Theorem eupthfi

Description: Any graph with an Eulerian path is of finite size, i.e. with a finite number of edges. (Contributed by Mario Carneiro, 7-Apr-2015) (Revised by AV, 18-Feb-2021)

	Ref	Expression
Hypothesis	eupths.i	|- I = ( iEdg ` G )
Assertion	eupthfi	|- ( F ( EulerPaths ` G ) P -> dom I e. Fin )

Proof

Step	Hyp	Ref	Expression
1		eupths.i	|- I = ( iEdg ` G )
2		fzofi	|- ( 0 ..^ ( # ` F ) ) e. Fin
3	1	eupthf1o	|- ( F ( EulerPaths ` G ) P -> F : ( 0 ..^ ( # ` F ) ) -1-1-onto-> dom I )
4		ovex	|- ( 0 ..^ ( # ` F ) ) e. _V
5	4	f1oen	|- ( F : ( 0 ..^ ( # ` F ) ) -1-1-onto-> dom I -> ( 0 ..^ ( # ` F ) ) ~~ dom I )
6		ensym	|- ( ( 0 ..^ ( # ` F ) ) ~~ dom I -> dom I ~~ ( 0 ..^ ( # ` F ) ) )
7	3 5 6	3syl	|- ( F ( EulerPaths ` G ) P -> dom I ~~ ( 0 ..^ ( # ` F ) ) )
8		enfii	|- ( ( ( 0 ..^ ( # ` F ) ) e. Fin /\ dom I ~~ ( 0 ..^ ( # ` F ) ) ) -> dom I e. Fin )
9	2 7 8	sylancr	|- ( F ( EulerPaths ` G ) P -> dom I e. Fin )