Theorem wlkop 30190

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 30188	. 2 ⊢ Rel (Walks‘𝐺)
2		1st2nd 8039	. 2 ⊢ ((Rel (Walks‘𝐺) ∧ 𝑊 ∈ (Walks‘𝐺)) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
3	1, 2	mpan 703	1 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ⟨cop 4590  Rel wrel 5656  ‘cfv 6531  1^st c1st 7988  2^nd c2nd 7989  Walkscwlks 30159

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fv 6539  df-1st 7990  df-2nd 7991  df-wlks 30162

This theorem is used by:  wlkcpr  30191  wlkeq  30196  clwlkcompbp  30351  clwlkclwwlkflem  30577  wlkl0  30950