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Theorem wlkop 29379
Description: A walk is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 1-Jan-2021.)
Assertion
Ref Expression
wlkop (π‘Š ∈ (Walksβ€˜πΊ) β†’ π‘Š = ⟨(1st β€˜π‘Š), (2nd β€˜π‘Š)⟩)

Proof of Theorem wlkop
StepHypRef Expression
1 relwlk 29377 . 2 Rel (Walksβ€˜πΊ)
2 1st2nd 8019 . 2 ((Rel (Walksβ€˜πΊ) ∧ π‘Š ∈ (Walksβ€˜πΊ)) β†’ π‘Š = ⟨(1st β€˜π‘Š), (2nd β€˜π‘Š)⟩)
31, 2mpan 687 1 (π‘Š ∈ (Walksβ€˜πΊ) β†’ π‘Š = ⟨(1st β€˜π‘Š), (2nd β€˜π‘Š)⟩)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4627  Rel wrel 5672  β€˜cfv 6534  1st c1st 7967  2nd c2nd 7968  Walkscwlks 29347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fv 6542  df-1st 7969  df-2nd 7970  df-wlks 29350
This theorem is referenced by:  wlkcpr  29380  wlkeq  29385  clwlkcompbp  29533  clwlkclwwlkflem  29751  wlkl0  30114
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