defpred S₁[ Element of NAT , set ] means $2 = F₁($1);
A1: for x being Element of NAT ex y being Element of NAT st S₁[x,y]

proof

let x be Element of NAT ; :: thesis: ex y being Element of NAT st S₁[x,y]
reconsider y = F₁(x) as Element of NAT by ORDINAL1:def 12;
take y ; :: thesis: S₁[x,y]
thus S₁[x,y] ; :: thesis: verum

end;

consider f being Function of NAT,NAT such that
A2: for x being Element of NAT holds S₁[x,f . x] from FUNCT_2:sch 3(A1);
take f ; :: thesis: for x being natural number holds f . x = F₁(x)
let c be natural number ; :: thesis: f . c = F₁(c)
reconsider c = c as Element of NAT by ORDINAL1:def 12;
S₁[c,f . c] by A2;
hence f . c = F₁(c) ; :: thesis: verum