bnj1446

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	bnj1446.1	\|- B = { d \| ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
	bnj1446.2	\|- Y = <. x , ( f \|` _pred ( x , A , R ) ) >.
	bnj1446.3	\|- C = { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
	bnj1446.4	\|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
	bnj1446.5	\|- D = { x e. A \| -. E. f ta }
	bnj1446.6	\|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
	bnj1446.7	\|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
	bnj1446.8	\|- ( ta' <-> [. y / x ]. ta )
	bnj1446.9	\|- H = { f \| E. y e. _pred ( x , A , R ) ta' }
	bnj1446.10	\|- P = U. H
	bnj1446.11	\|- Z = <. x , ( P \|` _pred ( x , A , R ) ) >.
	bnj1446.12	\|- Q = ( P u. { <. x , ( G ` Z ) >. } )
	bnj1446.13	\|- W = <. z , ( Q \|` _pred ( z , A , R ) ) >.
Assertion	bnj1446	\|- ( ( Q ` z ) = ( G ` W ) -> A. d ( Q ` z ) = ( G ` W ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1446.1	\|- B = { d \| ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2		bnj1446.2	\|- Y = <. x , ( f \|` _pred ( x , A , R ) ) >.
3		bnj1446.3	\|- C = { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4		bnj1446.4	\|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5		bnj1446.5	\|- D = { x e. A \| -. E. f ta }
6		bnj1446.6	\|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7		bnj1446.7	\|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8		bnj1446.8	\|- ( ta' <-> [. y / x ]. ta )
9		bnj1446.9	\|- H = { f \| E. y e. _pred ( x , A , R ) ta' }
10		bnj1446.10	\|- P = U. H
11		bnj1446.11	\|- Z = <. x , ( P \|` _pred ( x , A , R ) ) >.
12		bnj1446.12	\|- Q = ( P u. { <. x , ( G ` Z ) >. } )
13		bnj1446.13	\|- W = <. z , ( Q \|` _pred ( z , A , R ) ) >.
14		nfcv	\|- F/_ d _pred ( x , A , R )
15		nfcv	\|- F/_ d y
16		nfre1	\|- F/ d E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) )
17	16	nfab	\|- F/_ d { f \| E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
18	3 17	nfcxfr	\|- F/_ d C
19	18	nfcri	\|- F/ d f e. C
20		nfv	\|- F/ d dom f = ( { x } u. _trCl ( x , A , R ) )
21	19 20	nfan	\|- F/ d ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) )
22	4 21	nfxfr	\|- F/ d ta
23	15 22	nfsbcw	\|- F/ d [. y / x ]. ta
24	8 23	nfxfr	\|- F/ d ta'
25	14 24	nfrexw	\|- F/ d E. y e. _pred ( x , A , R ) ta'
26	25	nfab	\|- F/_ d { f \| E. y e. _pred ( x , A , R ) ta' }
27	9 26	nfcxfr	\|- F/_ d H
28	27	nfuni	\|- F/_ d U. H
29	10 28	nfcxfr	\|- F/_ d P
30		nfcv	\|- F/_ d x
31		nfcv	\|- F/_ d G
32	29 14	nfres	\|- F/_ d ( P \|` _pred ( x , A , R ) )
33	30 32	nfop	\|- F/_ d <. x , ( P \|` _pred ( x , A , R ) ) >.
34	11 33	nfcxfr	\|- F/_ d Z
35	31 34	nffv	\|- F/_ d ( G ` Z )
36	30 35	nfop	\|- F/_ d <. x , ( G ` Z ) >.
37	36	nfsn	\|- F/_ d { <. x , ( G ` Z ) >. }
38	29 37	nfun	\|- F/_ d ( P u. { <. x , ( G ` Z ) >. } )
39	12 38	nfcxfr	\|- F/_ d Q
40		nfcv	\|- F/_ d z
41	39 40	nffv	\|- F/_ d ( Q ` z )
42		nfcv	\|- F/_ d _pred ( z , A , R )
43	39 42	nfres	\|- F/_ d ( Q \|` _pred ( z , A , R ) )
44	40 43	nfop	\|- F/_ d <. z , ( Q \|` _pred ( z , A , R ) ) >.
45	13 44	nfcxfr	\|- F/_ d W
46	31 45	nffv	\|- F/_ d ( G ` W )
47	41 46	nfeq	\|- F/ d ( Q ` z ) = ( G ` W )
48	47	nf5ri	\|- ( ( Q ` z ) = ( G ` W ) -> A. d ( Q ` z ) = ( G ` W ) )

Metamath Proof Explorer

Theorem bnj1446

Proof