bnj1448

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	bnj1448.1	\|- B = { d \| ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
	bnj1448.2	\|- Y = <. x , ( f \|` _pred ( x , A , R ) ) >.
	bnj1448.3	\|- C = { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
	bnj1448.4	\|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
	bnj1448.5	\|- D = { x e. A \| -. E. f ta }
	bnj1448.6	\|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
	bnj1448.7	\|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
	bnj1448.8	\|- ( ta' <-> [. y / x ]. ta )
	bnj1448.9	\|- H = { f \| E. y e. _pred ( x , A , R ) ta' }
	bnj1448.10	\|- P = U. H
	bnj1448.11	\|- Z = <. x , ( P \|` _pred ( x , A , R ) ) >.
	bnj1448.12	\|- Q = ( P u. { <. x , ( G ` Z ) >. } )
	bnj1448.13	\|- W = <. z , ( Q \|` _pred ( z , A , R ) ) >.
Assertion	bnj1448	\|- ( ( Q ` z ) = ( G ` W ) -> A. f ( Q ` z ) = ( G ` W ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1448.1	\|- B = { d \| ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2		bnj1448.2	\|- Y = <. x , ( f \|` _pred ( x , A , R ) ) >.
3		bnj1448.3	\|- C = { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4		bnj1448.4	\|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5		bnj1448.5	\|- D = { x e. A \| -. E. f ta }
6		bnj1448.6	\|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7		bnj1448.7	\|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8		bnj1448.8	\|- ( ta' <-> [. y / x ]. ta )
9		bnj1448.9	\|- H = { f \| E. y e. _pred ( x , A , R ) ta' }
10		bnj1448.10	\|- P = U. H
11		bnj1448.11	\|- Z = <. x , ( P \|` _pred ( x , A , R ) ) >.
12		bnj1448.12	\|- Q = ( P u. { <. x , ( G ` Z ) >. } )
13		bnj1448.13	\|- W = <. z , ( Q \|` _pred ( z , A , R ) ) >.
14	9	bnj1317	\|- ( w e. H -> A. f w e. H )
15	14	nfcii	\|- F/_ f H
16	15	nfuni	\|- F/_ f U. H
17	10 16	nfcxfr	\|- F/_ f P
18		nfcv	\|- F/_ f x
19		nfcv	\|- F/_ f G
20		nfcv	\|- F/_ f _pred ( x , A , R )
21	17 20	nfres	\|- F/_ f ( P \|` _pred ( x , A , R ) )
22	18 21	nfop	\|- F/_ f <. x , ( P \|` _pred ( x , A , R ) ) >.
23	11 22	nfcxfr	\|- F/_ f Z
24	19 23	nffv	\|- F/_ f ( G ` Z )
25	18 24	nfop	\|- F/_ f <. x , ( G ` Z ) >.
26	25	nfsn	\|- F/_ f { <. x , ( G ` Z ) >. }
27	17 26	nfun	\|- F/_ f ( P u. { <. x , ( G ` Z ) >. } )
28	12 27	nfcxfr	\|- F/_ f Q
29		nfcv	\|- F/_ f z
30	28 29	nffv	\|- F/_ f ( Q ` z )
31		nfcv	\|- F/_ f _pred ( z , A , R )
32	28 31	nfres	\|- F/_ f ( Q \|` _pred ( z , A , R ) )
33	29 32	nfop	\|- F/_ f <. z , ( Q \|` _pred ( z , A , R ) ) >.
34	13 33	nfcxfr	\|- F/_ f W
35	19 34	nffv	\|- F/_ f ( G ` W )
36	30 35	nfeq	\|- F/ f ( Q ` z ) = ( G ` W )
37	36	nf5ri	\|- ( ( Q ` z ) = ( G ` W ) -> A. f ( Q ` z ) = ( G ` W ) )

Metamath Proof Explorer

Theorem bnj1448

Proof