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Theorem wlkop 25850
Description: A walk (in an undirected simple graph) is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
Assertion
Ref Expression
wlkop (𝑊 ∈ (𝑉 Walks 𝐸) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)

Proof of Theorem wlkop
StepHypRef Expression
1 relwlk 25840 . 2 Rel (𝑉 Walks 𝐸)
2 1st2nd 7083 . 2 ((Rel (𝑉 Walks 𝐸) ∧ 𝑊 ∈ (𝑉 Walks 𝐸)) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
31, 2mpan 701 1 (𝑊 ∈ (𝑉 Walks 𝐸) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  cop 4130  Rel wrel 5033  cfv 5790  (class class class)co 6527  1st c1st 7035  2nd c2nd 7036   Walks cwalk 25820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-wlk 25830
This theorem is referenced by:  wlkcpr  25851  2wlkeq  26029
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