Theorem bnj544

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	bnj544.1	|- ( ph' <-> ( f ` (/) ) = _pred ( x , A , R ) )
	bnj544.2	|- ( ps' <-> A. i e. _om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
	bnj544.3	|- D = ( _om \ { (/) } )
	bnj544.4	|- G = ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } )
	bnj544.5	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
	bnj544.6	|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
Assertion	bnj544	|- ( ( R _FrSe A /\ ta /\ si ) -> G Fn n )

Proof

Step	Hyp	Ref	Expression
1		bnj544.1	|- ( ph' <-> ( f ` (/) ) = _pred ( x , A , R ) )
2		bnj544.2	|- ( ps' <-> A. i e. _om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3		bnj544.3	|- D = ( _om \ { (/) } )
4		bnj544.4	|- G = ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } )
5		bnj544.5	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
6		bnj544.6	|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
7	3	bnj923	|- ( m e. D -> m e. _om )
8	7	3anim1i	|- ( ( m e. D /\ n = suc m /\ p e. m ) -> ( m e. _om /\ n = suc m /\ p e. m ) )
9	6 8	sylbi	|- ( si -> ( m e. _om /\ n = suc m /\ p e. m ) )
10		biid	|- ( ( m e. _om /\ n = suc m /\ p e. m ) <-> ( m e. _om /\ n = suc m /\ p e. m ) )
11	1 2 4 5 10	bnj543	|- ( ( R _FrSe A /\ ta /\ ( m e. _om /\ n = suc m /\ p e. m ) ) -> G Fn n )
12	9 11	syl3an3	|- ( ( R _FrSe A /\ ta /\ si ) -> G Fn n )