bnj1447

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	bnj1447.1	\|- B = { d \| ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
	bnj1447.2	\|- Y = <. x , ( f \|` _pred ( x , A , R ) ) >.
	bnj1447.3	\|- C = { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
	bnj1447.4	\|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
	bnj1447.5	\|- D = { x e. A \| -. E. f ta }
	bnj1447.6	\|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
	bnj1447.7	\|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
	bnj1447.8	\|- ( ta' <-> [. y / x ]. ta )
	bnj1447.9	\|- H = { f \| E. y e. _pred ( x , A , R ) ta' }
	bnj1447.10	\|- P = U. H
	bnj1447.11	\|- Z = <. x , ( P \|` _pred ( x , A , R ) ) >.
	bnj1447.12	\|- Q = ( P u. { <. x , ( G ` Z ) >. } )
	bnj1447.13	\|- W = <. z , ( Q \|` _pred ( z , A , R ) ) >.
Assertion	bnj1447	\|- ( ( Q ` z ) = ( G ` W ) -> A. y ( Q ` z ) = ( G ` W ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1447.1	\|- B = { d \| ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2		bnj1447.2	\|- Y = <. x , ( f \|` _pred ( x , A , R ) ) >.
3		bnj1447.3	\|- C = { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4		bnj1447.4	\|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5		bnj1447.5	\|- D = { x e. A \| -. E. f ta }
6		bnj1447.6	\|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7		bnj1447.7	\|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8		bnj1447.8	\|- ( ta' <-> [. y / x ]. ta )
9		bnj1447.9	\|- H = { f \| E. y e. _pred ( x , A , R ) ta' }
10		bnj1447.10	\|- P = U. H
11		bnj1447.11	\|- Z = <. x , ( P \|` _pred ( x , A , R ) ) >.
12		bnj1447.12	\|- Q = ( P u. { <. x , ( G ` Z ) >. } )
13		bnj1447.13	\|- W = <. z , ( Q \|` _pred ( z , A , R ) ) >.
14		nfre1	\|- F/ y E. y e. _pred ( x , A , R ) ta'
15	14	nfab	\|- F/_ y { f \| E. y e. _pred ( x , A , R ) ta' }
16	9 15	nfcxfr	\|- F/_ y H
17	16	nfuni	\|- F/_ y U. H
18	10 17	nfcxfr	\|- F/_ y P
19		nfcv	\|- F/_ y x
20		nfcv	\|- F/_ y G
21		nfcv	\|- F/_ y _pred ( x , A , R )
22	18 21	nfres	\|- F/_ y ( P \|` _pred ( x , A , R ) )
23	19 22	nfop	\|- F/_ y <. x , ( P \|` _pred ( x , A , R ) ) >.
24	11 23	nfcxfr	\|- F/_ y Z
25	20 24	nffv	\|- F/_ y ( G ` Z )
26	19 25	nfop	\|- F/_ y <. x , ( G ` Z ) >.
27	26	nfsn	\|- F/_ y { <. x , ( G ` Z ) >. }
28	18 27	nfun	\|- F/_ y ( P u. { <. x , ( G ` Z ) >. } )
29	12 28	nfcxfr	\|- F/_ y Q
30		nfcv	\|- F/_ y z
31	29 30	nffv	\|- F/_ y ( Q ` z )
32		nfcv	\|- F/_ y _pred ( z , A , R )
33	29 32	nfres	\|- F/_ y ( Q \|` _pred ( z , A , R ) )
34	30 33	nfop	\|- F/_ y <. z , ( Q \|` _pred ( z , A , R ) ) >.
35	13 34	nfcxfr	\|- F/_ y W
36	20 35	nffv	\|- F/_ y ( G ` W )
37	31 36	nfeq	\|- F/ y ( Q ` z ) = ( G ` W )
38	37	nf5ri	\|- ( ( Q ` z ) = ( G ` W ) -> A. y ( Q ` z ) = ( G ` W ) )

Metamath Proof Explorer

Theorem bnj1447

Proof