Theorem cnvfi

Description: If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014)

	Ref	Expression
Assertion	cnvfi	⊢ ( 𝐴 ∈ Fin → ◡ 𝐴 ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		cnvcnvss	⊢ ◡ ◡ 𝐴 ⊆ 𝐴
2		ssfi	⊢ ( ( 𝐴 ∈ Fin ∧ ◡ ◡ 𝐴 ⊆ 𝐴 ) → ◡ ◡ 𝐴 ∈ Fin )
3	1 2	mpan2	⊢ ( 𝐴 ∈ Fin → ◡ ◡ 𝐴 ∈ Fin )
4		relcnv	⊢ Rel ◡ 𝐴
5		cnvexg	⊢ ( 𝐴 ∈ Fin → ◡ 𝐴 ∈ V )
6		cnven	⊢ ( ( Rel ◡ 𝐴 ∧ ◡ 𝐴 ∈ V ) → ◡ 𝐴 ≈ ◡ ◡ 𝐴 )
7	4 5 6	sylancr	⊢ ( 𝐴 ∈ Fin → ◡ 𝐴 ≈ ◡ ◡ 𝐴 )
8		enfii	⊢ ( ( ◡ ◡ 𝐴 ∈ Fin ∧ ◡ 𝐴 ≈ ◡ ◡ 𝐴 ) → ◡ 𝐴 ∈ Fin )
9	3 7 8	syl2anc	⊢ ( 𝐴 ∈ Fin → ◡ 𝐴 ∈ Fin )