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Theorem iswwlksnon 27683
 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 ∧ (𝑤𝑁) = 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.
StepHypRef Expression
1 0ov 7182 . . 3 (𝐴𝐵) = ∅
2 df-wwlksnon 27662 . . . . 5 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))
32mpondm0 7377 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksNOn 𝐺) = ∅)
43oveqd 7162 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴𝐵))
5 df-wwlksn 27661 . . . . . 6 WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
65mpondm0 7377 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
76rabeqdv 3433 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
8 rab0 4294 . . . 4 {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = ∅
97, 8eqtrdi 2849 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = ∅)
101, 4, 93eqtr4a 2859 . 2 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
11 iswwlksnon.v . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
1211wwlksnon 27681 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
1312adantr 484 . . . . . . 7 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
1413oveqd 7162 . . . . . 6 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)})𝐵))
15 eqid 2798 . . . . . . . 8 (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)})
1615mpondm0 7377 . . . . . . 7 (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)})𝐵) = ∅)
1716adantl 485 . . . . . 6 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)})𝐵) = ∅)
1814, 17eqtrd 2833 . . . . 5 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
1918ex 416 . . . 4 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅))
204, 1eqtrdi 2849 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
2120a1d 25 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅))
2219, 21pm2.61i 185 . . 3 (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
2311wwlknllvtx 27676 . . . . . . 7 (𝑤 ∈ (𝑁 WWalksN 𝐺) → ((𝑤‘0) ∈ 𝑉 ∧ (𝑤𝑁) ∈ 𝑉))
24 eleq1 2877 . . . . . . . . 9 (𝐴 = (𝑤‘0) → (𝐴𝑉 ↔ (𝑤‘0) ∈ 𝑉))
2524eqcoms 2806 . . . . . . . 8 ((𝑤‘0) = 𝐴 → (𝐴𝑉 ↔ (𝑤‘0) ∈ 𝑉))
26 eleq1 2877 . . . . . . . . 9 (𝐵 = (𝑤𝑁) → (𝐵𝑉 ↔ (𝑤𝑁) ∈ 𝑉))
2726eqcoms 2806 . . . . . . . 8 ((𝑤𝑁) = 𝐵 → (𝐵𝑉 ↔ (𝑤𝑁) ∈ 𝑉))
2825, 27bi2anan9 638 . . . . . . 7 (((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵) → ((𝐴𝑉𝐵𝑉) ↔ ((𝑤‘0) ∈ 𝑉 ∧ (𝑤𝑁) ∈ 𝑉)))
2923, 28syl5ibrcom 250 . . . . . 6 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵) → (𝐴𝑉𝐵𝑉)))
3029con3rr3 158 . . . . 5 (¬ (𝐴𝑉𝐵𝑉) → (𝑤 ∈ (𝑁 WWalksN 𝐺) → ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)))
3130ralrimiv 3148 . . . 4 (¬ (𝐴𝑉𝐵𝑉) → ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵))
32 rabeq0 4295 . . . 4 ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = ∅ ↔ ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵))
3331, 32sylibr 237 . . 3 (¬ (𝐴𝑉𝐵𝑉) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = ∅)
3422, 33eqtr4d 2836 . 2 (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
3512adantr 484 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
36 eqeq2 2810 . . . . . 6 (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴))
37 eqeq2 2810 . . . . . 6 (𝑏 = 𝐵 → ((𝑤𝑁) = 𝑏 ↔ (𝑤𝑁) = 𝐵))
3836, 37bi2anan9 638 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)))
3938rabbidv 3428 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
4039adantl 485 . . 3 ((((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
41 simprl 770 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → 𝐴𝑉)
42 simprr 772 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → 𝐵𝑉)
43 ovex 7178 . . . . 5 (𝑁 WWalksN 𝐺) ∈ V
4443rabex 5203 . . . 4 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} ∈ V
4544a1i 11 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} ∈ V)
4635, 40, 41, 42, 45ovmpod 7292 . 2 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
4710, 34, 46ecase 1029 1 (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3442  ∅c0 4246  ‘cfv 6332  (class class class)co 7145   ∈ cmpo 7147  0cc0 10544  1c1 10545   + caddc 10547  ℕ0cn0 11903  ♯chash 13706  Vtxcvtx 26833  WWalkscwwlks 27655   WWalksN cwwlksn 27656   WWalksNOn cwwlksnon 27657 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-er 8290  df-map 8409  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-n0 11904  df-z 11990  df-uz 12252  df-fz 12906  df-fzo 13049  df-hash 13707  df-word 13878  df-wwlks 27660  df-wwlksn 27661  df-wwlksnon 27662 This theorem is referenced by:  wwlknon  27687  wwlksnonfi  27750  wpthswwlks2on  27791  clwwlknclwwlkdif  27808  clwwlknclwwlkdifnum  27809  numclwwlkqhash  28204
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