Theorem bnj125

Description: Technical lemma for bnj150 . 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	bnj125.1	|- ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
	bnj125.2	|- ( ph' <-> [. 1o / n ]. ph )
	bnj125.3	|- ( ph" <-> [. F / f ]. ph' )
	bnj125.4	|- F = { <. (/) , _pred ( x , A , R ) >. }
Assertion	bnj125	|- ( ph" <-> ( F ` (/) ) = _pred ( x , A , R ) )

Proof

Step	Hyp	Ref	Expression
1		bnj125.1	|- ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
2		bnj125.2	|- ( ph' <-> [. 1o / n ]. ph )
3		bnj125.3	|- ( ph" <-> [. F / f ]. ph' )
4		bnj125.4	|- F = { <. (/) , _pred ( x , A , R ) >. }
5	2	sbcbii	|- ( [. F / f ]. ph' <-> [. F / f ]. [. 1o / n ]. ph )
6		bnj105	|- 1o e. _V
7	1 6	bnj91	|- ( [. 1o / n ]. ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
8	7	sbcbii	|- ( [. F / f ]. [. 1o / n ]. ph <-> [. F / f ]. ( f ` (/) ) = _pred ( x , A , R ) )
9	4	bnj95	|- F e. _V
10		fveq1	|- ( f = F -> ( f ` (/) ) = ( F ` (/) ) )
11	10	eqeq1d	|- ( f = F -> ( ( f ` (/) ) = _pred ( x , A , R ) <-> ( F ` (/) ) = _pred ( x , A , R ) ) )
12	9 11	sbcie	|- ( [. F / f ]. ( f ` (/) ) = _pred ( x , A , R ) <-> ( F ` (/) ) = _pred ( x , A , R ) )
13	8 12	bitri	|- ( [. F / f ]. [. 1o / n ]. ph <-> ( F ` (/) ) = _pred ( x , A , R ) )
14	5 13	bitri	|- ( [. F / f ]. ph' <-> ( F ` (/) ) = _pred ( x , A , R ) )
15	3 14	bitri	|- ( ph" <-> ( F ` (/) ) = _pred ( x , A , R ) )