Metamath Proof Explorer


Theorem om2uz0i

Description: The mapping G is a one-to-one mapping from _om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number C (normally 0 for the upper integers NN0 or 1 for the upper integers NN ), 1 maps to C + 1, etc. This theorem shows the value of G at ordinal natural number zero. (This series of theorems generalizes an earlier series for NN0 contributed by Raph Levien, 10-Apr-2004.) (Contributed by NM, 3-Oct-2004) (Revised by Mario Carneiro, 13-Sep-2013)

Ref Expression
Hypotheses om2uz.1
|- C e. ZZ
om2uz.2
|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` _om )
Assertion om2uz0i
|- ( G ` (/) ) = C

Proof

Step Hyp Ref Expression
1 om2uz.1
 |-  C e. ZZ
2 om2uz.2
 |-  G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` _om )
3 2 fveq1i
 |-  ( G ` (/) ) = ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` _om ) ` (/) )
4 fr0g
 |-  ( C e. ZZ -> ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` _om ) ` (/) ) = C )
5 1 4 ax-mp
 |-  ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` _om ) ` (/) ) = C
6 3 5 eqtri
 |-  ( G ` (/) ) = C