Metamath Proof Explorer

Theorem wwlksonvtx

Description: If a word W represents a walk of length 2 on two classes A and C , these classes are vertices. (Contributed by AV, 14-Mar-2022)

Ref Expression
Hypothesis wwlksonvtx.v 
|- V = ( Vtx ` G )
Assertion wwlksonvtx 
|- ( W e. ( A ( N WWalksNOn G ) C ) -> ( A e. V /\ C e. V ) )

Proof

Step Hyp Ref Expression
1 wwlksonvtx.v 
 |-  V = ( Vtx ` G )
2 fvex 
 |-  ( Vtx ` g ) e. _V
3 2 2 pm3.2i 
 |-  ( ( Vtx ` g ) e. _V /\ ( Vtx ` g ) e. _V )
4 3 rgen2w 
 |-  A. n e. NN0 A. g e. _V ( ( Vtx ` g ) e. _V /\ ( Vtx ` g ) e. _V )
5 df-wwlksnon 
 |-  WWalksNOn = ( n e. NN0 , g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { w e. ( n WWalksN g ) | ( ( w ` 0 ) = a /\ ( w ` n ) = b ) } ) )
6 fveq2 
 |-  ( g = G -> ( Vtx ` g ) = ( Vtx ` G ) )
7 6 6 jca 
 |-  ( g = G -> ( ( Vtx ` g ) = ( Vtx ` G ) /\ ( Vtx ` g ) = ( Vtx ` G ) ) )
8 7 adantl 
 |-  ( ( n = N /\ g = G ) -> ( ( Vtx ` g ) = ( Vtx ` G ) /\ ( Vtx ` g ) = ( Vtx ` G ) ) )
9 5 8 el2mpocl 
 |-  ( A. n e. NN0 A. g e. _V ( ( Vtx ` g ) e. _V /\ ( Vtx ` g ) e. _V ) -> ( W e. ( A ( N WWalksNOn G ) C ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) ) )
10 4 9 ax-mp 
 |-  ( W e. ( A ( N WWalksNOn G ) C ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) )
11 1 eleq2i 
 |-  ( A e. V <-> A e. ( Vtx ` G ) )
12 1 eleq2i 
 |-  ( C e. V <-> C e. ( Vtx ` G ) )
13 11 12 anbi12i 
 |-  ( ( A e. V /\ C e. V ) <-> ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) )
14 13 biimpri 
 |-  ( ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) -> ( A e. V /\ C e. V ) )
15 10 14 simpl2im 
 |-  ( W e. ( A ( N WWalksNOn G ) C ) -> ( A e. V /\ C e. V ) )