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Metamath Proof Explorer

Theorem wlkp1lem3

Description: Lemma for wlkp1 . (Contributed by AV, 6-Mar-2021)

Ref Expression
Hypotheses wlkp1.v 
|- V = ( Vtx ` G )
wlkp1.i 
|- I = ( iEdg ` G )
wlkp1.f 
|- ( ph -> Fun I )
wlkp1.a 
|- ( ph -> I e. Fin )
wlkp1.b 
|- ( ph -> B e. W )
wlkp1.c 
|- ( ph -> C e. V )
wlkp1.d 
|- ( ph -> -. B e. dom I )
wlkp1.w 
|- ( ph -> F ( Walks ` G ) P )
wlkp1.n 
|- N = ( # ` F )
wlkp1.e 
|- ( ph -> E e. ( Edg ` G ) )
wlkp1.x 
|- ( ph -> { ( P ` N ) , C } C_ E )
wlkp1.u 
|- ( ph -> ( iEdg ` S ) = ( I u. { <. B , E >. } ) )
wlkp1.h 
|- H = ( F u. { <. N , B >. } )
Assertion wlkp1lem3 
|- ( ph -> ( ( iEdg ` S ) ` ( H ` N ) ) = ( ( I u. { <. B , E >. } ) ` B ) )

Proof

Step Hyp Ref Expression
1 wlkp1.v 
 |-  V = ( Vtx ` G )
2 wlkp1.i 
 |-  I = ( iEdg ` G )
3 wlkp1.f 
 |-  ( ph -> Fun I )
4 wlkp1.a 
 |-  ( ph -> I e. Fin )
5 wlkp1.b 
 |-  ( ph -> B e. W )
6 wlkp1.c 
 |-  ( ph -> C e. V )
7 wlkp1.d 
 |-  ( ph -> -. B e. dom I )
8 wlkp1.w 
 |-  ( ph -> F ( Walks ` G ) P )
9 wlkp1.n 
 |-  N = ( # ` F )
10 wlkp1.e 
 |-  ( ph -> E e. ( Edg ` G ) )
11 wlkp1.x 
 |-  ( ph -> { ( P ` N ) , C } C_ E )
12 wlkp1.u 
 |-  ( ph -> ( iEdg ` S ) = ( I u. { <. B , E >. } ) )
13 wlkp1.h 
 |-  H = ( F u. { <. N , B >. } )
14 13 a1i 
 |-  ( ph -> H = ( F u. { <. N , B >. } ) )
15 14 fveq1d 
 |-  ( ph -> ( H ` N ) = ( ( F u. { <. N , B >. } ) ` N ) )
16 9 fvexi 
 |-  N e. _V
17 2 wlkf 
 |-  ( F ( Walks ` G ) P -> F e. Word dom I )
18 lencl 
 |-  ( F e. Word dom I -> ( # ` F ) e. NN0 )
19 wrddm 
 |-  ( F e. Word dom I -> dom F = ( 0 ..^ ( # ` F ) ) )
20 fzonel 
 |-  -. ( # ` F ) e. ( 0 ..^ ( # ` F ) )
21 9 a1i 
 |-  ( ( ( # ` F ) e. NN0 /\ dom F = ( 0 ..^ ( # ` F ) ) ) -> N = ( # ` F ) )
22 simpr 
 |-  ( ( ( # ` F ) e. NN0 /\ dom F = ( 0 ..^ ( # ` F ) ) ) -> dom F = ( 0 ..^ ( # ` F ) ) )
23 21 22 eleq12d 
 |-  ( ( ( # ` F ) e. NN0 /\ dom F = ( 0 ..^ ( # ` F ) ) ) -> ( N e. dom F <-> ( # ` F ) e. ( 0 ..^ ( # ` F ) ) ) )
24 20 23 mtbiri 
 |-  ( ( ( # ` F ) e. NN0 /\ dom F = ( 0 ..^ ( # ` F ) ) ) -> -. N e. dom F )
25 18 19 24 syl2anc 
 |-  ( F e. Word dom I -> -. N e. dom F )
26 8 17 25 3syl 
 |-  ( ph -> -. N e. dom F )
27 fsnunfv 
 |-  ( ( N e. _V /\ B e. W /\ -. N e. dom F ) -> ( ( F u. { <. N , B >. } ) ` N ) = B )
28 16 5 26 27 mp3an2i 
 |-  ( ph -> ( ( F u. { <. N , B >. } ) ` N ) = B )
29 15 28 eqtrd 
 |-  ( ph -> ( H ` N ) = B )
30 12 29 fveq12d 
 |-  ( ph -> ( ( iEdg ` S ) ` ( H ` N ) ) = ( ( I u. { <. B , E >. } ) ` B ) )