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Metamath Proof Explorer

Theorem bnj1127

Description: Property of _trCl . (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Assertion bnj1127 
|- ( Y e. _trCl ( X , A , R ) -> Y e. A )

Proof

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 )