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Metamath Proof Explorer

Theorem wwlksnon

Description: The set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018) (Revised by AV, 11-May-2021)

Ref Expression
Hypothesis wwlksnon.v 
|- V = ( Vtx ` G )
Assertion wwlksnon 
|- ( ( N e. NN0 /\ G e. U ) -> ( N WWalksNOn G ) = ( a e. V , b e. V |-> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` N ) = b ) } ) )

Proof

Step Hyp Ref Expression
1 wwlksnon.v 
 |-  V = ( Vtx ` G )
2 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 ) } ) )
3 2 a1i 
 |-  ( ( N e. NN0 /\ G e. U ) -> 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 ) } ) ) )
4 fveq2 
 |-  ( g = G -> ( Vtx ` g ) = ( Vtx ` G ) )
5 4 1 eqtr4di 
 |-  ( g = G -> ( Vtx ` g ) = V )
6 5 adantl 
 |-  ( ( n = N /\ g = G ) -> ( Vtx ` g ) = V )
7 oveq12 
 |-  ( ( n = N /\ g = G ) -> ( n WWalksN g ) = ( N WWalksN G ) )
8 fveqeq2 
 |-  ( n = N -> ( ( w ` n ) = b <-> ( w ` N ) = b ) )
9 8 anbi2d 
 |-  ( n = N -> ( ( ( w ` 0 ) = a /\ ( w ` n ) = b ) <-> ( ( w ` 0 ) = a /\ ( w ` N ) = b ) ) )
10 9 adantr 
 |-  ( ( n = N /\ g = G ) -> ( ( ( w ` 0 ) = a /\ ( w ` n ) = b ) <-> ( ( w ` 0 ) = a /\ ( w ` N ) = b ) ) )
11 7 10 rabeqbidv 
 |-  ( ( n = N /\ g = G ) -> { w e. ( n WWalksN g ) | ( ( w ` 0 ) = a /\ ( w ` n ) = b ) } = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` N ) = b ) } )
12 6 6 11 mpoeq123dv 
 |-  ( ( n = N /\ g = G ) -> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { w e. ( n WWalksN g ) | ( ( w ` 0 ) = a /\ ( w ` n ) = b ) } ) = ( a e. V , b e. V |-> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` N ) = b ) } ) )
13 12 adantl 
 |-  ( ( ( N e. NN0 /\ G e. U ) /\ ( n = N /\ g = G ) ) -> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { w e. ( n WWalksN g ) | ( ( w ` 0 ) = a /\ ( w ` n ) = b ) } ) = ( a e. V , b e. V |-> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` N ) = b ) } ) )
14 simpl 
 |-  ( ( N e. NN0 /\ G e. U ) -> N e. NN0 )
15 elex 
 |-  ( G e. U -> G e. _V )
16 15 adantl 
 |-  ( ( N e. NN0 /\ G e. U ) -> G e. _V )
17 1 fvexi 
 |-  V e. _V
18 17 17 mpoex 
 |-  ( a e. V , b e. V |-> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` N ) = b ) } ) e. _V
19 18 a1i 
 |-  ( ( N e. NN0 /\ G e. U ) -> ( a e. V , b e. V |-> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` N ) = b ) } ) e. _V )
20 3 13 14 16 19 ovmpod 
 |-  ( ( N e. NN0 /\ G e. U ) -> ( N WWalksNOn G ) = ( a e. V , b e. V |-> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` N ) = b ) } ) )