dlwwlknondlwlknonen - Metamath Proof Explorer

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Theorem dlwwlknondlwlknonen 29018

Description: The sets of the two representations of double loops of a fixed length on a fixed vertex are equinumerous. (Contributed by AV, 30-May-2022.) (Proof shortened by AV, 3-Nov-2022.)

**Hypotheses**
Ref	Expression
dlwwlknondlwlknonbij.v	⊢ 𝑉 = (Vtx‘𝐺)
dlwwlknondlwlknonbij.w	⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
dlwwlknondlwlknonbij.d	⊢ 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}

**Assertion**
Ref	Expression
dlwwlknondlwlknonen	⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → 𝑊 ≈ 𝐷)

Distinct variable groups: 𝑤,𝐺 𝑤,𝑁 𝑤,𝑋 Allowed substitution hints: 𝐷(𝑤) 𝑉(𝑤) 𝑊(𝑤)

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

Step	Hyp	Ref	Expression
1		dlwwlknondlwlknonbij.w	. . 3 ⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
2		fvex 6838	. . 3 ⊢ (ClWalks‘𝐺) ∈ V
3	1, 2	rabex2 5278	. 2 ⊢ 𝑊 ∈ V
4		dlwwlknondlwlknonbij.d	. . 3 ⊢ 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}
5		ovex 7370	. . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V
6	4, 5	rabex2 5278	. 2 ⊢ 𝐷 ∈ V
7		dlwwlknondlwlknonbij.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
8		eqid 2736	. . 3 ⊢ (𝑐 ∈ 𝑊 ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) = (𝑐 ∈ 𝑊 ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))))
9	7, 1, 4, 8	dlwwlknondlwlknonf1o 29017	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑐 ∈ 𝑊 ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))):𝑊–1-1-onto→𝐷)
10		f1oen2g 8829	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐷 ∈ V ∧ (𝑐 ∈ 𝑊 ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))):𝑊–1-1-onto→𝐷) → 𝑊 ≈ 𝐷)
11	3, 6, 9, 10	mp3an12i 1464	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → 𝑊 ≈ 𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 {crab 3403 Vcvv 3441 class class class wbr 5092 ↦ cmpt 5175 –1-1-onto→wf1o 6478 ‘cfv 6479 (class class class)co 7337 1^st c1st 7897 2^nd c2nd 7898 ≈ cen 8801 0cc0 10972 − cmin 11306 2c2 12129 ℤ_≥cuz 12683 ♯chash 14145 prefix cpfx 14481 Vtxcvtx 27655 USPGraphcuspgr 27807 ClWalkscclwlks 28426 ClWWalksNOncclwwlknon 28739

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 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049

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 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-2o 8368 df-oadd 8371 df-er 8569 df-map 8688 df-pm 8689 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-dju 9758 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-n0 12335 df-xnn0 12407 df-z 12421 df-uz 12684 df-rp 12832 df-fz 13341 df-fzo 13484 df-hash 14146 df-word 14318 df-lsw 14366 df-concat 14374 df-s1 14400 df-substr 14452 df-pfx 14482 df-edg 27707 df-uhgr 27717 df-upgr 27741 df-uspgr 27809 df-wlks 28255 df-clwlks 28427 df-clwwlk 28634 df-clwwlkn 28677 df-clwwlknon 28740

This theorem is referenced by: numclwlk1lem2 29022

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