now :: thesis: for b₁ being [:(dom p),J:] -defined Function st b₁ = uncurry p holds
b₁ is total

set j = the Element of J;
dom B = J by PARTFUN1:def 2;
then B . the Element of J in rng B by FUNCT_1:def 3;
then reconsider S = union (rng B) as non empty set by XBOOLE_1:3, ZFMISC_1:74;
rng p c= Funcs (J,S)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng p or x in Funcs (J,S) )
A1: rng p c= product B by FINSEQ_1:def 4;
assume x in rng p ; :: thesis: x in Funcs (J,S)
then consider f being Function such that
A2: x = f and
A3: dom f = dom B and
A4: for y being object st y in dom B holds
f . y in B . y by A1, CARD_3:def 5;
A5: rng f c= S

proof

let z be object ; :: according to TARSKI:def 3 :: thesis: ( not z in rng f or z in S )
assume z in rng f ; :: thesis: z in S
then consider y being object such that
A6: y in dom f and
A7: z = f . y by FUNCT_1:def 3;
B . y in rng B by A3, A6, FUNCT_1:def 3;
then B . y c= union (rng B) by ZFMISC_1:74;
hence z in S by A3, A4, A6, A7; :: thesis: verum

end;

dom B = J by PARTFUN1:def 2;
hence x in Funcs (J,S) by A2, A3, A5, FUNCT_2:def 2; :: thesis: verum

end;

hence for b₁ being [:(dom p),J:] -defined Function st b₁ = uncurry p holds
b₁ is total by PARTFUN1:def 2, FUNCT_5:26; :: thesis: verum

end;

hence for b₁ being [:(dom p),J:] -defined Function st b₁ = uncurry p holds
b₁ is total ; :: thesis: verum