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Metamath Proof Explorer

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 ) )