Theorem wwlksnextbij 29625

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→{𝑛 ∈ 𝑉 ∣ {(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.

Step	Hyp	Ref	Expression
1		ovexd 7436	. . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑁 + 1) WWalksN 𝐺) ∈ V)
2		rabexg 5321	. . 3 ⊢ (((𝑁 + 1) WWalksN 𝐺) ∈ V → {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ∈ V)
3		mptexg 7214	. . 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)
4	1, 2, 3	3syl 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 2724	. . . 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 4730	. . . . . 6 ⊢ (𝑛 = 𝑝 → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), 𝑝})
9	8	eleq1d 2810	. . . . 5 ⊢ (𝑛 = 𝑝 → ({(lastS‘𝑊), 𝑛} ∈ 𝐸 ↔ {(lastS‘𝑊), 𝑝} ∈ 𝐸))
10	9	cbvrabv 3434	. . . 4 ⊢ {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} = {𝑝 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑝} ∈ 𝐸}
11		fveqeq2 6890	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((♯‘𝑡) = (𝑁 + 2) ↔ (♯‘𝑤) = (𝑁 + 2)))
12		oveq1 7408	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (𝑡 prefix (𝑁 + 1)) = (𝑤 prefix (𝑁 + 1)))
13	12	eqeq1d 2726	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((𝑡 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑤 prefix (𝑁 + 1)) = 𝑊))
14		fveq2 6881	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → (lastS‘𝑡) = (lastS‘𝑤))
15	14	preq2d 4736	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → {(lastS‘𝑊), (lastS‘𝑡)} = {(lastS‘𝑊), (lastS‘𝑤)})
16	15	eleq1d 2810	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ({(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))
17	11, 13, 16	3anbi123d 1432	. . . . . 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)))
18	17	cbvrabv 3434	. . . . 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)}
19	18	mpteq1i 5234	. . . 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))
20	5, 6, 7, 10, 19	wwlksnextbij0 29624	. . 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→{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
21		eqid 2724	. . . . . . 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)}
22	5, 6, 21	wwlksnextwrd 29620	. . . . . 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)})
23	22	eqcomd 2730	. . . . 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)})
24	23	mpteq1d 5233	. . . 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)))
25	5, 6, 7	wwlksnextwrd 29620	. . . . 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)})
26	25	eqcomd 2730	. . . 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 2725	. . . 4 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
28	24, 26, 27	f1oeq123d 6817	. . 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→{𝑛 ∈ 𝑉 ∣ {(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→{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}))
29	20, 28	mpbird 257	. 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→{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
30		f1oeq1 6811	. 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→{𝑛 ∈ 𝑉 ∣ {(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→{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}))
31	4, 29, 30	spcedv 3580	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→{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {crab 3424  Vcvv 3466  {cpr 4622   ↦ cmpt 5221  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7401  1c1 11107   + caddc 11109  2c2 12264  ♯chash 14287  Word cword 14461  lastSclsw 14509   prefix cpfx 14617  Vtxcvtx 28725  Edgcedg 28776   WWalksN cwwlksn 29549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-n0 12470  df-xnn0 12542  df-z 12556  df-uz 12820  df-rp 12972  df-fz 13482  df-fzo 13625  df-hash 14288  df-word 14462  df-lsw 14510  df-concat 14518  df-s1 14543  df-substr 14588  df-pfx 14618  df-wwlks 29553  df-wwlksn 29554

This theorem is referenced by:  wwlksnexthasheq  29626