Theorem rusgrnumwrdl2 29744

Description: In a k-regular simple graph, the number of edges resp. walks of length 1 (represented as words of length 2) starting at a fixed vertex is k. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 6-May-2021.)

Hypothesis

Ref	Expression
rusgrnumwrdl2.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
rusgrnumwrdl2	⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = 𝐾)

Distinct variable groups:   𝑤,𝐺   𝑤,𝑃   𝑤,𝑉

Allowed substitution hint:   𝐾(𝑤)

Proof of Theorem rusgrnumwrdl2

Dummy variables 𝑓 𝑝 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rusgrnumwrdl2.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	fvexi 6876	. . . . 5 ⊢ 𝑉 ∈ V
3	2	wrdexi 14533	. . . 4 ⊢ Word 𝑉 ∈ V
4	3	rabex 5292	. . 3 ⊢ {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))} ∈ V
5	2	a1i 11	. . . 4 ⊢ (𝐺 RegUSGraph 𝐾 → 𝑉 ∈ V)
6		wrd2f1tovbij 14967	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑃 ∈ 𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}–1-1-onto→{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)})
7	5, 6	sylan 589	. . 3 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}–1-1-onto→{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)})
8		hasheqf1oi 14358	. . 3 ⊢ ({𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))} ∈ V → (∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}–1-1-onto→{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)} → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = (♯‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)})))
9	4, 7, 8	mpsyl 68	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = (♯‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}))
10	1	rusgrpropadjvtx 29743	. . . 4 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝 ∈ 𝑉 (♯‘{𝑠 ∈ 𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
11		preq1 4689	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → {𝑝, 𝑠} = {𝑃, 𝑠})
12	11	eleq1d 2846	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ({𝑝, 𝑠} ∈ (Edg‘𝐺) ↔ {𝑃, 𝑠} ∈ (Edg‘𝐺)))
13	12	rabbidv 3420	. . . . . . 7 ⊢ (𝑝 = 𝑃 → {𝑠 ∈ 𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)} = {𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)})
14	13	fveqeq2d 6870	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((♯‘{𝑠 ∈ 𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾 ↔ (♯‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
15	14	rspccv 3577	. . . . 5 ⊢ (∀𝑝 ∈ 𝑉 (♯‘{𝑠 ∈ 𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾 → (𝑃 ∈ 𝑉 → (♯‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
16	15	3ad2ant3 1147	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝 ∈ 𝑉 (♯‘{𝑠 ∈ 𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾) → (𝑃 ∈ 𝑉 → (♯‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
17	10, 16	syl 17	. . 3 ⊢ (𝐺 RegUSGraph 𝐾 → (𝑃 ∈ 𝑉 → (♯‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
18	17	imp 410	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑠 ∈ 𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾)
19	9, 18	eqtrd 2796	1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = 𝐾)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  {crab 3413  Vcvv 3453  {cpr 4581   class class class wbr 5097  –1-1-onto→wf1o 6515  ‘cfv 6516  0cc0 11067  1c1 11068  2c2 12266  ℕ₀^*cxnn0 12548  ♯chash 14337  Word cword 14520  Vtxcvtx 29154  Edgcedg 29205  USGraphcusgr 29307   RegUSGraph crusgr 29714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-2o 8432  df-oadd 8435  df-er 8672  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9853  df-card 9891  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-n0 12476  df-xnn0 12549  df-z 12563  df-uz 12834  df-xadd 13109  df-fz 13507  df-fzo 13654  df-hash 14338  df-word 14521  df-edg 29206  df-uhgr 29216  df-ushgr 29217  df-upgr 29240  df-umgr 29241  df-uspgr 29308  df-usgr 29309  df-nbgr 29491  df-vtxdg 29624  df-rgr 29715  df-rusgr 29716

This theorem is referenced by:  rusgrnumwwlkl1  30128  clwwlknon2num  30264