Theorem iswwlksnon 29883

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 7468	. . 3 ⊢ (𝐴∅𝐵) = ∅
2		df-wwlksnon 29862	. . . . 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 7673	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksNOn 𝐺) = ∅)
4	3	oveqd 7448	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴∅𝐵))
5		df-wwlksn 29861	. . . . . 6 ⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
6	5	mpondm0 7673	. . . . 5 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
7	6	rabeqdv 3449	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)})
8		rab0 4392	. . . 4 ⊢ {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅
9	7, 8	eqtrdi 2791	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅)
10	1, 4, 9	3eqtr4a 2801	. 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 29881	. . . . . . . 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 7448	. . . . . 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 2735	. . . . . . . 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 7673	. . . . . . 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 2775	. . . . 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 2791	. . . . 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 29876	. . . . . . 7 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → ((𝑤‘0) ∈ 𝑉 ∧ (𝑤‘𝑁) ∈ 𝑉))
24		eleq1 2827	. . . . . . . . 9 ⊢ (𝐴 = (𝑤‘0) → (𝐴 ∈ 𝑉 ↔ (𝑤‘0) ∈ 𝑉))
25	24	eqcoms 2743	. . . . . . . 8 ⊢ ((𝑤‘0) = 𝐴 → (𝐴 ∈ 𝑉 ↔ (𝑤‘0) ∈ 𝑉))
26		eleq1 2827	. . . . . . . . 9 ⊢ (𝐵 = (𝑤‘𝑁) → (𝐵 ∈ 𝑉 ↔ (𝑤‘𝑁) ∈ 𝑉))
27	26	eqcoms 2743	. . . . . . . 8 ⊢ ((𝑤‘𝑁) = 𝐵 → (𝐵 ∈ 𝑉 ↔ (𝑤‘𝑁) ∈ 𝑉))
28	25, 27	bi2anan9 638	. . . . . . 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 3143	. . . 4 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵))
32		rabeq0 4394	. . . 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 2778	. 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 2747	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴))
37		eqeq2 2747	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑤‘𝑁) = 𝑏 ↔ (𝑤‘𝑁) = 𝐵))
38	36, 37	bi2anan9 638	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)))
39	38	rabbidv 3441	. . . 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 771	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
42		simprr 773	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
43		ovex 7464	. . . . 5 ⊢ (𝑁 WWalksN 𝐺) ∈ V
44	43	rabex 5345	. . . 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 7585	. 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 2106  ∀wral 3059  {crab 3433  Vcvv 3478  ∅c0 4339  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  0cc0 11153  1c1 11154   + caddc 11156  ℕ₀cn0 12524  ♯chash 14366  Vtxcvtx 29028  WWalkscwwlks 29855   WWalksN cwwlksn 29856   WWalksNOn cwwlksnon 29857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-wwlks 29860  df-wwlksn 29861  df-wwlksnon 29862

This theorem is referenced by:  wwlknon  29887  wwlksnonfi  29950  wpthswwlks2on  29991  clwwlknclwwlkdif  30008  clwwlknclwwlkdifnum  30009  numclwwlkqhash  30404