9899 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-n0 12478 df-xnn0 12550 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-hash 14296 df-word 14470 df-lsw 14518 df-concat 14526 df-s1 14551 df-substr 14596 df-pfx 14626 df-edg 28572 df-uhgr 28582 df-upgr 28606 df-uspgr 28674 df-wlks 29120 df-clwlks 29292 df-clwwlk 29499 df-clwwlkn 29542

This theorem is referenced by: clwlksndivn 29603

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