Theorem wwlksnexthasheq 29892

Description: The number of the extensions of a walk (as word) by an edge equals the number of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 23-Aug-2018.) (Revised by AV, 19-Apr-2021.) (Revised by AV, 27-Oct-2022.)

Hypotheses

Ref	Expression
wwlksnexthasheq.v	⊢ 𝑉 = (Vtx‘𝐺)
wwlksnexthasheq.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
wwlksnexthasheq	⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}) = (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}))

Distinct variable groups:   𝑛,𝐸,𝑤   𝑤,𝐺   𝑤,𝑁   𝑛,𝑉,𝑤   𝑛,𝑊,𝑤

Allowed substitution hints:   𝐺(𝑛)   𝑁(𝑛)

Proof of Theorem wwlksnexthasheq

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7388	. . 3 ⊢ ((𝑁 + 1) WWalksN 𝐺) ∈ V
2	1	rabex 5281	. 2 ⊢ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ∈ V
3		wwlksnexthasheq.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
4		wwlksnexthasheq.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
5	3, 4	wwlksnextbij 29891	. 2 ⊢ (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})
6		hasheqf1oi 14268	. 2 ⊢ ({𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ∈ V → (∃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} → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}) = (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})))
7	2, 5, 6	mpsyl 68	1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}) = (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {crab 3397  Vcvv 3438  {cpr 4579  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7355  1c1 11017   + caddc 11019  ♯chash 14247  lastSclsw 14479   prefix cpfx 14588  Vtxcvtx 28985  Edgcedg 29036   WWalksN cwwlksn 29815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11072  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092  ax-pre-mulgt0 11093

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-map 8761  df-en 8879  df-dom 8880  df-sdom 8881  df-fin 8882  df-card 9842  df-pnf 11158  df-mnf 11159  df-xr 11160  df-ltxr 11161  df-le 11162  df-sub 11356  df-neg 11357  df-nn 12136  df-2 12198  df-n0 12392  df-xnn0 12465  df-z 12479  df-uz 12743  df-rp 12901  df-fz 13418  df-fzo 13565  df-hash 14248  df-word 14431  df-lsw 14480  df-concat 14488  df-s1 14514  df-substr 14559  df-pfx 14589  df-wwlks 29819  df-wwlksn 29820

This theorem is referenced by:  rusgrnumwwlks  29966