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Theorem wlkcpr 24302
Description: A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
Assertion
Ref Expression
wlkcpr  |-  ( W  e.  ( V Walks  E
)  <->  ( 1st `  W
) ( V Walks  E
) ( 2nd `  W
) )

Proof of Theorem wlkcpr
StepHypRef Expression
1 wlkop 24301 . . 3  |-  ( W  e.  ( V Walks  E
)  ->  W  =  <. ( 1st `  W
) ,  ( 2nd `  W ) >. )
2 eleq1 2539 . . . 4  |-  ( W  =  <. ( 1st `  W
) ,  ( 2nd `  W ) >.  ->  ( W  e.  ( V Walks  E )  <->  <. ( 1st `  W
) ,  ( 2nd `  W ) >.  e.  ( V Walks  E ) ) )
3 df-br 4448 . . . . 5  |-  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  <->  <. ( 1st `  W
) ,  ( 2nd `  W ) >.  e.  ( V Walks  E ) )
43biimpri 206 . . . 4  |-  ( <.
( 1st `  W
) ,  ( 2nd `  W ) >.  e.  ( V Walks  E )  -> 
( 1st `  W
) ( V Walks  E
) ( 2nd `  W
) )
52, 4syl6bi 228 . . 3  |-  ( W  =  <. ( 1st `  W
) ,  ( 2nd `  W ) >.  ->  ( W  e.  ( V Walks  E )  ->  ( 1st `  W ) ( V Walks 
E ) ( 2nd `  W ) ) )
61, 5mpcom 36 . 2  |-  ( W  e.  ( V Walks  E
)  ->  ( 1st `  W ) ( V Walks 
E ) ( 2nd `  W ) )
7 wlkn0 24300 . . . 4  |-  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  ->  ( 2nd `  W )  =/=  (/) )
8 2ndnpr 6790 . . . . 5  |-  ( -.  W  e.  ( _V 
X.  _V )  ->  ( 2nd `  W )  =  (/) )
98necon3ai 2695 . . . 4  |-  ( ( 2nd `  W )  =/=  (/)  ->  -.  -.  W  e.  ( _V  X.  _V ) )
107, 9syl 16 . . 3  |-  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  ->  -.  -.  W  e.  ( _V  X.  _V ) )
11 notnot 291 . . . 4  |-  ( W  e.  ( _V  X.  _V )  <->  -.  -.  W  e.  ( _V  X.  _V ) )
12 1st2nd2 6822 . . . . . . . 8  |-  ( W  e.  ( _V  X.  _V )  ->  W  = 
<. ( 1st `  W
) ,  ( 2nd `  W ) >. )
1312eqcomd 2475 . . . . . . 7  |-  ( W  e.  ( _V  X.  _V )  ->  <. ( 1st `  W ) ,  ( 2nd `  W
) >.  =  W )
1413eleq1d 2536 . . . . . 6  |-  ( W  e.  ( _V  X.  _V )  ->  ( <.
( 1st `  W
) ,  ( 2nd `  W ) >.  e.  ( V Walks  E )  <->  W  e.  ( V Walks  E ) ) )
1514biimpd 207 . . . . 5  |-  ( W  e.  ( _V  X.  _V )  ->  ( <.
( 1st `  W
) ,  ( 2nd `  W ) >.  e.  ( V Walks  E )  ->  W  e.  ( V Walks  E ) ) )
163, 15syl5bi 217 . . . 4  |-  ( W  e.  ( _V  X.  _V )  ->  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  ->  W  e.  ( V Walks  E ) ) )
1711, 16sylbir 213 . . 3  |-  ( -. 
-.  W  e.  ( _V  X.  _V )  ->  ( ( 1st `  W
) ( V Walks  E
) ( 2nd `  W
)  ->  W  e.  ( V Walks  E ) ) )
1810, 17mpcom 36 . 2  |-  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  ->  W  e.  ( V Walks  E ) )
196, 18impbii 188 1  |-  ( W  e.  ( V Walks  E
)  <->  ( 1st `  W
) ( V Walks  E
) ( 2nd `  W
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785   <.cop 4033   class class class wbr 4447    X. cxp 4997   ` cfv 5588  (class class class)co 6285   1stc1st 6783   2ndc2nd 6784   Walks cwalk 24271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-hash 12375  df-word 12509  df-wlk 24281
This theorem is referenced by:  vfwlkniswwlkn  24479  wlkv0  24533  wlk0  24534  wlkc  32044
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