Step |
Hyp |
Ref |
Expression |
1 |
|
dfimafn |
|- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A ( F ` x ) = y } ) |
2 |
1
|
inteqd |
|- ( ( Fun F /\ A C_ dom F ) -> |^| ( F " A ) = |^| { y | E. x e. A ( F ` x ) = y } ) |
3 |
|
fvex |
|- ( F ` x ) e. _V |
4 |
3
|
rgenw |
|- A. x e. A ( F ` x ) e. _V |
5 |
|
iinabrex |
|- ( A. x e. A ( F ` x ) e. _V -> |^|_ x e. A ( F ` x ) = |^| { y | E. x e. A y = ( F ` x ) } ) |
6 |
4 5
|
ax-mp |
|- |^|_ x e. A ( F ` x ) = |^| { y | E. x e. A y = ( F ` x ) } |
7 |
|
eqcom |
|- ( ( F ` x ) = y <-> y = ( F ` x ) ) |
8 |
7
|
rexbii |
|- ( E. x e. A ( F ` x ) = y <-> E. x e. A y = ( F ` x ) ) |
9 |
8
|
abbii |
|- { y | E. x e. A ( F ` x ) = y } = { y | E. x e. A y = ( F ` x ) } |
10 |
9
|
inteqi |
|- |^| { y | E. x e. A ( F ` x ) = y } = |^| { y | E. x e. A y = ( F ` x ) } |
11 |
6 10
|
eqtr4i |
|- |^|_ x e. A ( F ` x ) = |^| { y | E. x e. A ( F ` x ) = y } |
12 |
2 11
|
eqtr4di |
|- ( ( Fun F /\ A C_ dom F ) -> |^| ( F " A ) = |^|_ x e. A ( F ` x ) ) |