Step |
Hyp |
Ref |
Expression |
1 |
|
difss |
|- ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) ) C_ ( _V X. _V ) |
2 |
|
df-rel |
|- ( Rel ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) ) <-> ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) ) C_ ( _V X. _V ) ) |
3 |
1 2
|
mpbir |
|- Rel ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) ) |
4 |
|
df-image |
|- Image A = ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) ) |
5 |
4
|
releqi |
|- ( Rel Image A <-> Rel ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) ) ) |
6 |
3 5
|
mpbir |
|- Rel Image A |
7 |
|
vex |
|- x e. _V |
8 |
|
vex |
|- y e. _V |
9 |
7 8
|
brimage |
|- ( x Image A y <-> y = ( A " x ) ) |
10 |
|
vex |
|- z e. _V |
11 |
7 10
|
brimage |
|- ( x Image A z <-> z = ( A " x ) ) |
12 |
|
eqtr3 |
|- ( ( y = ( A " x ) /\ z = ( A " x ) ) -> y = z ) |
13 |
9 11 12
|
syl2anb |
|- ( ( x Image A y /\ x Image A z ) -> y = z ) |
14 |
13
|
gen2 |
|- A. y A. z ( ( x Image A y /\ x Image A z ) -> y = z ) |
15 |
14
|
ax-gen |
|- A. x A. y A. z ( ( x Image A y /\ x Image A z ) -> y = z ) |
16 |
|
dffun2 |
|- ( Fun Image A <-> ( Rel Image A /\ A. x A. y A. z ( ( x Image A y /\ x Image A z ) -> y = z ) ) ) |
17 |
6 15 16
|
mpbir2an |
|- Fun Image A |