Theorem hashginv

Description: The converse of G maps the size function's value to card . (Contributed by Paul Chapman, 22-Jun-2011) (Revised by Mario Carneiro, 15-Sep-2013)

	Ref	Expression
Hypothesis	hashgval.1	|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om )
Assertion	hashginv	|- ( A e. Fin -> ( `' G ` ( # ` A ) ) = ( card ` A ) )

Proof

Step	Hyp	Ref	Expression
1		hashgval.1	|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om )
2		ficardom	|- ( A e. Fin -> ( card ` A ) e. _om )
3	1	hashgval	|- ( A e. Fin -> ( G ` ( card ` A ) ) = ( # ` A ) )
4	1	hashgf1o	|- G : _om -1-1-onto-> NN0
5		f1ocnvfv	|- ( ( G : _om -1-1-onto-> NN0 /\ ( card ` A ) e. _om ) -> ( ( G ` ( card ` A ) ) = ( # ` A ) -> ( `' G ` ( # ` A ) ) = ( card ` A ) ) )
6	4 5	mpan	|- ( ( card ` A ) e. _om -> ( ( G ` ( card ` A ) ) = ( # ` A ) -> ( `' G ` ( # ` A ) ) = ( card ` A ) ) )
7	2 3 6	sylc	|- ( A e. Fin -> ( `' G ` ( # ` A ) ) = ( card ` A ) )