Theorem wlkop 16313

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 e. (Walks ` G ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )

Proof of Theorem wlkop

Step	Hyp	Ref	Expression
1		relwlk 16312	. 2 |- Rel (Walks ` G )
2		1st2nd 6366	. 2 |- ( ( Rel (Walks ` G ) /\ W e. (Walks ` G ) ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )
3	1, 2	mpan 424	1 |- ( W e. (Walks ` G ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   <.cop 3685   Rel wrel 4745   ` cfv 5343   1stc1st 6323   2ndc2nd 6324  Walkscwlks 16282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-br 4103  df-opab 4165  df-mpt 4166  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-iota 5303  df-fun 5345  df-fv 5351  df-1st 6325  df-2nd 6326  df-wlks 16283

This theorem is referenced by:  wlkelvv  16314  wlkcprim  16315