Step |
Hyp |
Ref |
Expression |
1 |
|
fnwe2.su |
|- ( z = ( F ` x ) -> S = U ) |
2 |
|
fnwe2.t |
|- T = { <. x , y >. | ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x U y ) ) } |
3 |
|
vex |
|- a e. _V |
4 |
|
vex |
|- b e. _V |
5 |
|
fveq2 |
|- ( x = a -> ( F ` x ) = ( F ` a ) ) |
6 |
|
fveq2 |
|- ( y = b -> ( F ` y ) = ( F ` b ) ) |
7 |
5 6
|
breqan12d |
|- ( ( x = a /\ y = b ) -> ( ( F ` x ) R ( F ` y ) <-> ( F ` a ) R ( F ` b ) ) ) |
8 |
5 6
|
eqeqan12d |
|- ( ( x = a /\ y = b ) -> ( ( F ` x ) = ( F ` y ) <-> ( F ` a ) = ( F ` b ) ) ) |
9 |
|
simpl |
|- ( ( x = a /\ y = b ) -> x = a ) |
10 |
|
fvex |
|- ( F ` x ) e. _V |
11 |
10 1
|
csbie |
|- [_ ( F ` x ) / z ]_ S = U |
12 |
5
|
csbeq1d |
|- ( x = a -> [_ ( F ` x ) / z ]_ S = [_ ( F ` a ) / z ]_ S ) |
13 |
11 12
|
eqtr3id |
|- ( x = a -> U = [_ ( F ` a ) / z ]_ S ) |
14 |
13
|
adantr |
|- ( ( x = a /\ y = b ) -> U = [_ ( F ` a ) / z ]_ S ) |
15 |
|
simpr |
|- ( ( x = a /\ y = b ) -> y = b ) |
16 |
9 14 15
|
breq123d |
|- ( ( x = a /\ y = b ) -> ( x U y <-> a [_ ( F ` a ) / z ]_ S b ) ) |
17 |
8 16
|
anbi12d |
|- ( ( x = a /\ y = b ) -> ( ( ( F ` x ) = ( F ` y ) /\ x U y ) <-> ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) ) |
18 |
7 17
|
orbi12d |
|- ( ( x = a /\ y = b ) -> ( ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x U y ) ) <-> ( ( F ` a ) R ( F ` b ) \/ ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) ) ) |
19 |
3 4 18 2
|
braba |
|- ( a T b <-> ( ( F ` a ) R ( F ` b ) \/ ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) ) |