Theorem wlkop 29568

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	⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)

Proof of Theorem wlkop

Step	Hyp	Ref	Expression
1		relwlk 29566	. 2 ⊢ Rel (Walks‘𝐺)
2		1st2nd 8055	. 2 ⊢ ((Rel (Walks‘𝐺) ∧ 𝑊 ∈ (Walks‘𝐺)) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
3	1, 2	mpan 688	1 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ⟨cop 4639  Rel wrel 5689  ‘cfv 6556  1^st c1st 8003  2^nd c2nd 8004  Walkscwlks 29536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fv 6564  df-1st 8005  df-2nd 8006  df-wlks 29539

This theorem is referenced by:  wlkcpr  29569  wlkeq  29574  clwlkcompbp  29722  clwlkclwwlkflem  29940  wlkl0  30303