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Theorem wlkop 29441
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 (π‘Š ∈ (Walksβ€˜πΊ) β†’ π‘Š = ⟨(1st β€˜π‘Š), (2nd β€˜π‘Š)⟩)

Proof of Theorem wlkop
StepHypRef Expression
1 relwlk 29439 . 2 Rel (Walksβ€˜πΊ)
2 1st2nd 8043 . 2 ((Rel (Walksβ€˜πΊ) ∧ π‘Š ∈ (Walksβ€˜πΊ)) β†’ π‘Š = ⟨(1st β€˜π‘Š), (2nd β€˜π‘Š)⟩)
31, 2mpan 689 1 (π‘Š ∈ (Walksβ€˜πΊ) β†’ π‘Š = ⟨(1st β€˜π‘Š), (2nd β€˜π‘Š)⟩)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  βŸ¨cop 4635  Rel wrel 5683  β€˜cfv 6548  1st c1st 7991  2nd c2nd 7992  Walkscwlks 29409
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 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556  df-1st 7993  df-2nd 7994  df-wlks 29412
This theorem is referenced by:  wlkcpr  29442  wlkeq  29447  clwlkcompbp  29595  clwlkclwwlkflem  29813  wlkl0  30176
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