Theorem wlkop 29774

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 29772	. 2 ⊢ Rel (Walks‘𝐺)
2		1st2nd 8016	. 2 ⊢ ((Rel (Walks‘𝐺) ∧ 𝑊 ∈ (Walks‘𝐺)) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
3	1, 2	mpan 700	1 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ⟨cop 4587  Rel wrel 5650  ‘cfv 6517  1^st c1st 7964  2^nd c2nd 7965  Walkscwlks 29743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fv 6525  df-1st 7966  df-2nd 7967  df-wlks 29746

This theorem is referenced by:  wlkcpr  29775  wlkeq  29780  clwlkcompbp  29928  clwlkclwwlkflem  30152  wlkl0  30515