dlwwlknondlwlknonen - Metamath Proof Explorer

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Theorem dlwwlknondlwlknonen 30455

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 6849	. . 3 ⊢ (ClWalks‘𝐺) ∈ V
3	1, 2	rabex2 5279	. 2 ⊢ 𝑊 ∈ V
4		dlwwlknondlwlknonbij.d	. . 3 ⊢ 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}
5		ovex 7395	. . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V
6	4, 5	rabex2 5279	. 2 ⊢ 𝐷 ∈ V
7		dlwwlknondlwlknonbij.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
8		eqid 2737	. . 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 30454	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑐 ∈ 𝑊 ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))):𝑊–1-1-onto→𝐷)
10		f1oen2g 8910	. 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 1468	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → 𝑊 ≈ 𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {crab 3390 Vcvv 3430 class class class wbr 5086 ↦ cmpt 5167 –1-1-onto→wf1o 6493 ‘cfv 6494 (class class class)co 7362 1^st c1st 7935 2^nd c2nd 7936 ≈ cen 8885 0cc0 11033 − cmin 11372 2c2 12231 ℤ_≥cuz 12783 ♯chash 14287 prefix cpfx 14628 Vtxcvtx 29083 USPGraphcuspgr 29235 ClWalkscclwlks 29857 ClWWalksNOncclwwlknon 30176

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 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 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-oadd 8404 df-er 8638 df-map 8770 df-pm 8771 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 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-rp 12938 df-fz 13457 df-fzo 13604 df-hash 14288 df-word 14471 df-lsw 14520 df-concat 14528 df-s1 14554 df-substr 14599 df-pfx 14629 df-edg 29135 df-uhgr 29145 df-upgr 29169 df-uspgr 29237 df-wlks 29687 df-clwlks 29858 df-clwwlk 30071 df-clwwlkn 30114 df-clwwlknon 30177

This theorem is referenced by: numclwlk1lem2 30459

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