Step |
Hyp |
Ref |
Expression |
1 |
|
eqcom |
|- ( y = ( F ` B ) <-> ( F ` B ) = y ) |
2 |
|
fnbrfvb |
|- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = y <-> B F y ) ) |
3 |
1 2
|
bitrid |
|- ( ( F Fn A /\ B e. A ) -> ( y = ( F ` B ) <-> B F y ) ) |
4 |
3
|
abbidv |
|- ( ( F Fn A /\ B e. A ) -> { y | y = ( F ` B ) } = { y | B F y } ) |
5 |
|
df-sn |
|- { ( F ` B ) } = { y | y = ( F ` B ) } |
6 |
5
|
a1i |
|- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = { y | y = ( F ` B ) } ) |
7 |
|
fnrel |
|- ( F Fn A -> Rel F ) |
8 |
|
relimasn |
|- ( Rel F -> ( F " { B } ) = { y | B F y } ) |
9 |
7 8
|
syl |
|- ( F Fn A -> ( F " { B } ) = { y | B F y } ) |
10 |
9
|
adantr |
|- ( ( F Fn A /\ B e. A ) -> ( F " { B } ) = { y | B F y } ) |
11 |
4 6 10
|
3eqtr4d |
|- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) ) |