Theorem wwlksnextbij 29970

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 7402	. . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑁 + 1) WWalksN 𝐺) ∈ V)
2		rabexg 5278	. . 3 ⊢ (((𝑁 + 1) WWalksN 𝐺) ∈ V → {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ∈ V)
3		mptexg 7176	. . 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 2736	. . . 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 4678	. . . . . 6 ⊢ (𝑛 = 𝑝 → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), 𝑝})
9	8	eleq1d 2821	. . . . 5 ⊢ (𝑛 = 𝑝 → ({(lastS‘𝑊), 𝑛} ∈ 𝐸 ↔ {(lastS‘𝑊), 𝑝} ∈ 𝐸))
10	9	cbvrabv 3399	. . . 4 ⊢ {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} = {𝑝 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑝} ∈ 𝐸}
11		fveqeq2 6849	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((♯‘𝑡) = (𝑁 + 2) ↔ (♯‘𝑤) = (𝑁 + 2)))
12		oveq1 7374	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (𝑡 prefix (𝑁 + 1)) = (𝑤 prefix (𝑁 + 1)))
13	12	eqeq1d 2738	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((𝑡 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑤 prefix (𝑁 + 1)) = 𝑊))
14		fveq2 6840	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → (lastS‘𝑡) = (lastS‘𝑤))
15	14	preq2d 4684	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → {(lastS‘𝑊), (lastS‘𝑡)} = {(lastS‘𝑊), (lastS‘𝑤)})
16	15	eleq1d 2821	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ({(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))
17	11, 13, 16	3anbi123d 1439	. . . . . 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 3399	. . . . 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 5176	. . . 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 29969	. . 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 2736	. . . . . . 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 29965	. . . . . 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 2742	. . . . 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 5175	. . . 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 29965	. . . . 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 2742	. . . 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 2737	. . . 4 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
28	24, 26, 27	f1oeq123d 6774	. . 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 6768	. 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 3540	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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {crab 3389  Vcvv 3429  {cpr 4569   ↦ cmpt 5166  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  1c1 11039   + caddc 11041  2c2 12236  ♯chash 14292  Word cword 14475  lastSclsw 14524   prefix cpfx 14633  Vtxcvtx 29065  Edgcedg 29116   WWalksN cwwlksn 29894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-lsw 14525  df-concat 14533  df-s1 14559  df-substr 14604  df-pfx 14634  df-wwlks 29898  df-wwlksn 29899

This theorem is referenced by:  wwlksnexthasheq  29971