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Metamath Proof Explorer

Theorem 2trld

Description: Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017) (Revised by AV, 24-Jan-2021) (Revised by AV, 24-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

Ref Expression
Hypotheses 2wlkd.p 
|- P = <" A B C ">
2wlkd.f 
|- F = <" J K ">
2wlkd.s 
|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
2wlkd.n 
|- ( ph -> ( A =/= B /\ B =/= C ) )
2wlkd.e 
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
2wlkd.v 
|- V = ( Vtx ` G )
2wlkd.i 
|- I = ( iEdg ` G )
2trld.n 
|- ( ph -> J =/= K )
Assertion 2trld 
|- ( ph -> F ( Trails ` G ) P )

Proof

Step Hyp Ref Expression
1 2wlkd.p 
 |-  P = <" A B C ">
2 2wlkd.f 
 |-  F = <" J K ">
3 2wlkd.s 
 |-  ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
4 2wlkd.n 
 |-  ( ph -> ( A =/= B /\ B =/= C ) )
5 2wlkd.e 
 |-  ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
6 2wlkd.v 
 |-  V = ( Vtx ` G )
7 2wlkd.i 
 |-  I = ( iEdg ` G )
8 2trld.n 
 |-  ( ph -> J =/= K )
9 1 2 3 4 5 6 7 2wlkd 
 |-  ( ph -> F ( Walks ` G ) P )
10 1 2 3 4 5 2wlkdlem7 
 |-  ( ph -> ( J e. _V /\ K e. _V ) )
11 df-3an 
 |-  ( ( J e. _V /\ K e. _V /\ J =/= K ) <-> ( ( J e. _V /\ K e. _V ) /\ J =/= K ) )
12 10 8 11 sylanbrc 
 |-  ( ph -> ( J e. _V /\ K e. _V /\ J =/= K ) )
13 funcnvs2 
 |-  ( ( J e. _V /\ K e. _V /\ J =/= K ) -> Fun `' <" J K "> )
14 12 13 syl 
 |-  ( ph -> Fun `' <" J K "> )
15 2 cnveqi 
 |-  `' F = `' <" J K ">
16 15 funeqi 
 |-  ( Fun `' F <-> Fun `' <" J K "> )
17 14 16 sylibr 
 |-  ( ph -> Fun `' F )
18 istrl 
 |-  ( F ( Trails ` G ) P <-> ( F ( Walks ` G ) P /\ Fun `' F ) )
19 9 17 18 sylanbrc 
 |-  ( ph -> F ( Trails ` G ) P )