Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( 0 ..^ 0 ) = ( 0 ..^ 0 ) |
2 |
1
|
naryrcl |
|- ( F e. ( 0 -aryF X ) -> ( 0 e. NN0 /\ X e. _V ) ) |
3 |
|
0aryfvalel |
|- ( X e. _V -> ( F e. ( 0 -aryF X ) <-> E. x e. X F = { <. (/) , x >. } ) ) |
4 |
|
0ex |
|- (/) e. _V |
5 |
|
fvsng |
|- ( ( (/) e. _V /\ x e. X ) -> ( { <. (/) , x >. } ` (/) ) = x ) |
6 |
4 5
|
mpan |
|- ( x e. X -> ( { <. (/) , x >. } ` (/) ) = x ) |
7 |
|
fveq1 |
|- ( F = { <. (/) , x >. } -> ( F ` (/) ) = ( { <. (/) , x >. } ` (/) ) ) |
8 |
7
|
eqeq1d |
|- ( F = { <. (/) , x >. } -> ( ( F ` (/) ) = x <-> ( { <. (/) , x >. } ` (/) ) = x ) ) |
9 |
6 8
|
syl5ibrcom |
|- ( x e. X -> ( F = { <. (/) , x >. } -> ( F ` (/) ) = x ) ) |
10 |
9
|
reximia |
|- ( E. x e. X F = { <. (/) , x >. } -> E. x e. X ( F ` (/) ) = x ) |
11 |
3 10
|
syl6bi |
|- ( X e. _V -> ( F e. ( 0 -aryF X ) -> E. x e. X ( F ` (/) ) = x ) ) |
12 |
11
|
adantl |
|- ( ( 0 e. NN0 /\ X e. _V ) -> ( F e. ( 0 -aryF X ) -> E. x e. X ( F ` (/) ) = x ) ) |
13 |
2 12
|
mpcom |
|- ( F e. ( 0 -aryF X ) -> E. x e. X ( F ` (/) ) = x ) |