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 ) ) >. ) )

Metamath Proof Explorer

Theorem bnj1500

Proof