Theorem konigsbergssiedgwpr

Description: Each subset of the indexed edges of the Königsberg graph G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021)

	Ref	Expression
Hypotheses	konigsberg.v	|- V = ( 0 ... 3 )
	konigsberg.e	|- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">
	konigsberg.g	|- G = <. V , E >.
Assertion	konigsbergssiedgwpr	|- ( ( A e. Word _V /\ B e. Word _V /\ E = ( A ++ B ) ) -> A e. Word { x e. ~P V | ( # ` x ) = 2 } )

Proof

Step	Hyp	Ref	Expression
1		konigsberg.v	|- V = ( 0 ... 3 )
2		konigsberg.e	|- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">
3		konigsberg.g	|- G = <. V , E >.
4	1 2 3	konigsbergiedgw	|- E e. Word { x e. ~P V | ( # ` x ) = 2 }
5	4	jctr	|- ( E = ( A ++ B ) -> ( E = ( A ++ B ) /\ E e. Word { x e. ~P V | ( # ` x ) = 2 } ) )
6		ccatrcl1	|- ( ( A e. Word _V /\ B e. Word _V /\ ( E = ( A ++ B ) /\ E e. Word { x e. ~P V | ( # ` x ) = 2 } ) ) -> A e. Word { x e. ~P V | ( # ` x ) = 2 } )
7	5 6	syl3an3	|- ( ( A e. Word _V /\ B e. Word _V /\ E = ( A ++ B ) ) -> A e. Word { x e. ~P V | ( # ` x ) = 2 } )