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Metamath Proof Explorer

Theorem wspthsnon

Description: The set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018) (Revised by AV, 11-May-2021)

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

Proof

Step Hyp Ref Expression
1 wwlksnon.v 
 |-  V = ( Vtx ` G )
2 df-wspthsnon 
 |-  WSPathsNOn = ( n e. NN0 , g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { w e. ( a ( n WWalksNOn g ) b ) | E. f f ( a ( SPathsOn ` g ) b ) w } ) )
3 2 a1i 
 |-  ( ( N e. NN0 /\ G e. U ) -> WSPathsNOn = ( n e. NN0 , g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { w e. ( a ( n WWalksNOn g ) b ) | E. f f ( a ( SPathsOn ` g ) b ) w } ) ) )
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 WWalksNOn g ) = ( N WWalksNOn G ) )
8 7 oveqd 
 |-  ( ( n = N /\ g = G ) -> ( a ( n WWalksNOn g ) b ) = ( a ( N WWalksNOn G ) b ) )
9 fveq2 
 |-  ( g = G -> ( SPathsOn ` g ) = ( SPathsOn ` G ) )
10 9 oveqd 
 |-  ( g = G -> ( a ( SPathsOn ` g ) b ) = ( a ( SPathsOn ` G ) b ) )
11 10 breqd 
 |-  ( g = G -> ( f ( a ( SPathsOn ` g ) b ) w <-> f ( a ( SPathsOn ` G ) b ) w ) )
12 11 adantl 
 |-  ( ( n = N /\ g = G ) -> ( f ( a ( SPathsOn ` g ) b ) w <-> f ( a ( SPathsOn ` G ) b ) w ) )
13 12 exbidv 
 |-  ( ( n = N /\ g = G ) -> ( E. f f ( a ( SPathsOn ` g ) b ) w <-> E. f f ( a ( SPathsOn ` G ) b ) w ) )
14 8 13 rabeqbidv 
 |-  ( ( n = N /\ g = G ) -> { w e. ( a ( n WWalksNOn g ) b ) | E. f f ( a ( SPathsOn ` g ) b ) w } = { w e. ( a ( N WWalksNOn G ) b ) | E. f f ( a ( SPathsOn ` G ) b ) w } )
15 6 6 14 mpoeq123dv 
 |-  ( ( n = N /\ g = G ) -> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { w e. ( a ( n WWalksNOn g ) b ) | E. f f ( a ( SPathsOn ` g ) b ) w } ) = ( a e. V , b e. V |-> { w e. ( a ( N WWalksNOn G ) b ) | E. f f ( a ( SPathsOn ` G ) b ) w } ) )
16 15 adantl 
 |-  ( ( ( N e. NN0 /\ G e. U ) /\ ( n = N /\ g = G ) ) -> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { w e. ( a ( n WWalksNOn g ) b ) | E. f f ( a ( SPathsOn ` g ) b ) w } ) = ( a e. V , b e. V |-> { w e. ( a ( N WWalksNOn G ) b ) | E. f f ( a ( SPathsOn ` G ) b ) w } ) )
17 simpl 
 |-  ( ( N e. NN0 /\ G e. U ) -> N e. NN0 )
18 elex 
 |-  ( G e. U -> G e. _V )
19 18 adantl 
 |-  ( ( N e. NN0 /\ G e. U ) -> G e. _V )
20 1 fvexi 
 |-  V e. _V
21 20 20 mpoex 
 |-  ( a e. V , b e. V |-> { w e. ( a ( N WWalksNOn G ) b ) | E. f f ( a ( SPathsOn ` G ) b ) w } ) e. _V
22 21 a1i 
 |-  ( ( N e. NN0 /\ G e. U ) -> ( a e. V , b e. V |-> { w e. ( a ( N WWalksNOn G ) b ) | E. f f ( a ( SPathsOn ` G ) b ) w } ) e. _V )
23 3 16 17 19 22 ovmpod 
 |-  ( ( N e. NN0 /\ G e. U ) -> ( N WSPathsNOn G ) = ( a e. V , b e. V |-> { w e. ( a ( N WWalksNOn G ) b ) | E. f f ( a ( SPathsOn ` G ) b ) w } ) )