Theorem bnj917

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj917.1	|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
	bnj917.2	|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
	bnj917.3	|- D = ( _om \ { (/) } )
	bnj917.4	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
	bnj917.5	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
Assertion	bnj917	|- ( y e. _trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj917.1	|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2		bnj917.2	|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3		bnj917.3	|- D = ( _om \ { (/) } )
4		bnj917.4	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
5		bnj917.5	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
6		biid	|- ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ph /\ ps ) )
7	1 2 3 4 6	bnj916	|- ( y e. _trCl ( X , A , R ) -> E. f E. n E. i ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ i e. n /\ y e. ( f ` i ) ) )
8		bnj252	|- ( ( n e. D /\ f Fn n /\ ph /\ ps ) <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
9	5 8	bitri	|- ( ch <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
10	9	3anbi1i	|- ( ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) /\ i e. n /\ y e. ( f ` i ) ) )
11		bnj253	|- ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ i e. n /\ y e. ( f ` i ) ) <-> ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) /\ i e. n /\ y e. ( f ` i ) ) )
12	10 11	bitr4i	|- ( ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ i e. n /\ y e. ( f ` i ) ) )
13	12	3exbii	|- ( E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> E. f E. n E. i ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ i e. n /\ y e. ( f ` i ) ) )
14	7 13	sylibr	|- ( y e. _trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) )