Metamath Proof Explorer


Theorem trlsegvdeglem3

Description: Lemma for trlsegvdeg . (Contributed by AV, 20-Feb-2021)

Ref Expression
Hypotheses trlsegvdeg.v
|- V = ( Vtx ` G )
trlsegvdeg.i
|- I = ( iEdg ` G )
trlsegvdeg.f
|- ( ph -> Fun I )
trlsegvdeg.n
|- ( ph -> N e. ( 0 ..^ ( # ` F ) ) )
trlsegvdeg.u
|- ( ph -> U e. V )
trlsegvdeg.w
|- ( ph -> F ( Trails ` G ) P )
trlsegvdeg.vx
|- ( ph -> ( Vtx ` X ) = V )
trlsegvdeg.vy
|- ( ph -> ( Vtx ` Y ) = V )
trlsegvdeg.vz
|- ( ph -> ( Vtx ` Z ) = V )
trlsegvdeg.ix
|- ( ph -> ( iEdg ` X ) = ( I |` ( F " ( 0 ..^ N ) ) ) )
trlsegvdeg.iy
|- ( ph -> ( iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
trlsegvdeg.iz
|- ( ph -> ( iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
Assertion trlsegvdeglem3
|- ( ph -> Fun ( iEdg ` Y ) )

Proof

Step Hyp Ref Expression
1 trlsegvdeg.v
 |-  V = ( Vtx ` G )
2 trlsegvdeg.i
 |-  I = ( iEdg ` G )
3 trlsegvdeg.f
 |-  ( ph -> Fun I )
4 trlsegvdeg.n
 |-  ( ph -> N e. ( 0 ..^ ( # ` F ) ) )
5 trlsegvdeg.u
 |-  ( ph -> U e. V )
6 trlsegvdeg.w
 |-  ( ph -> F ( Trails ` G ) P )
7 trlsegvdeg.vx
 |-  ( ph -> ( Vtx ` X ) = V )
8 trlsegvdeg.vy
 |-  ( ph -> ( Vtx ` Y ) = V )
9 trlsegvdeg.vz
 |-  ( ph -> ( Vtx ` Z ) = V )
10 trlsegvdeg.ix
 |-  ( ph -> ( iEdg ` X ) = ( I |` ( F " ( 0 ..^ N ) ) ) )
11 trlsegvdeg.iy
 |-  ( ph -> ( iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
12 trlsegvdeg.iz
 |-  ( ph -> ( iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
13 fvex
 |-  ( F ` N ) e. _V
14 fvex
 |-  ( I ` ( F ` N ) ) e. _V
15 13 14 pm3.2i
 |-  ( ( F ` N ) e. _V /\ ( I ` ( F ` N ) ) e. _V )
16 funsng
 |-  ( ( ( F ` N ) e. _V /\ ( I ` ( F ` N ) ) e. _V ) -> Fun { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
17 15 16 mp1i
 |-  ( ph -> Fun { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
18 11 funeqd
 |-  ( ph -> ( Fun ( iEdg ` Y ) <-> Fun { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )
19 17 18 mpbird
 |-  ( ph -> Fun ( iEdg ` Y ) )