Theorem bnj579

Description: Technical lemma for bnj852 . 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	bnj579.1	|- ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
	bnj579.2	|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
	bnj579.3	|- D = ( _om \ { (/) } )
Assertion	bnj579	|- ( n e. D -> E* f ( f Fn n /\ ph /\ ps ) )

Proof

Step	Hyp	Ref	Expression
1		bnj579.1	|- ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
2		bnj579.2	|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3		bnj579.3	|- D = ( _om \ { (/) } )
4		biid	|- ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ph /\ ps ) )
5		biid	|- ( [. g / f ]. ph <-> [. g / f ]. ph )
6		biid	|- ( [. g / f ]. ps <-> [. g / f ]. ps )
7		biid	|- ( [. g / f ]. ( f Fn n /\ ph /\ ps ) <-> [. g / f ]. ( f Fn n /\ ph /\ ps ) )
8		biid	|- ( ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ [. g / f ]. ( f Fn n /\ ph /\ ps ) ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ [. g / f ]. ( f Fn n /\ ph /\ ps ) ) -> ( f ` j ) = ( g ` j ) ) )
9		biid	|- ( A. k e. n ( k _E j -> [. k / j ]. ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ [. g / f ]. ( f Fn n /\ ph /\ ps ) ) -> ( f ` j ) = ( g ` j ) ) ) <-> A. k e. n ( k _E j -> [. k / j ]. ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ [. g / f ]. ( f Fn n /\ ph /\ ps ) ) -> ( f ` j ) = ( g ` j ) ) ) )
10	1 2 4 5 6 7 3 8 9	bnj580	|- ( n e. D -> E* f ( f Fn n /\ ph /\ ps ) )