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Metamath Proof Explorer

Theorem cardfz

Description: The cardinality of a finite set of sequential integers. (See om2uz0i for a description of the hypothesis.) (Contributed by NM, 7-Nov-2008) (Revised by Mario Carneiro, 15-Sep-2013)

Ref Expression
Hypothesis fzennn.1 
|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om )
Assertion cardfz 
|- ( N e. NN0 -> ( card ` ( 1 ... N ) ) = ( `' G ` N ) )

Proof

Step Hyp Ref Expression
1 fzennn.1 
 |-  G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om )
2 1 fzennn 
 |-  ( N e. NN0 -> ( 1 ... N ) ~~ ( `' G ` N ) )
3 carden2b 
 |-  ( ( 1 ... N ) ~~ ( `' G ` N ) -> ( card ` ( 1 ... N ) ) = ( card ` ( `' G ` N ) ) )
4 2 3 syl 
 |-  ( N e. NN0 -> ( card ` ( 1 ... N ) ) = ( card ` ( `' G ` N ) ) )
5 0z 
 |-  0 e. ZZ
6 5 1 om2uzf1oi 
 |-  G : _om -1-1-onto-> ( ZZ>= ` 0 )
7 elnn0uz 
 |-  ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) )
8 7 biimpi 
 |-  ( N e. NN0 -> N e. ( ZZ>= ` 0 ) )
9 f1ocnvdm 
 |-  ( ( G : _om -1-1-onto-> ( ZZ>= ` 0 ) /\ N e. ( ZZ>= ` 0 ) ) -> ( `' G ` N ) e. _om )
10 6 8 9 sylancr 
 |-  ( N e. NN0 -> ( `' G ` N ) e. _om )
11 cardnn 
 |-  ( ( `' G ` N ) e. _om -> ( card ` ( `' G ` N ) ) = ( `' G ` N ) )
12 10 11 syl 
 |-  ( N e. NN0 -> ( card ` ( `' G ` N ) ) = ( `' G ` N ) )
13 4 12 eqtrd 
 |-  ( N e. NN0 -> ( card ` ( 1 ... N ) ) = ( `' G ` N ) )