Theorem bnj1522

Description: Well-founded recursion, part 3 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	bnj1522.1	|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
	bnj1522.2	|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
	bnj1522.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
	bnj1522.4	|- F = U. C
Assertion	bnj1522	|- ( ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) -> F = H )

Proof

Step	Hyp	Ref	Expression
1		bnj1522.1	|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2		bnj1522.2	|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3		bnj1522.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4		bnj1522.4	|- F = U. C
5		biid	|- ( ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) <-> ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) )
6		biid	|- ( ( ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) /\ F =/= H ) <-> ( ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) /\ F =/= H ) )
7		biid	|- ( ( ( ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) /\ F =/= H ) /\ x e. A /\ ( F ` x ) =/= ( H ` x ) ) <-> ( ( ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) /\ F =/= H ) /\ x e. A /\ ( F ` x ) =/= ( H ` x ) ) )
8		eqid	|- { x e. A | ( F ` x ) =/= ( H ` x ) } = { x e. A | ( F ` x ) =/= ( H ` x ) }
9		biid	|- ( ( ( ( ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) /\ F =/= H ) /\ x e. A /\ ( F ` x ) =/= ( H ` x ) ) /\ y e. { x e. A | ( F ` x ) =/= ( H ` x ) } /\ A. z e. { x e. A | ( F ` x ) =/= ( H ` x ) } -. z R y ) <-> ( ( ( ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) /\ F =/= H ) /\ x e. A /\ ( F ` x ) =/= ( H ` x ) ) /\ y e. { x e. A | ( F ` x ) =/= ( H ` x ) } /\ A. z e. { x e. A | ( F ` x ) =/= ( H ` x ) } -. z R y ) )
10	1 2 3 4 5 6 7 8 9	bnj1523	|- ( ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` _pred ( x , A , R ) ) >. ) ) -> F = H )