Theorem wwlksnextbij 30318

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 7454	. . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑁 + 1) WWalksN 𝐺) ∈ V)
2		rabexg 5310	. . 3 ⊢ (((𝑁 + 1) WWalksN 𝐺) ∈ V → {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ∈ V)
3		mptexg 7226	. . 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 19	. 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 2765	. . . 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 4702	. . . . . 6 ⊢ (𝑛 = 𝑝 → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), 𝑝})
9	8	eleq1d 2850	. . . . 5 ⊢ (𝑛 = 𝑝 → ({(lastS‘𝑊), 𝑛} ∈ 𝐸 ↔ {(lastS‘𝑊), 𝑝} ∈ 𝐸))
10	9	cbvrabv 3428	. . . 4 ⊢ {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} = {𝑝 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑝} ∈ 𝐸}
11		fveqeq2 6894	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((♯‘𝑡) = (𝑁 + 2) ↔ (♯‘𝑤) = (𝑁 + 2)))
12		oveq1 7426	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (𝑡 prefix (𝑁 + 1)) = (𝑤 prefix (𝑁 + 1)))
13	12	eqeq1d 2767	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((𝑡 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑤 prefix (𝑁 + 1)) = 𝑊))
14		fveq2 6885	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → (lastS‘𝑡) = (lastS‘𝑤))
15	14	preq2d 4708	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → {(lastS‘𝑊), (lastS‘𝑡)} = {(lastS‘𝑊), (lastS‘𝑤)})
16	15	eleq1d 2850	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ({(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))
17	11, 13, 16	3anbi123d 1464	. . . . . 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 3428	. . . . 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 5204	. . . 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 30317	. . 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 2765	. . . . . . 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 30313	. . . . . 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 2771	. . . . 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 5203	. . . 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 30313	. . . . 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 2771	. . . 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 2766	. . . 4 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
28	24, 26, 27	f1oeq123d 6818	. . 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 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→{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
30		f1oeq1 6812	. 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 3559	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})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {crab 3418  Vcvv 3457  {cpr 4593   ↦ cmpt 5194  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7419  1c1 11116   + caddc 11118  2c2 12310  ♯chash 14384  Word cword 14568  lastSclsw 14617   prefix cpfx 14730  Vtxcvtx 29401  Edgcedg 29452   WWalksN cwwlksn 30242

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-card 9941  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-n0 12520  df-xnn0 12593  df-z 12607  df-uz 12879  df-rp 13033  df-fz 13552  df-fzo 13700  df-hash 14385  df-word 14569  df-lsw 14618  df-concat 14626  df-s1 14653  df-substr 14699  df-pfx 14731  df-wwlks 30246  df-wwlksn 30247

This theorem is used by:  wwlksnexthasheq  30319