Theorem iswwlksnon 29921

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 7404	. . 3 ⊢ (𝐴∅𝐵) = ∅
2		df-wwlksnon 29900	. . . . 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 7607	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksNOn 𝐺) = ∅)
4	3	oveqd 7384	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴∅𝐵))
5		df-wwlksn 29899	. . . . . 6 ⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
6	5	mpondm0 7607	. . . . 5 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
7	6	rabeqdv 3404	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)})
8		rab0 4326	. . . 4 ⊢ {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅
9	7, 8	eqtrdi 2787	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅)
10	1, 4, 9	3eqtr4a 2797	. 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 29919	. . . . . . . 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 7384	. . . . . 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 2736	. . . . . . . 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 7607	. . . . . . 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 2771	. . . . 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 2787	. . . . 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 29914	. . . . . . 7 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → ((𝑤‘0) ∈ 𝑉 ∧ (𝑤‘𝑁) ∈ 𝑉))
24		eleq1 2824	. . . . . . . . 9 ⊢ (𝐴 = (𝑤‘0) → (𝐴 ∈ 𝑉 ↔ (𝑤‘0) ∈ 𝑉))
25	24	eqcoms 2744	. . . . . . . 8 ⊢ ((𝑤‘0) = 𝐴 → (𝐴 ∈ 𝑉 ↔ (𝑤‘0) ∈ 𝑉))
26		eleq1 2824	. . . . . . . . 9 ⊢ (𝐵 = (𝑤‘𝑁) → (𝐵 ∈ 𝑉 ↔ (𝑤‘𝑁) ∈ 𝑉))
27	26	eqcoms 2744	. . . . . . . 8 ⊢ ((𝑤‘𝑁) = 𝐵 → (𝐵 ∈ 𝑉 ↔ (𝑤‘𝑁) ∈ 𝑉))
28	25, 27	bi2anan9 639	. . . . . . 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 3128	. . . 4 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵))
32		rabeq0 4328	. . . 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 2774	. 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 2748	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴))
37		eqeq2 2748	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑤‘𝑁) = 𝑏 ↔ (𝑤‘𝑁) = 𝐵))
38	36, 37	bi2anan9 639	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)))
39	38	rabbidv 3396	. . . 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 7400	. . . . 5 ⊢ (𝑁 WWalksN 𝐺) ∈ V
44	43	rabex 5280	. . . 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 7519	. 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 1034	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 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389  Vcvv 3429  ∅c0 4273  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  0cc0 11038  1c1 11039   + caddc 11041  ℕ₀cn0 12437  ♯chash 14292  Vtxcvtx 29065  WWalkscwwlks 29893   WWalksN cwwlksn 29894   WWalksNOn cwwlksnon 29895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-wwlks 29898  df-wwlksn 29899  df-wwlksnon 29900

This theorem is referenced by:  wwlknon  29925  wwlksnonfi  29988  wpthswwlks2on  30032  clwwlknclwwlkdif  30049  clwwlknclwwlkdifnum  30050  numclwwlkqhash  30445