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Theorem wwlksnextbij 29687
Description: There is a bijection between the extensions of a walk (as word) by an edge and the set of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
Hypotheses
Ref Expression
wwlksnextbij.v 𝑉 = (Vtxβ€˜πΊ)
wwlksnextbij.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
wwlksnextbij (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ βˆƒπ‘“ 𝑓:{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}–1-1-ontoβ†’{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸})
Distinct variable groups:   𝑓,𝐸,𝑛,𝑀   𝑓,𝐺,𝑀   𝑓,𝑁,𝑀   𝑓,𝑉,𝑛,𝑀   𝑓,π‘Š,𝑛,𝑀
Allowed substitution hints:   𝐺(𝑛)   𝑁(𝑛)

Proof of Theorem wwlksnextbij
Dummy variables 𝑝 𝑑 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovexd 7449 . . 3 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ ((𝑁 + 1) WWalksN 𝐺) ∈ V)
2 rabexg 5327 . . 3 (((𝑁 + 1) WWalksN 𝐺) ∈ V β†’ {𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ∈ V)
3 mptexg 7227 . . 3 ({𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ∈ V β†’ (π‘₯ ∈ {𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ↦ (lastSβ€˜π‘₯)) ∈ V)
41, 2, 33syl 18 . 2 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (π‘₯ ∈ {𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ↦ (lastSβ€˜π‘₯)) ∈ V)
5 wwlksnextbij.v . . . 4 𝑉 = (Vtxβ€˜πΊ)
6 wwlksnextbij.e . . . 4 𝐸 = (Edgβ€˜πΊ)
7 eqid 2727 . . . 4 {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)} = {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}
8 preq2 4734 . . . . . 6 (𝑛 = 𝑝 β†’ {(lastSβ€˜π‘Š), 𝑛} = {(lastSβ€˜π‘Š), 𝑝})
98eleq1d 2813 . . . . 5 (𝑛 = 𝑝 β†’ ({(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), 𝑝} ∈ 𝐸))
109cbvrabv 3437 . . . 4 {𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸} = {𝑝 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑝} ∈ 𝐸}
11 fveqeq2 6900 . . . . . . 7 (𝑑 = 𝑀 β†’ ((β™―β€˜π‘‘) = (𝑁 + 2) ↔ (β™―β€˜π‘€) = (𝑁 + 2)))
12 oveq1 7421 . . . . . . . 8 (𝑑 = 𝑀 β†’ (𝑑 prefix (𝑁 + 1)) = (𝑀 prefix (𝑁 + 1)))
1312eqeq1d 2729 . . . . . . 7 (𝑑 = 𝑀 β†’ ((𝑑 prefix (𝑁 + 1)) = π‘Š ↔ (𝑀 prefix (𝑁 + 1)) = π‘Š))
14 fveq2 6891 . . . . . . . . 9 (𝑑 = 𝑀 β†’ (lastSβ€˜π‘‘) = (lastSβ€˜π‘€))
1514preq2d 4740 . . . . . . . 8 (𝑑 = 𝑀 β†’ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} = {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)})
1615eleq1d 2813 . . . . . . 7 (𝑑 = 𝑀 β†’ ({(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸))
1711, 13, 163anbi123d 1433 . . . . . 6 (𝑑 = 𝑀 β†’ (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸) ↔ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)))
1817cbvrabv 3437 . . . . 5 {𝑑 ∈ Word 𝑉 ∣ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} = {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}
1918mpteq1i 5238 . . . 4 (π‘₯ ∈ {𝑑 ∈ Word 𝑉 ∣ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ↦ (lastSβ€˜π‘₯)) = (π‘₯ ∈ {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)} ↦ (lastSβ€˜π‘₯))
205, 6, 7, 10, 19wwlksnextbij0 29686 . . 3 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (π‘₯ ∈ {𝑑 ∈ Word 𝑉 ∣ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ↦ (lastSβ€˜π‘₯)):{𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}–1-1-ontoβ†’{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸})
21 eqid 2727 . . . . . . 7 {𝑑 ∈ Word 𝑉 ∣ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} = {𝑑 ∈ Word 𝑉 ∣ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)}
225, 6, 21wwlksnextwrd 29682 . . . . . 6 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ {𝑑 ∈ Word 𝑉 ∣ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} = {𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)})
2322eqcomd 2733 . . . . 5 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ {𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} = {𝑑 ∈ Word 𝑉 ∣ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)})
2423mpteq1d 5237 . . . 4 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (π‘₯ ∈ {𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ↦ (lastSβ€˜π‘₯)) = (π‘₯ ∈ {𝑑 ∈ Word 𝑉 ∣ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ↦ (lastSβ€˜π‘₯)))
255, 6, 7wwlksnextwrd 29682 . . . . 5 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)} = {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)})
2625eqcomd 2733 . . . 4 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)} = {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)})
27 eqidd 2728 . . . 4 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ {𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸} = {𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸})
2824, 26, 27f1oeq123d 6827 . . 3 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ ((π‘₯ ∈ {𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ↦ (lastSβ€˜π‘₯)):{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}–1-1-ontoβ†’{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸} ↔ (π‘₯ ∈ {𝑑 ∈ Word 𝑉 ∣ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ↦ (lastSβ€˜π‘₯)):{𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}–1-1-ontoβ†’{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸}))
2920, 28mpbird 257 . 2 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (π‘₯ ∈ {𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ↦ (lastSβ€˜π‘₯)):{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}–1-1-ontoβ†’{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸})
30 f1oeq1 6821 . 2 (𝑓 = (π‘₯ ∈ {𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ↦ (lastSβ€˜π‘₯)) β†’ (𝑓:{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}–1-1-ontoβ†’{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸} ↔ (π‘₯ ∈ {𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)} ↦ (lastSβ€˜π‘₯)):{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}–1-1-ontoβ†’{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸}))
314, 29, 30spcedv 3583 1 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ βˆƒπ‘“ 𝑓:{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}–1-1-ontoβ†’{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099  {crab 3427  Vcvv 3469  {cpr 4626   ↦ cmpt 5225  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414  1c1 11125   + caddc 11127  2c2 12283  β™―chash 14307  Word cword 14482  lastSclsw 14530   prefix cpfx 14638  Vtxcvtx 28783  Edgcedg 28834   WWalksN cwwlksn 29611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-map 8836  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-n0 12489  df-xnn0 12561  df-z 12575  df-uz 12839  df-rp 12993  df-fz 13503  df-fzo 13646  df-hash 14308  df-word 14483  df-lsw 14531  df-concat 14539  df-s1 14564  df-substr 14609  df-pfx 14639  df-wwlks 29615  df-wwlksn 29616
This theorem is referenced by:  wwlksnexthasheq  29688
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