Theorem wlkcprim 16505

Description: A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Revised by Jim Kingdon, 1-Feb-2026.)

Assertion

Ref	Expression
wlkcprim	|- ( W e. (Walks ` G ) -> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) )

Proof of Theorem wlkcprim

Step	Hyp	Ref	Expression
1		wlkop 16503	. . . 4 |- ( W e. (Walks ` G ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )
2	1	eleq1d 2307	. . 3 |- ( W e. (Walks ` G ) -> ( W e. (Walks ` G ) <-> <. ( 1st ` W ) , ( 2nd ` W ) >. e. (Walks ` G ) ) )
3	2	ibi 176	. 2 |- ( W e. (Walks ` G ) -> <. ( 1st ` W ) , ( 2nd ` W ) >. e. (Walks ` G ) )
4		df-br 4126	. 2 |- ( ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) <-> <. ( 1st ` W ) , ( 2nd ` W ) >. e. (Walks ` G ) )
5	3, 4	sylibr 134	1 |- ( W e. (Walks ` G ) -> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) )

Syntax hints:   -> wi 4   e. wcel 2209   <.cop 3708   class class class wbr 4125   ` cfv 5372   1stc1st 6362   2ndc2nd 6363  Walkscwlks 16472

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fv 5380  df-1st 6364  df-2nd 6365  df-wlks 16473

This theorem is referenced by:  wlk2f  16506  wlkcompim  16507  wlkeq  16509  upgrwlkcompim  16517  uspgr2wlkeqi  16522  wlkv0  16524  g0wlk0  16525