now

consider j being Element of J;
dom B = J by PARTFUN1:def 4;
then B . j in rng B by FUNCT_1:def 5;
then reconsider S = union (rng B) as non empty set by XBOOLE_1:3, ZFMISC_1:92;
rng p c= Funcs J,S

proof

let x be set ; :: 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 set 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 set ; :: 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 set such that
A6: y in dom f and
A7: z = f . y by FUNCT_1:def 5;
B . y in rng B by A3, A6, FUNCT_1:def 5;
then A8: B . y c= union (rng B) by ZFMISC_1:92;
z in B . y by A3, A4, A6, A7;
hence z in S by A8; :: thesis: verum

end;

dom B = J by PARTFUN1:def 4;
hence x in Funcs J,S by A2, A3, A5, FUNCT_2:def 2; :: thesis: verum

end;

then dom (uncurry p) = [:(dom p),J:] by FUNCT_5:33;
hence for b₁ being [:(dom p),J:] -defined Function st b₁ = uncurry p holds
b₁ is total by PARTFUN1:def 4; :: thesis: verum

end;

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