Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - add the Axiom of Power Sets
Functions
abrexco
Next ⟩
imaiun
Metamath Proof Explorer
Ascii
Unicode
Theorem
abrexco
Description:
Composition of two image maps
C ( y )
and
B ( w )
.
(Contributed by
NM
, 27-May-2013)
Ref
Expression
Hypotheses
abrexco.1
⊢
B
∈
V
abrexco.2
⊢
y
=
B
→
C
=
D
Assertion
abrexco
⊢
x
|
∃
y
∈
z
|
∃
w
∈
A
z
=
B
x
=
C
=
x
|
∃
w
∈
A
x
=
D
Proof
Step
Hyp
Ref
Expression
1
abrexco.1
⊢
B
∈
V
2
abrexco.2
⊢
y
=
B
→
C
=
D
3
df-rex
⊢
∃
y
∈
z
|
∃
w
∈
A
z
=
B
x
=
C
↔
∃
y
y
∈
z
|
∃
w
∈
A
z
=
B
∧
x
=
C
4
vex
⊢
y
∈
V
5
eqeq1
⊢
z
=
y
→
z
=
B
↔
y
=
B
6
5
rexbidv
⊢
z
=
y
→
∃
w
∈
A
z
=
B
↔
∃
w
∈
A
y
=
B
7
4
6
elab
⊢
y
∈
z
|
∃
w
∈
A
z
=
B
↔
∃
w
∈
A
y
=
B
8
7
anbi1i
⊢
y
∈
z
|
∃
w
∈
A
z
=
B
∧
x
=
C
↔
∃
w
∈
A
y
=
B
∧
x
=
C
9
r19.41v
⊢
∃
w
∈
A
y
=
B
∧
x
=
C
↔
∃
w
∈
A
y
=
B
∧
x
=
C
10
8
9
bitr4i
⊢
y
∈
z
|
∃
w
∈
A
z
=
B
∧
x
=
C
↔
∃
w
∈
A
y
=
B
∧
x
=
C
11
10
exbii
⊢
∃
y
y
∈
z
|
∃
w
∈
A
z
=
B
∧
x
=
C
↔
∃
y
∃
w
∈
A
y
=
B
∧
x
=
C
12
3
11
bitri
⊢
∃
y
∈
z
|
∃
w
∈
A
z
=
B
x
=
C
↔
∃
y
∃
w
∈
A
y
=
B
∧
x
=
C
13
rexcom4
⊢
∃
w
∈
A
∃
y
y
=
B
∧
x
=
C
↔
∃
y
∃
w
∈
A
y
=
B
∧
x
=
C
14
12
13
bitr4i
⊢
∃
y
∈
z
|
∃
w
∈
A
z
=
B
x
=
C
↔
∃
w
∈
A
∃
y
y
=
B
∧
x
=
C
15
2
eqeq2d
⊢
y
=
B
→
x
=
C
↔
x
=
D
16
1
15
ceqsexv
⊢
∃
y
y
=
B
∧
x
=
C
↔
x
=
D
17
16
rexbii
⊢
∃
w
∈
A
∃
y
y
=
B
∧
x
=
C
↔
∃
w
∈
A
x
=
D
18
14
17
bitri
⊢
∃
y
∈
z
|
∃
w
∈
A
z
=
B
x
=
C
↔
∃
w
∈
A
x
=
D
19
18
abbii
⊢
x
|
∃
y
∈
z
|
∃
w
∈
A
z
=
B
x
=
C
=
x
|
∃
w
∈
A
x
=
D