Step |
Hyp |
Ref |
Expression |
1 |
|
biid |
|- ( ( f ` (/) ) = _pred ( X , A , R ) <-> ( f ` (/) ) = _pred ( X , A , R ) ) |
2 |
|
biid |
|- ( A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) |
3 |
|
eqid |
|- ( _om \ { (/) } ) = ( _om \ { (/) } ) |
4 |
|
eqid |
|- { f | E. n e. ( _om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = _pred ( X , A , R ) /\ A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) } = { f | E. n e. ( _om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = _pred ( X , A , R ) /\ A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) } |
5 |
|
biid |
|- ( ( n e. ( _om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = _pred ( X , A , R ) /\ A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) <-> ( n e. ( _om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = _pred ( X , A , R ) /\ A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) ) |
6 |
|
biid |
|- ( ( ( n e. ( _om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = _pred ( X , A , R ) /\ A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) -> ( f ` i ) C_ A ) <-> ( ( n e. ( _om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = _pred ( X , A , R ) /\ A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) -> ( f ` i ) C_ A ) ) |
7 |
|
biid |
|- ( A. j e. n ( j _E i -> [. j / i ]. ( ( n e. ( _om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = _pred ( X , A , R ) /\ A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) -> ( f ` i ) C_ A ) ) <-> A. j e. n ( j _E i -> [. j / i ]. ( ( n e. ( _om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = _pred ( X , A , R ) /\ A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) -> ( f ` i ) C_ A ) ) ) |
8 |
|
biid |
|- ( [. j / i ]. ( f ` (/) ) = _pred ( X , A , R ) <-> [. j / i ]. ( f ` (/) ) = _pred ( X , A , R ) ) |
9 |
|
biid |
|- ( [. j / i ]. A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) <-> [. j / i ]. A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) |
10 |
|
biid |
|- ( [. j / i ]. ( n e. ( _om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = _pred ( X , A , R ) /\ A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) <-> [. j / i ]. ( n e. ( _om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = _pred ( X , A , R ) /\ A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) ) |
11 |
|
biid |
|- ( [. j / i ]. ( ( n e. ( _om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = _pred ( X , A , R ) /\ A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) -> ( f ` i ) C_ A ) <-> [. j / i ]. ( ( n e. ( _om \ { (/) } ) /\ f Fn n /\ ( f ` (/) ) = _pred ( X , A , R ) /\ A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) -> ( f ` i ) C_ A ) ) |
12 |
1 2 3 4 5 6 7 8 9 10 11
|
bnj1128 |
|- ( Y e. _trCl ( X , A , R ) -> Y e. A ) |