Library
▹
Essentials
▹
Functions
Proposition 1.5.23.
Let
X
be a set,
A
⊆
X
,
Y
be a set,
f
:
X
→
Y
be a
function
. Then:
f
∣
A
f
(
A
)
is surjective
Proof.
Let
y
∈
f
(
A
)
. Then ∃
x
∈
A
:
f
∣
A
f
(
A
)
(
x
)
=
y
:
y
∈
f
(
A
)
⇒
def
∃
a
∈
A
:
y
=
f
(
a
)
Take
x
:
=
a
.
f
∣
A
f
(
A
)
(
a
)
=
f
(
a
)
=
y
.