Theorem wlkelvv 16361

Description: A walk is an ordered pair. (Contributed by Jim Kingdon, 2-Feb-2026.)

Assertion

Ref	Expression
wlkelvv	|- ( W e. (Walks ` G ) -> W e. ( _V X. _V ) )

Proof of Theorem wlkelvv

Step	Hyp	Ref	Expression
1		wlkop 16360	. 2 |- ( W e. (Walks ` G ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )
2		1stexg 6363	. . 3 |- ( W e. (Walks ` G ) -> ( 1st ` W ) e. _V )
3		2ndexg 6364	. . 3 |- ( W e. (Walks ` G ) -> ( 2nd ` W ) e. _V )
4	2, 3	opelxpd 4784	. 2 |- ( W e. (Walks ` G ) -> <. ( 1st ` W ) , ( 2nd ` W ) >. e. ( _V X. _V ) )
5	1, 4	eqeltrd 2311	1 |- ( W e. (Walks ` G ) -> W e. ( _V X. _V ) )

Syntax hints:   -> wi 4   e. wcel 2205   _Vcvv 2815   <.cop 3694   X. cxp 4749   ` cfv 5354   1stc1st 6334   2ndc2nd 6335  Walkscwlks 16329

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fo 5360  df-fv 5362  df-1st 6336  df-2nd 6337  df-wlks 16330

This theorem is referenced by:  wlkeq  16366