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Theorem wwlknon 41053
 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 41051 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})
32eleq2d 2673 . 2 ((𝐴𝑉𝐵𝑉) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ 𝑊 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)}))
4 fveq1 6102 . . . . . 6 (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
54eqeq1d 2612 . . . . 5 (𝑤 = 𝑊 → ((𝑤‘0) = 𝐴 ↔ (𝑊‘0) = 𝐴))
6 fveq1 6102 . . . . . 6 (𝑤 = 𝑊 → (𝑤𝑁) = (𝑊𝑁))
76eqeq1d 2612 . . . . 5 (𝑤 = 𝑊 → ((𝑤𝑁) = 𝐵 ↔ (𝑊𝑁) = 𝐵))
85, 7anbi12d 743 . . . 4 (𝑤 = 𝑊 → (((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵) ↔ ((𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))
98elrab 3331 . . 3 (𝑊 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))
10 3anass 1035 . . 3 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵) ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))
119, 10bitr4i 266 . 2 (𝑊 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)} ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵))
123, 11syl6bb 275 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  ‘cfv 5804  (class class class)co 6549  0cc0 9815  Vtxcvtx 25673   WWalkSN cwwlksn 41029   WWalksNOn cwwlksnon 41030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-wwlksn 41034  df-wwlksnon 41035 This theorem is referenced by:  wwlksnwwlksnon  41121  wspthsnwspthsnon  41122  wspthsnonn0vne  41124  elwwlks2ons3  41159  s3wwlks2on  41160  wpthswwlks2on  41164  elwspths2spth  41171
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