bnj1525

Description: Technical lemma for bnj1522 . 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	bnj1525.1	\|- B = { d \| ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
	bnj1525.2	\|- Y = <. x , ( f \|` _pred ( x , A , R ) ) >.
	bnj1525.3	\|- C = { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
	bnj1525.4	\|- F = U. C
	bnj1525.5	\|- ( ph <-> ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H \|` _pred ( x , A , R ) ) >. ) ) )
	bnj1525.6	\|- ( ps <-> ( ph /\ F =/= H ) )
Assertion	bnj1525	\|- ( ps -> A. x ps )

Proof

Step	Hyp	Ref	Expression
1		bnj1525.1	\|- B = { d \| ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2		bnj1525.2	\|- Y = <. x , ( f \|` _pred ( x , A , R ) ) >.
3		bnj1525.3	\|- C = { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4		bnj1525.4	\|- F = U. C
5		bnj1525.5	\|- ( ph <-> ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H \|` _pred ( x , A , R ) ) >. ) ) )
6		bnj1525.6	\|- ( ps <-> ( ph /\ F =/= H ) )
7		nfv	\|- F/ x R _FrSe A
8		nfv	\|- F/ x H Fn A
9		nfra1	\|- F/ x A. x e. A ( H ` x ) = ( G ` <. x , ( H \|` _pred ( x , A , R ) ) >. )
10	7 8 9	nf3an	\|- F/ x ( R _FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H \|` _pred ( x , A , R ) ) >. ) )
11	5 10	nfxfr	\|- F/ x ph
12	1	bnj1309	\|- ( w e. B -> A. x w e. B )
13	3 12	bnj1307	\|- ( w e. C -> A. x w e. C )
14	13	nfcii	\|- F/_ x C
15	14	nfuni	\|- F/_ x U. C
16	4 15	nfcxfr	\|- F/_ x F
17		nfcv	\|- F/_ x H
18	16 17	nfne	\|- F/ x F =/= H
19	11 18	nfan	\|- F/ x ( ph /\ F =/= H )
20	6 19	nfxfr	\|- F/ x ps
21	20	nf5ri	\|- ( ps -> A. x ps )

Metamath Proof Explorer

Theorem bnj1525

Proof