Step |
Hyp |
Ref |
Expression |
1 |
|
dfoprab2 |
|- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } |
2 |
1
|
rneqi |
|- ran { <. <. x , y >. , z >. | ph } = ran { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } |
3 |
|
rnopab |
|- ran { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } = { z | E. w E. x E. y ( w = <. x , y >. /\ ph ) } |
4 |
|
exrot3 |
|- ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y E. w ( w = <. x , y >. /\ ph ) ) |
5 |
|
opex |
|- <. x , y >. e. _V |
6 |
5
|
isseti |
|- E. w w = <. x , y >. |
7 |
|
19.41v |
|- ( E. w ( w = <. x , y >. /\ ph ) <-> ( E. w w = <. x , y >. /\ ph ) ) |
8 |
6 7
|
mpbiran |
|- ( E. w ( w = <. x , y >. /\ ph ) <-> ph ) |
9 |
8
|
2exbii |
|- ( E. x E. y E. w ( w = <. x , y >. /\ ph ) <-> E. x E. y ph ) |
10 |
4 9
|
bitri |
|- ( E. w E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y ph ) |
11 |
10
|
abbii |
|- { z | E. w E. x E. y ( w = <. x , y >. /\ ph ) } = { z | E. x E. y ph } |
12 |
2 3 11
|
3eqtri |
|- ran { <. <. x , y >. , z >. | ph } = { z | E. x E. y ph } |