Theorem wlkop 27995

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 27993	. 2 ⊢ Rel (Walks‘𝐺)
2		1st2nd 7880	. 2 ⊢ ((Rel (Walks‘𝐺) ∧ 𝑊 ∈ (Walks‘𝐺)) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
3	1, 2	mpan 687	1 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ⟨cop 4567  Rel wrel 5594  ‘cfv 6433  1^st c1st 7829  2^nd c2nd 7830  Walkscwlks 27963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831  df-2nd 7832  df-wlks 27966

This theorem is referenced by:  wlkcpr  27996  wlkeq  28001  clwlkcompbp  28150  clwlkclwwlkflem  28368  wlkl0  28731