Theorem wlkop 16589

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 16588	. 2 |- Rel (Walks ` G )
2		1st2nd 6415	. 2 |- ( ( Rel (Walks ` G ) /\ W e. (Walks ` G ) ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )
3	1, 2	mpan 428	1 |- ( W e. (Walks ` G ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   <.cop 3712   Rel wrel 4779   ` cfv 5377   1stc1st 6372   2ndc2nd 6373  Walkscwlks 16558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fv 5385  df-1st 6374  df-2nd 6375  df-wlks 16559

This theorem is used by:  wlkelvv  16590  wlkcprim  16591