Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } |
2 |
1
|
tfrlem3 |
|- { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } = { g | E. z e. On ( g Fn z /\ A. v e. z ( g ` v ) = ( G ` ( g |` v ) ) ) } |
3 |
|
fveq2 |
|- ( v = w -> ( g ` v ) = ( g ` w ) ) |
4 |
|
reseq2 |
|- ( v = w -> ( g |` v ) = ( g |` w ) ) |
5 |
4
|
fveq2d |
|- ( v = w -> ( G ` ( g |` v ) ) = ( G ` ( g |` w ) ) ) |
6 |
3 5
|
eqeq12d |
|- ( v = w -> ( ( g ` v ) = ( G ` ( g |` v ) ) <-> ( g ` w ) = ( G ` ( g |` w ) ) ) ) |
7 |
6
|
cbvralvw |
|- ( A. v e. z ( g ` v ) = ( G ` ( g |` v ) ) <-> A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) |
8 |
7
|
anbi2i |
|- ( ( g Fn z /\ A. v e. z ( g ` v ) = ( G ` ( g |` v ) ) ) <-> ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) ) |
9 |
8
|
rexbii |
|- ( E. z e. On ( g Fn z /\ A. v e. z ( g ` v ) = ( G ` ( g |` v ) ) ) <-> E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) ) |
10 |
9
|
abbii |
|- { g | E. z e. On ( g Fn z /\ A. v e. z ( g ` v ) = ( G ` ( g |` v ) ) ) } = { g | E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) } |
11 |
2 10
|
eqtri |
|- { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } = { g | E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) } |