dlwwlknondlwlknonen - Metamath Proof Explorer

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Theorem dlwwlknondlwlknonen 28730

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 6787	. . 3 ⊢ (ClWalks‘𝐺) ∈ V
3	1, 2	rabex2 5258	. 2 ⊢ 𝑊 ∈ V
4		dlwwlknondlwlknonbij.d	. . 3 ⊢ 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}
5		ovex 7308	. . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V
6	4, 5	rabex2 5258	. 2 ⊢ 𝐷 ∈ V
7		dlwwlknondlwlknonbij.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
8		eqid 2738	. . 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 28729	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑐 ∈ 𝑊 ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))):𝑊–1-1-onto→𝐷)
10		f1oen2g 8756	. 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 1539 ∈ wcel 2106 {crab 3068 Vcvv 3432 class class class wbr 5074 ↦ cmpt 5157 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 ≈ cen 8730 0cc0 10871 − cmin 11205 2c2 12028 ℤ_≥cuz 12582 ♯chash 14044 prefix cpfx 14383 Vtxcvtx 27366 USPGraphcuspgr 27518 ClWalkscclwlks 28138 ClWWalksNOncclwwlknon 28451

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

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 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-hash 14045 df-word 14218 df-lsw 14266 df-concat 14274 df-s1 14301 df-substr 14354 df-pfx 14384 df-edg 27418 df-uhgr 27428 df-upgr 27452 df-uspgr 27520 df-wlks 27966 df-clwlks 28139 df-clwwlk 28346 df-clwwlkn 28389 df-clwwlknon 28452

This theorem is referenced by: numclwlk1lem2 28734

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