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Theorem wwlknon 26645
 Description: An element of 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.)
Hypothesis
Ref Expression
wwlknon.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
wwlknon ((𝐴𝑉𝐵𝑉) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))

Proof of Theorem wwlknon
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 wwlknon.v . . . 4 𝑉 = (Vtx‘𝐺)
21iswwlksnon 26643 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
32eleq2d 2684 . 2 ((𝐴𝑉𝐵𝑉) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ 𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)}))
4 fveq1 6157 . . . . . 6 (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
54eqeq1d 2623 . . . . 5 (𝑤 = 𝑊 → ((𝑤‘0) = 𝐴 ↔ (𝑊‘0) = 𝐴))
6 fveq1 6157 . . . . . 6 (𝑤 = 𝑊 → (𝑤𝑁) = (𝑊𝑁))
76eqeq1d 2623 . . . . 5 (𝑤 = 𝑊 → ((𝑤𝑁) = 𝐵 ↔ (𝑊𝑁) = 𝐵))
85, 7anbi12d 746 . . . 4 (𝑤 = 𝑊 → (((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵) ↔ ((𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))
98elrab 3351 . . 3 (𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))
10 3anass 1040 . . 3 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))
119, 10bitr4i 267 . 2 (𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵))
123, 11syl6bb 276 1 ((𝐴𝑉𝐵𝑉) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  {crab 2912  ‘cfv 5857  (class class class)co 6615  0cc0 9896  Vtxcvtx 25808   WWalksN cwwlksn 26621   WWalksNOn cwwlksnon 26622 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-wwlksn 26626  df-wwlksnon 26627 This theorem is referenced by:  wwlksnwwlksnon  26713  wspthsnwspthsnon  26714  wspthsnonn0vne  26716  elwwlks2ons3  26751  s3wwlks2on  26752  wpthswwlks2on  26756  elwspths2spth  26763
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