Theorem bnj1307

Description: Technical lemma for bnj60 . 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	bnj1307.1	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
	bnj1307.2	|- ( w e. B -> A. x w e. B )
Assertion	bnj1307	|- ( w e. C -> A. x w e. C )

Proof

Step	Hyp	Ref	Expression
1		bnj1307.1	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
2		bnj1307.2	|- ( w e. B -> A. x w e. B )
3	2	nfcii	|- F/_ x B
4		nfv	|- F/ x f Fn d
5		nfra1	|- F/ x A. x e. d ( f ` x ) = ( G ` Y )
6	4 5	nfan	|- F/ x ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) )
7	3 6	nfrex	|- F/ x E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) )
8	7	nfab	|- F/_ x { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
9	1 8	nfcxfr	|- F/_ x C
10	9	nfcrii	|- ( w e. C -> A. x w e. C )