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Theorem iswwlksnon 29074
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 7433 . . 3 (𝐴𝐵) = ∅
2 df-wwlksnon 29053 . . . . 5 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))
32mpondm0 7634 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksNOn 𝐺) = ∅)
43oveqd 7413 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴𝐵))
5 df-wwlksn 29052 . . . . . 6 WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
65mpondm0 7634 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
76rabeqdv 3448 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
8 rab0 4380 . . . 4 {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = ∅
97, 8eqtrdi 2789 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = ∅)
101, 4, 93eqtr4a 2799 . 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 29072 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
1312adantr 482 . . . . . . 7 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
1413oveqd 7413 . . . . . 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 2733 . . . . . . . 8 (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)})
1615mpondm0 7634 . . . . . . 7 (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)})𝐵) = ∅)
1716adantl 483 . . . . . 6 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)})𝐵) = ∅)
1814, 17eqtrd 2773 . . . . 5 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
1918ex 414 . . . 4 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅))
204, 1eqtrdi 2789 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
2120a1d 25 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅))
2219, 21pm2.61i 182 . . 3 (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
2311wwlknllvtx 29067 . . . . . . 7 (𝑤 ∈ (𝑁 WWalksN 𝐺) → ((𝑤‘0) ∈ 𝑉 ∧ (𝑤𝑁) ∈ 𝑉))
24 eleq1 2822 . . . . . . . . 9 (𝐴 = (𝑤‘0) → (𝐴𝑉 ↔ (𝑤‘0) ∈ 𝑉))
2524eqcoms 2741 . . . . . . . 8 ((𝑤‘0) = 𝐴 → (𝐴𝑉 ↔ (𝑤‘0) ∈ 𝑉))
26 eleq1 2822 . . . . . . . . 9 (𝐵 = (𝑤𝑁) → (𝐵𝑉 ↔ (𝑤𝑁) ∈ 𝑉))
2726eqcoms 2741 . . . . . . . 8 ((𝑤𝑁) = 𝐵 → (𝐵𝑉 ↔ (𝑤𝑁) ∈ 𝑉))
2825, 27bi2anan9 638 . . . . . . 7 (((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵) → ((𝐴𝑉𝐵𝑉) ↔ ((𝑤‘0) ∈ 𝑉 ∧ (𝑤𝑁) ∈ 𝑉)))
2923, 28syl5ibrcom 246 . . . . . 6 (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵) → (𝐴𝑉𝐵𝑉)))
3029con3rr3 155 . . . . 5 (¬ (𝐴𝑉𝐵𝑉) → (𝑤 ∈ (𝑁 WWalksN 𝐺) → ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)))
3130ralrimiv 3146 . . . 4 (¬ (𝐴𝑉𝐵𝑉) → ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵))
32 rabeq0 4382 . . . 4 ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = ∅ ↔ ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵))
3331, 32sylibr 233 . . 3 (¬ (𝐴𝑉𝐵𝑉) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} = ∅)
3422, 33eqtr4d 2776 . 2 (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
3512adantr 482 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
36 eqeq2 2745 . . . . . 6 (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴))
37 eqeq2 2745 . . . . . 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 3441 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
4039adantl 483 . . 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 7429 . . . . 5 (𝑁 WWalksN 𝐺) ∈ V
4443rabex 5328 . . . 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 7547 . 2 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
4710, 34, 46ecase 1032 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 205  wa 397   = wceq 1542  wcel 2107  wral 3062  {crab 3433  Vcvv 3475  c0 4320  cfv 6535  (class class class)co 7396  cmpo 7398  0cc0 11097  1c1 11098   + caddc 11100  0cn0 12459  chash 14277  Vtxcvtx 28223  WWalkscwwlks 29046   WWalksN cwwlksn 29047   WWalksNOn cwwlksnon 29048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-mulcom 11161  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173  ax-pre-mulgt0 11174
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7843  df-1st 7962  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8691  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-card 9921  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-sub 11433  df-neg 11434  df-nn 12200  df-n0 12460  df-z 12546  df-uz 12810  df-fz 13472  df-fzo 13615  df-hash 14278  df-word 14452  df-wwlks 29051  df-wwlksn 29052  df-wwlksnon 29053
This theorem is referenced by:  wwlknon  29078  wwlksnonfi  29141  wpthswwlks2on  29182  clwwlknclwwlkdif  29199  clwwlknclwwlkdifnum  29200  numclwwlkqhash  29595
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