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Theorem iswwlkn 26212
 Description: Properties of a word to represent a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Assertion
Ref Expression
iswwlkn ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (𝑁 + 1))))

Proof of Theorem iswwlkn
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 wwlkn 26210 . . 3 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → ((𝑉 WWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 WWalks 𝐸) ∣ (#‘𝑤) = (𝑁 + 1)})
21eleq2d 2673 . 2 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ 𝑊 ∈ {𝑤 ∈ (𝑉 WWalks 𝐸) ∣ (#‘𝑤) = (𝑁 + 1)}))
3 fveq2 6103 . . . 4 (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
43eqeq1d 2612 . . 3 (𝑤 = 𝑊 → ((#‘𝑤) = (𝑁 + 1) ↔ (#‘𝑊) = (𝑁 + 1)))
54elrab 3331 . 2 (𝑊 ∈ {𝑤 ∈ (𝑉 WWalks 𝐸) ∣ (#‘𝑤) = (𝑁 + 1)} ↔ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (𝑁 + 1)))
62, 5syl6bb 275 1 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (𝑁 + 1))))
 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  1c1 9816   + caddc 9818  ℕ0cn0 11169  #chash 12979   WWalks cwwlk 26205   WWalksN cwwlkn 26206 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  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  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-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898  df-n0 11170  df-wwlkn 26208 This theorem is referenced by:  wwlknimp  26215  wwlkn0  26217  wlklniswwlkn1  26227  wlklniswwlkn2  26228  wwlkiswwlkn  26230  vfwlkniswwlkn  26234  wwlknred  26251  wwlknext  26252  wwlkextproplem3  26271  clwwlkel  26321  clwwlkf  26322  wwlksubclwwlk  26332  rusgranumwlkl1  26474  rusgra0edg  26482
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