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 e. Fin -> `' A e. Fin )

Proof

Step	Hyp	Ref	Expression
1		cnvcnvss	|- `' `' A C_ A
2		ssfi	|- ( ( A e. Fin /\ `' `' A C_ A ) -> `' `' A e. Fin )
3	1 2	mpan2	|- ( A e. Fin -> `' `' A e. Fin )
4		relcnv	|- Rel `' A
5		cnvexg	|- ( A e. Fin -> `' A e. _V )
6		cnven	|- ( ( Rel `' A /\ `' A e. _V ) -> `' A ~~ `' `' A )
7	4 5 6	sylancr	|- ( A e. Fin -> `' A ~~ `' `' A )
8		enfii	|- ( ( `' `' A e. Fin /\ `' A ~~ `' `' A ) -> `' A e. Fin )
9	3 7 8	syl2anc	|- ( A e. Fin -> `' A e. Fin )