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Theorem wwlksnextbij 27840
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 = (Vtx‘𝐺)
wwlksnextbij.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
wwlksnextbij (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
Distinct variable groups:   𝑓,𝐸,𝑛,𝑤   𝑓,𝐺,𝑤   𝑓,𝑁,𝑤   𝑓,𝑉,𝑛,𝑤   𝑓,𝑊,𝑛,𝑤
Allowed substitution hints:   𝐺(𝑛)   𝑁(𝑛)

Proof of Theorem wwlksnextbij
Dummy variables 𝑝 𝑡 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovexd 7205 . . 3 (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑁 + 1) WWalksN 𝐺) ∈ V)
2 rabexg 5199 . . 3 (((𝑁 + 1) WWalksN 𝐺) ∈ V → {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ∈ V)
3 mptexg 6994 . . 3 ({𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ∈ V → (𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ↦ (lastS‘𝑥)) ∈ V)
41, 2, 33syl 18 . 2 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ↦ (lastS‘𝑥)) ∈ V)
5 wwlksnextbij.v . . . 4 𝑉 = (Vtx‘𝐺)
6 wwlksnextbij.e . . . 4 𝐸 = (Edg‘𝐺)
7 eqid 2738 . . . 4 {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
8 preq2 4625 . . . . . 6 (𝑛 = 𝑝 → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), 𝑝})
98eleq1d 2817 . . . . 5 (𝑛 = 𝑝 → ({(lastS‘𝑊), 𝑛} ∈ 𝐸 ↔ {(lastS‘𝑊), 𝑝} ∈ 𝐸))
109cbvrabv 3393 . . . 4 {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} = {𝑝𝑉 ∣ {(lastS‘𝑊), 𝑝} ∈ 𝐸}
11 fveqeq2 6683 . . . . . . 7 (𝑡 = 𝑤 → ((♯‘𝑡) = (𝑁 + 2) ↔ (♯‘𝑤) = (𝑁 + 2)))
12 oveq1 7177 . . . . . . . 8 (𝑡 = 𝑤 → (𝑡 prefix (𝑁 + 1)) = (𝑤 prefix (𝑁 + 1)))
1312eqeq1d 2740 . . . . . . 7 (𝑡 = 𝑤 → ((𝑡 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑤 prefix (𝑁 + 1)) = 𝑊))
14 fveq2 6674 . . . . . . . . 9 (𝑡 = 𝑤 → (lastS‘𝑡) = (lastS‘𝑤))
1514preq2d 4631 . . . . . . . 8 (𝑡 = 𝑤 → {(lastS‘𝑊), (lastS‘𝑡)} = {(lastS‘𝑊), (lastS‘𝑤)})
1615eleq1d 2817 . . . . . . 7 (𝑡 = 𝑤 → ({(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))
1711, 13, 163anbi123d 1437 . . . . . 6 (𝑡 = 𝑤 → (((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸) ↔ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)))
1817cbvrabv 3393 . . . . 5 {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
1918mpteq1i 5120 . . . 4 (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ↦ (lastS‘𝑥)) = (𝑥 ∈ {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ↦ (lastS‘𝑥))
205, 6, 7, 10, 19wwlksnextbij0 27839 . . 3 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ↦ (lastS‘𝑥)):{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
21 eqid 2738 . . . . . . 7 {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} = {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)}
225, 6, 21wwlksnextwrd 27835 . . . . . 6 (𝑊 ∈ (𝑁 WWalksN 𝐺) → {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} = {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)})
2322eqcomd 2744 . . . . 5 (𝑊 ∈ (𝑁 WWalksN 𝐺) → {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} = {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)})
2423mpteq1d 5119 . . . 4 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ↦ (lastS‘𝑥)) = (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ↦ (lastS‘𝑥)))
255, 6, 7wwlksnextwrd 27835 . . . . 5 (𝑊 ∈ (𝑁 WWalksN 𝐺) → {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})
2625eqcomd 2744 . . . 4 (𝑊 ∈ (𝑁 WWalksN 𝐺) → {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})
27 eqidd 2739 . . . 4 (𝑊 ∈ (𝑁 WWalksN 𝐺) → {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
2824, 26, 27f1oeq123d 6612 . . 3 (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ↦ (lastS‘𝑥)):{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} ↔ (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ↦ (lastS‘𝑥)):{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}))
2920, 28mpbird 260 . 2 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ↦ (lastS‘𝑥)):{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
30 f1oeq1 6606 . 2 (𝑓 = (𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ↦ (lastS‘𝑥)) → (𝑓:{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} ↔ (𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ↦ (lastS‘𝑥)):{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}))
314, 29, 30spcedv 3502 1 (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  {crab 3057  Vcvv 3398  {cpr 4518  cmpt 5110  1-1-ontowf1o 6338  cfv 6339  (class class class)co 7170  1c1 10616   + caddc 10618  2c2 11771  chash 13782  Word cword 13955  lastSclsw 14003   prefix cpfx 14121  Vtxcvtx 26941  Edgcedg 26992   WWalksN cwwlksn 27764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-n0 11977  df-xnn0 12049  df-z 12063  df-uz 12325  df-rp 12473  df-fz 12982  df-fzo 13125  df-hash 13783  df-word 13956  df-lsw 14004  df-concat 14012  df-s1 14039  df-substr 14092  df-pfx 14122  df-wwlks 27768  df-wwlksn 27769
This theorem is referenced by:  wwlksnexthasheq  27841
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