Theorem iswwlksnon 29886

Description: The set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)

Hypothesis

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

Assertion

Ref	Expression
iswwlksnon	⊢ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)}

Distinct variable groups:   𝑤,𝐴   𝑤,𝐵   𝑤,𝐺   𝑤,𝑁   𝑤,𝑉

Proof of Theorem iswwlksnon

Dummy variables 𝑎 𝑏 𝑔 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ov 7485	. . 3 ⊢ (𝐴∅𝐵) = ∅
2		df-wwlksnon 29865	. . . . 5 ⊢ WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)}))
3	2	mpondm0 7690	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksNOn 𝐺) = ∅)
4	3	oveqd 7465	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴∅𝐵))
5		df-wwlksn 29864	. . . . . 6 ⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
6	5	mpondm0 7690	. . . . 5 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
7	6	rabeqdv 3459	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)})
8		rab0 4409	. . . 4 ⊢ {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅
9	7, 8	eqtrdi 2796	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅)
10	1, 4, 9	3eqtr4a 2806	. 2 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)})
11		iswwlksnon.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
12	11	wwlksnon 29884	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)}))
13	12	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ ¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑁 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)}))
14	13	oveqd 7465	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ ¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})𝐵))
15		eqid 2740	. . . . . . . 8 ⊢ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})
16	15	mpondm0 7690	. . . . . . 7 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})𝐵) = ∅)
17	16	adantl 481	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ ¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})𝐵) = ∅)
18	14, 17	eqtrd 2780	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ ¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
19	18	ex 412	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅))
20	4, 1	eqtrdi 2796	. . . . 5 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
21	20	a1d 25	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅))
22	19, 21	pm2.61i 182	. . 3 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
23	11	wwlknllvtx 29879	. . . . . . 7 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → ((𝑤‘0) ∈ 𝑉 ∧ (𝑤‘𝑁) ∈ 𝑉))
24		eleq1 2832	. . . . . . . . 9 ⊢ (𝐴 = (𝑤‘0) → (𝐴 ∈ 𝑉 ↔ (𝑤‘0) ∈ 𝑉))
25	24	eqcoms 2748	. . . . . . . 8 ⊢ ((𝑤‘0) = 𝐴 → (𝐴 ∈ 𝑉 ↔ (𝑤‘0) ∈ 𝑉))
26		eleq1 2832	. . . . . . . . 9 ⊢ (𝐵 = (𝑤‘𝑁) → (𝐵 ∈ 𝑉 ↔ (𝑤‘𝑁) ∈ 𝑉))
27	26	eqcoms 2748	. . . . . . . 8 ⊢ ((𝑤‘𝑁) = 𝐵 → (𝐵 ∈ 𝑉 ↔ (𝑤‘𝑁) ∈ 𝑉))
28	25, 27	bi2anan9 637	. . . . . . 7 ⊢ (((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ↔ ((𝑤‘0) ∈ 𝑉 ∧ (𝑤‘𝑁) ∈ 𝑉)))
29	23, 28	syl5ibrcom 247	. . . . . 6 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
30	29	con3rr3 155	. . . . 5 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑤 ∈ (𝑁 WWalksN 𝐺) → ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)))
31	30	ralrimiv 3151	. . . 4 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵))
32		rabeq0 4411	. . . 4 ⊢ ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅ ↔ ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵))
33	31, 32	sylibr 234	. . 3 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅)
34	22, 33	eqtr4d 2783	. 2 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)})
35	12	adantr 480	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑁 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)}))
36		eqeq2 2752	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴))
37		eqeq2 2752	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑤‘𝑁) = 𝑏 ↔ (𝑤‘𝑁) = 𝐵))
38	36, 37	bi2anan9 637	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)))
39	38	rabbidv 3451	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)})
40	39	adantl 481	. . 3 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)})
41		simprl 770	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
42		simprr 772	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
43		ovex 7481	. . . . 5 ⊢ (𝑁 WWalksN 𝐺) ∈ V
44	43	rabex 5357	. . . 4 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} ∈ V
45	44	a1i 11	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} ∈ V)
46	35, 40, 41, 42, 45	ovmpod 7602	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)})
47	10, 34, 46	ecase 1033	1 ⊢ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488  ∅c0 4352  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  0cc0 11184  1c1 11185   + caddc 11187  ℕ₀cn0 12553  ♯chash 14379  Vtxcvtx 29031  WWalkscwwlks 29858   WWalksN cwwlksn 29859   WWalksNOn cwwlksnon 29860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-wwlks 29863  df-wwlksn 29864  df-wwlksnon 29865

This theorem is referenced by:  wwlknon  29890  wwlksnonfi  29953  wpthswwlks2on  29994  clwwlknclwwlkdif  30011  clwwlknclwwlkdifnum  30012  numclwwlkqhash  30407