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Theorem clwwlknonclwlknonen 29014

Description: The sets of the two representations of closed walks of a fixed positive length on a fixed vertex are equinumerous. (Contributed by AV, 27-May-2022.) (Proof shortened by AV, 3-Nov-2022.)

**Assertion**
Ref	Expression
clwwlknonclwlknonen	⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)𝑁))

Distinct variable groups: 𝑤,𝐺 𝑤,𝑁 𝑤,𝑋

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

Step	Hyp	Ref	Expression
1		fvex 6842	. . 3 ⊢ (ClWalks‘𝐺) ∈ V
2	1	rabex 5280	. 2 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ∈ V
3		ovex 7374	. 2 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V
4		eqid 2737	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2737	. . 3 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}
6		eqid 2737	. . 3 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))))
7	4, 5, 6	clwwlknonclwlknonf1o 29013	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
8		f1oen2g 8833	. 2 ⊢ (({𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ∈ V ∧ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V ∧ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁)) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)𝑁))
9	2, 3, 7, 8	mp3an12i 1465	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3404 Vcvv 3442 class class class wbr 5096 ↦ cmpt 5179 –1-1-onto→wf1o 6482 ‘cfv 6483 (class class class)co 7341 1^st c1st 7901 2^nd c2nd 7902 ≈ cen 8805 0cc0 10976 ℕcn 12078 ♯chash 14149 prefix cpfx 14481 Vtxcvtx 27654 USPGraphcuspgr 27806 ClWalkscclwlks 28425 ClWWalksNOncclwwlknon 28738

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2708 ax-rep 5233 ax-sep 5247 ax-nul 5254 ax-pow 5312 ax-pr 5376 ax-un 7654 ax-cnex 11032 ax-resscn 11033 ax-1cn 11034 ax-icn 11035 ax-addcl 11036 ax-addrcl 11037 ax-mulcl 11038 ax-mulrcl 11039 ax-mulcom 11040 ax-addass 11041 ax-mulass 11042 ax-distr 11043 ax-i2m1 11044 ax-1ne0 11045 ax-1rid 11046 ax-rnegex 11047 ax-rrecex 11048 ax-cnre 11049 ax-pre-lttri 11050 ax-pre-lttrn 11051 ax-pre-ltadd 11052 ax-pre-mulgt0 11053

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3731 df-csb 3847 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3920 df-nul 4274 df-if 4478 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4857 df-int 4899 df-iun 4947 df-br 5097 df-opab 5159 df-mpt 5180 df-tr 5214 df-id 5522 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5579 df-we 5581 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6242 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6435 df-fun 6485 df-fn 6486 df-f 6487 df-f1 6488 df-fo 6489 df-f1o 6490 df-fv 6491 df-riota 7297 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7785 df-1st 7903 df-2nd 7904 df-frecs 8171 df-wrecs 8202 df-recs 8276 df-rdg 8315 df-1o 8371 df-2o 8372 df-oadd 8375 df-er 8573 df-map 8692 df-pm 8693 df-en 8809 df-dom 8810 df-sdom 8811 df-fin 8812 df-dju 9762 df-card 9800 df-pnf 11116 df-mnf 11117 df-xr 11118 df-ltxr 11119 df-le 11120 df-sub 11312 df-neg 11313 df-nn 12079 df-2 12141 df-n0 12339 df-xnn0 12411 df-z 12425 df-uz 12688 df-rp 12836 df-fz 13345 df-fzo 13488 df-hash 14150 df-word 14322 df-lsw 14370 df-concat 14378 df-s1 14403 df-substr 14452 df-pfx 14482 df-edg 27706 df-uhgr 27716 df-upgr 27740 df-uspgr 27808 df-wlks 28254 df-clwlks 28426 df-clwwlk 28633 df-clwwlkn 28676 df-clwwlknon 28739

This theorem is referenced by: clwlknon2num 29019 numclwlk1lem2 29021

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