Theorem wwlksnextbij 29752

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 7448	. . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑁 + 1) WWalksN 𝐺) ∈ V)
2		rabexg 5329	. . 3 ⊢ (((𝑁 + 1) WWalksN 𝐺) ∈ V → {𝑡 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)} ∈ V)
3		mptexg 7227	. . 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 2725	. . . 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 4735	. . . . . 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 3430	. . . 4 ⊢ {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} = {𝑝 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑝} ∈ 𝐸}
11		fveqeq2 6899	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((♯‘𝑡) = (𝑁 + 2) ↔ (♯‘𝑤) = (𝑁 + 2)))
12		oveq1 7420	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (𝑡 prefix (𝑁 + 1)) = (𝑤 prefix (𝑁 + 1)))
13	12	eqeq1d 2727	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((𝑡 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑤 prefix (𝑁 + 1)) = 𝑊))
14		fveq2 6890	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → (lastS‘𝑡) = (lastS‘𝑤))
15	14	preq2d 4741	. . . . . . . 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 3430	. . . . 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 5240	. . . 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 29751	. . 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 2725	. . . . . . 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 29747	. . . . . 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 2731	. . . . 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 5239	. . . 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 29747	. . . . 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 2731	. . . 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 2726	. . . 4 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
28	24, 26, 27	f1oeq123d 6826	. . 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 256	. 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 6820	. 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 3579	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 394   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {crab 3419  Vcvv 3463  {cpr 4627   ↦ cmpt 5227  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7413  1c1 11134   + caddc 11136  2c2 12292  ♯chash 14316  Word cword 14491  lastSclsw 14539   prefix cpfx 14647  Vtxcvtx 28848  Edgcedg 28899   WWalksN cwwlksn 29676

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 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-n0 12498  df-xnn0 12570  df-z 12584  df-uz 12848  df-rp 13002  df-fz 13512  df-fzo 13655  df-hash 14317  df-word 14492  df-lsw 14540  df-concat 14548  df-s1 14573  df-substr 14618  df-pfx 14648  df-wwlks 29680  df-wwlksn 29681

This theorem is referenced by:  wwlksnexthasheq  29753