let G be _Graph; :: thesis: union (G .componentSet()) = the_Vertices_of G

now

let x be set ; :: thesis: ( ( x in union (G .componentSet()) implies x in the_Vertices_of G ) & ( x in the_Vertices_of G implies x in union (G .componentSet()) ) )
thus ( x in union (G .componentSet()) implies x in the_Vertices_of G ) ; :: thesis: ( x in the_Vertices_of G implies x in union (G .componentSet()) )
assume x in the_Vertices_of G ; :: thesis: x in union (G .componentSet())
then reconsider x9 = x as Vertex of G ;
set Y = G .reachableFrom x9;
( x in G .reachableFrom x9 & G .reachableFrom x9 in G .componentSet() ) by Def8, Lm1;
hence x in union (G .componentSet()) by TARSKI:def 4; :: thesis: verum

end;

hence union (G .componentSet()) = the_Vertices_of G by TARSKI:1; :: thesis: verum