let I be set ; :: thesis: for A being ManySortedSet of I
for B, C being non-empty ManySortedSet of I
for F being ManySortedFunction of A,[|B,C|] st F is "onto" holds
Mpr1 F is "onto"
let A be ManySortedSet of I; :: thesis: for B, C being non-empty ManySortedSet of I
for F being ManySortedFunction of A,[|B,C|] st F is "onto" holds
Mpr1 F is "onto"
let B, C be non-empty ManySortedSet of I; :: thesis: for F being ManySortedFunction of A,[|B,C|] st F is "onto" holds
Mpr1 F is "onto"
let F be ManySortedFunction of A,[|B,C|]; :: thesis: ( F is "onto" implies Mpr1 F is "onto" )
assume A1: F is "onto" ; :: thesis: Mpr1 F is "onto"
let i be set ; :: according to MSUALG_3:def 3 :: thesis: ( not i in I or rng ((Mpr1 F) . i) = B . i )
assume A2: i in I ; :: thesis: rng ((Mpr1 F) . i) = B . i
then reconsider m = (Mpr1 F) . i as Function of (A . i),(B . i) by PBOOLE:def 15;
rng m = B . i

proof

thus rng m c= B . i ; :: according to XBOOLE_0:def 10 :: thesis: B . i c= rng m
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in B . i or a in rng m )
assume A3: a in B . i ; :: thesis: a in rng m
consider z being object such that
A4: z in [|B,C|] . i by A2, XBOOLE_0:def 1;
set p = [a,(z `2)];
z in [:(B . i),(C . i):] by A2, A4, PBOOLE:def 16;
then A5: [a,(z `2)] `2 in C . i by MCART_1:10;
[a,(z `2)] `1 in B . i by A3;
then [a,(z `2)] in [:(B . i),(C . i):] by A5, ZFMISC_1:def 2;
then A6: [a,(z `2)] in [|B,C|] . i by A2, PBOOLE:def 16;
A7: dom m = A . i by A2, FUNCT_2:def 1;
A8: F . i is Function of (A . i),([|B,C|] . i) by A2, PBOOLE:def 15;
then A9: dom (F . i) = A . i by A2, FUNCT_2:def 1;
rng (F . i) = [|B,C|] . i by A1, A2;
then consider b being object such that
A10: b in A . i and
A11: (F . i) . b = [a,(z `2)] by A8, A6, FUNCT_2:11;
m . b = (pr1 (F . i)) . b by A2, Def1
.= [a,(z `2)] `1 by A10, A11, A9, MCART_1:def 12
.= a ;
hence a in rng m by A10, A7, FUNCT_1:def 3; :: thesis: verum

end;

hence rng ((Mpr1 F) . i) = B . i ; :: thesis: verum