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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 6506 | . . 3 ⊢ (ClWalks‘𝐺) ∈ V | |
3 | 1, 2 | rabex2 5087 | . 2 ⊢ 𝑊 ∈ V |
4 | dlwwlknondlwlknonbij.d | . . 3 ⊢ 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} | |
5 | ovex 7002 | . . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V | |
6 | 4, 5 | rabex2 5087 | . 2 ⊢ 𝐷 ∈ V |
7 | dlwwlknondlwlknonbij.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | eqid 2772 | . . 3 ⊢ (𝑐 ∈ 𝑊 ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))) = (𝑐 ∈ 𝑊 ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))) | |
9 | 7, 1, 4, 8 | dlwwlknondlwlknonf1o 27906 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑐 ∈ 𝑊 ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))):𝑊–1-1-onto→𝐷) |
10 | f1oen2g 8315 | . 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 1444 | 1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → 𝑊 ≈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 {crab 3086 Vcvv 3409 class class class wbr 4923 ↦ cmpt 5002 –1-1-onto→wf1o 6181 ‘cfv 6182 (class class class)co 6970 1st c1st 7492 2nd c2nd 7493 ≈ cen 8295 0cc0 10327 − cmin 10662 2c2 11488 ℤ≥cuz 12051 ♯chash 13498 prefix cpfx 13842 Vtxcvtx 26474 USPGraphcuspgr 26626 ClWalkscclwlks 27249 ClWWalksNOncclwwlknon 27605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-ifp 1044 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-pm 8201 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-dju 9116 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-n0 11701 df-xnn0 11773 df-z 11787 df-uz 12052 df-rp 12198 df-fz 12702 df-fzo 12843 df-hash 13499 df-word 13663 df-lsw 13716 df-concat 13724 df-s1 13749 df-substr 13794 df-pfx 13843 df-edg 26526 df-uhgr 26536 df-upgr 26560 df-uspgr 26628 df-wlks 27074 df-clwlks 27250 df-clwwlk 27478 df-clwwlkn 27530 df-clwwlknon 27606 |
This theorem is referenced by: numclwlk1lem2 27913 |
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