Theorem bnj1500

Description: Well-founded recursion, part 2 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf . (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1500.1	|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
	bnj1500.2	|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
	bnj1500.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
	bnj1500.4	|- F = U. C
Assertion	bnj1500	|- ( R _FrSe A -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` _pred ( x , A , R ) ) >. ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1500.1	|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2		bnj1500.2	|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3		bnj1500.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4		bnj1500.4	|- F = U. C
5		biid	|- ( ( R _FrSe A /\ x e. A ) <-> ( R _FrSe A /\ x e. A ) )
6		biid	|- ( ( ( R _FrSe A /\ x e. A ) /\ f e. C /\ x e. dom f ) <-> ( ( R _FrSe A /\ x e. A ) /\ f e. C /\ x e. dom f ) )
7		biid	|- ( ( ( ( R _FrSe A /\ x e. A ) /\ f e. C /\ x e. dom f ) /\ d e. B /\ dom f = d ) <-> ( ( ( R _FrSe A /\ x e. A ) /\ f e. C /\ x e. dom f ) /\ d e. B /\ dom f = d ) )
8	1 2 3 4 5 6 7	bnj1501	|- ( R _FrSe A -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` _pred ( x , A , R ) ) >. ) )