bnj1520

Description: Technical lemma for bnj1500 . 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	bnj1520.1	\|- B = { d \| ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
	bnj1520.2	\|- Y = <. x , ( f \|` _pred ( x , A , R ) ) >.
	bnj1520.3	\|- C = { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
	bnj1520.4	\|- F = U. C
Assertion	bnj1520	\|- ( ( F ` x ) = ( G ` <. x , ( F \|` _pred ( x , A , R ) ) >. ) -> A. f ( F ` x ) = ( G ` <. x , ( F \|` _pred ( x , A , R ) ) >. ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1520.1	\|- B = { d \| ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2		bnj1520.2	\|- Y = <. x , ( f \|` _pred ( x , A , R ) ) >.
3		bnj1520.3	\|- C = { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4		bnj1520.4	\|- F = U. C
5	3	bnj1317	\|- ( w e. C -> A. f w e. C )
6	5	nfcii	\|- F/_ f C
7	6	nfuni	\|- F/_ f U. C
8	4 7	nfcxfr	\|- F/_ f F
9		nfcv	\|- F/_ f x
10	8 9	nffv	\|- F/_ f ( F ` x )
11		nfcv	\|- F/_ f G
12		nfcv	\|- F/_ f _pred ( x , A , R )
13	8 12	nfres	\|- F/_ f ( F \|` _pred ( x , A , R ) )
14	9 13	nfop	\|- F/_ f <. x , ( F \|` _pred ( x , A , R ) ) >.
15	11 14	nffv	\|- F/_ f ( G ` <. x , ( F \|` _pred ( x , A , R ) ) >. )
16	10 15	nfeq	\|- F/ f ( F ` x ) = ( G ` <. x , ( F \|` _pred ( x , A , R ) ) >. )
17	16	nf5ri	\|- ( ( F ` x ) = ( G ` <. x , ( F \|` _pred ( x , A , R ) ) >. ) -> A. f ( F ` x ) = ( G ` <. x , ( F \|` _pred ( x , A , R ) ) >. ) )

Metamath Proof Explorer

Theorem bnj1520

Proof