Metamath Proof Explorer

Theorem fmtnorn

Description: A Fermat number is a function value of the enumeration of the Fermat numbers. (Contributed by AV, 3-Aug-2021)

Ref Expression
Assertion fmtnorn 
|- ( F e. ran FermatNo <-> E. n e. NN0 ( FermatNo ` n ) = F )

Proof

Step Hyp Ref Expression
1 ovex 
 |-  ( ( 2 ^ ( 2 ^ n ) ) + 1 ) e. _V
2 df-fmtno 
 |-  FermatNo = ( n e. NN0 |-> ( ( 2 ^ ( 2 ^ n ) ) + 1 ) )
3 1 2 fnmpti 
 |-  FermatNo Fn NN0
4 fvelrnb 
 |-  ( FermatNo Fn NN0 -> ( F e. ran FermatNo <-> E. n e. NN0 ( FermatNo ` n ) = F ) )
5 3 4 ax-mp 
 |-  ( F e. ran FermatNo <-> E. n e. NN0 ( FermatNo ` n ) = F )