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Mirrors > Home > MPE Home > Th. List > dlwwlknondlwlknonen | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
dlwwlknondlwlknonbij.v | ⊢ 𝑉 = (Vtx‘𝐺) |
dlwwlknondlwlknonbij.w | ⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} |
dlwwlknondlwlknonbij.d | ⊢ 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} |
Ref | Expression |
---|---|
dlwwlknondlwlknonen | ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → 𝑊 ≈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dlwwlknondlwlknonbij.w | . . 3 ⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} | |
2 | fvex 6677 | . . 3 ⊢ (ClWalks‘𝐺) ∈ V | |
3 | 1, 2 | rabex2 5229 | . 2 ⊢ 𝑊 ∈ V |
4 | dlwwlknondlwlknonbij.d | . . 3 ⊢ 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} | |
5 | ovex 7183 | . . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V | |
6 | 4, 5 | rabex2 5229 | . 2 ⊢ 𝐷 ∈ V |
7 | dlwwlknondlwlknonbij.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | eqid 2821 | . . 3 ⊢ (𝑐 ∈ 𝑊 ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))) = (𝑐 ∈ 𝑊 ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))) | |
9 | 7, 1, 4, 8 | dlwwlknondlwlknonf1o 28138 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑐 ∈ 𝑊 ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))):𝑊–1-1-onto→𝐷) |
10 | f1oen2g 8520 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝐷 ∈ V ∧ (𝑐 ∈ 𝑊 ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))):𝑊–1-1-onto→𝐷) → 𝑊 ≈ 𝐷) | |
11 | 3, 6, 9, 10 | mp3an12i 1461 | 1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → 𝑊 ≈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 class class class wbr 5058 ↦ cmpt 5138 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 ≈ cen 8500 0cc0 10531 − cmin 10864 2c2 11686 ℤ≥cuz 12237 ♯chash 13684 prefix cpfx 14026 Vtxcvtx 26775 USPGraphcuspgr 26927 ClWalkscclwlks 27545 ClWWalksNOncclwwlknon 27860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-lsw 13909 df-concat 13917 df-s1 13944 df-substr 13997 df-pfx 14027 df-edg 26827 df-uhgr 26837 df-upgr 26861 df-uspgr 26929 df-wlks 27375 df-clwlks 27546 df-clwwlk 27754 df-clwwlkn 27797 df-clwwlknon 27861 |
This theorem is referenced by: numclwlk1lem2 28143 |
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