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Metamath Proof Explorer

Theorem 3trld

Description: Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017) (Revised by AV, 8-Feb-2021) (Revised by AV, 24-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

Ref Expression
Hypotheses 3wlkd.p 
|- P = <" A B C D ">
3wlkd.f 
|- F = <" J K L ">
3wlkd.s 
|- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
3wlkd.n 
|- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
3wlkd.e 
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
3wlkd.v 
|- V = ( Vtx ` G )
3wlkd.i 
|- I = ( iEdg ` G )
3trld.n 
|- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )
Assertion 3trld 
|- ( ph -> F ( Trails ` G ) P )

Proof

Step Hyp Ref Expression
1 3wlkd.p 
 |-  P = <" A B C D ">
2 3wlkd.f 
 |-  F = <" J K L ">
3 3wlkd.s 
 |-  ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
4 3wlkd.n 
 |-  ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
5 3wlkd.e 
 |-  ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
6 3wlkd.v 
 |-  V = ( Vtx ` G )
7 3wlkd.i 
 |-  I = ( iEdg ` G )
8 3trld.n 
 |-  ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )
9 1 2 3 4 5 6 7 3wlkd 
 |-  ( ph -> F ( Walks ` G ) P )
10 1 2 3 4 5 3wlkdlem7 
 |-  ( ph -> ( J e. _V /\ K e. _V /\ L e. _V ) )
11 funcnvs3 
 |-  ( ( ( J e. _V /\ K e. _V /\ L e. _V ) /\ ( J =/= K /\ J =/= L /\ K =/= L ) ) -> Fun `' <" J K L "> )
12 10 8 11 syl2anc 
 |-  ( ph -> Fun `' <" J K L "> )
13 2 cnveqi 
 |-  `' F = `' <" J K L ">
14 13 funeqi 
 |-  ( Fun `' F <-> Fun `' <" J K L "> )
15 12 14 sylibr 
 |-  ( ph -> Fun `' F )
16 istrl 
 |-  ( F ( Trails ` G ) P <-> ( F ( Walks ` G ) P /\ Fun `' F ) )
17 9 15 16 sylanbrc 
 |-  ( ph -> F ( Trails ` G ) P )