Step |
Hyp |
Ref |
Expression |
1 |
|
df-pred |
|- Pred ( R , Pred ( R , A , X ) , X ) = ( Pred ( R , A , X ) i^i ( `' R " { X } ) ) |
2 |
|
df-pred |
|- Pred ( R , A , X ) = ( A i^i ( `' R " { X } ) ) |
3 |
|
inidm |
|- ( ( `' R " { X } ) i^i ( `' R " { X } ) ) = ( `' R " { X } ) |
4 |
3
|
ineq2i |
|- ( A i^i ( ( `' R " { X } ) i^i ( `' R " { X } ) ) ) = ( A i^i ( `' R " { X } ) ) |
5 |
2 4
|
eqtr4i |
|- Pred ( R , A , X ) = ( A i^i ( ( `' R " { X } ) i^i ( `' R " { X } ) ) ) |
6 |
|
inass |
|- ( ( A i^i ( `' R " { X } ) ) i^i ( `' R " { X } ) ) = ( A i^i ( ( `' R " { X } ) i^i ( `' R " { X } ) ) ) |
7 |
5 6
|
eqtr4i |
|- Pred ( R , A , X ) = ( ( A i^i ( `' R " { X } ) ) i^i ( `' R " { X } ) ) |
8 |
2
|
ineq1i |
|- ( Pred ( R , A , X ) i^i ( `' R " { X } ) ) = ( ( A i^i ( `' R " { X } ) ) i^i ( `' R " { X } ) ) |
9 |
7 8
|
eqtr4i |
|- Pred ( R , A , X ) = ( Pred ( R , A , X ) i^i ( `' R " { X } ) ) |
10 |
1 9
|
eqtr4i |
|- Pred ( R , Pred ( R , A , X ) , X ) = Pred ( R , A , X ) |