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 ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		eupths.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2		fzofi	⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∈ Fin
3	1	eupthf1o	⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1-onto→ dom 𝐼 )
4		ovex	⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∈ 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 ) ) ∈ Fin ∧ dom 𝐼 ≈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → dom 𝐼 ∈ Fin )
9	2 7 8	sylancr	⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → dom 𝐼 ∈ Fin )