Metamath Proof Explorer

Theorem wspthnonp

Description: Properties of a set being a simple path of a fixed length between two vertices as word. (Contributed by AV, 14-May-2021) (Proof shortened by AV, 15-Mar-2022)

Ref Expression
Hypothesis wspthnonp.v 
|- V = ( Vtx ` G )
Assertion wspthnonp 
|- ( W e. ( A ( N WSPathsNOn G ) B ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( 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 wspthnonp.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-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 } ) )
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 WSPathsNOn G ) B ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) ) ) )
10 4 9 ax-mp 
 |-  ( W e. ( A ( N WSPathsNOn G ) B ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) ) )
11 simprl 
 |-  ( ( W e. ( A ( N WSPathsNOn G ) B ) /\ ( ( N e. NN0 /\ G e. _V ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) ) ) -> ( N e. NN0 /\ G e. _V ) )
12 1 eleq2i 
 |-  ( A e. V <-> A e. ( Vtx ` G ) )
13 1 eleq2i 
 |-  ( B e. V <-> B e. ( Vtx ` G ) )
14 12 13 anbi12i 
 |-  ( ( A e. V /\ B e. V ) <-> ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) )
15 14 biimpri 
 |-  ( ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) -> ( A e. V /\ B e. V ) )
16 15 adantl 
 |-  ( ( ( N e. NN0 /\ G e. _V ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) ) -> ( A e. V /\ B e. V ) )
17 16 adantl 
 |-  ( ( W e. ( A ( N WSPathsNOn G ) B ) /\ ( ( N e. NN0 /\ G e. _V ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) ) ) -> ( A e. V /\ B e. V ) )
18 wspthnon 
 |-  ( W e. ( A ( N WSPathsNOn G ) B ) <-> ( W e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A ( SPathsOn ` G ) B ) W ) )
19 18 biimpi 
 |-  ( W e. ( A ( N WSPathsNOn G ) B ) -> ( W e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A ( SPathsOn ` G ) B ) W ) )
20 19 adantr 
 |-  ( ( W e. ( A ( N WSPathsNOn G ) B ) /\ ( ( 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 ) )
21 11 17 20 3jca 
 |-  ( ( W e. ( A ( N WSPathsNOn G ) B ) /\ ( ( N e. NN0 /\ G e. _V ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) ) ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. V /\ B e. V ) /\ ( W e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A ( SPathsOn ` G ) B ) W ) ) )
22 10 21 mpdan 
 |-  ( W e. ( A ( N WSPathsNOn G ) B ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. V /\ B e. V ) /\ ( W e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A ( SPathsOn ` G ) B ) W ) ) )