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Metamath Proof Explorer

Theorem bnj1445

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 bnj1445.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1445.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1445.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1445.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1445.5 
|- D = { x e. A | -. E. f ta }
bnj1445.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1445.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1445.8 
|- ( ta' <-> [. y / x ]. ta )
bnj1445.9 
|- H = { f | E. y e. _pred ( x , A , R ) ta' }
bnj1445.10 
|- P = U. H
bnj1445.11 
|- Z = <. x , ( P |` _pred ( x , A , R ) ) >.
bnj1445.12 
|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1445.13 
|- W = <. z , ( Q |` _pred ( z , A , R ) ) >.
bnj1445.14 
|- E = ( { x } u. _trCl ( x , A , R ) )
bnj1445.15 
|- ( ch -> P Fn _trCl ( x , A , R ) )
bnj1445.16 
|- ( ch -> Q Fn ( { x } u. _trCl ( x , A , R ) ) )
bnj1445.17 
|- ( th <-> ( ch /\ z e. E ) )
bnj1445.18 
|- ( et <-> ( th /\ z e. { x } ) )
bnj1445.19 
|- ( ze <-> ( th /\ z e. _trCl ( x , A , R ) ) )
bnj1445.20 
|- ( rh <-> ( ze /\ f e. H /\ z e. dom f ) )
bnj1445.21 
|- ( si <-> ( rh /\ y e. _pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
bnj1445.22 
|- ( ph <-> ( si /\ d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
bnj1445.23 
|- X = <. z , ( f |` _pred ( z , A , R ) ) >.
Assertion bnj1445 
|- ( si -> A. d si )

Proof

Step Hyp Ref Expression
1 bnj1445.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1445.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1445.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1445.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1445.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1445.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1445.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1445.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1445.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1445.10 
 |-  P = U. H
11 bnj1445.11 
 |-  Z = <. x , ( P |` _pred ( x , A , R ) ) >.
12 bnj1445.12 
 |-  Q = ( P u. { <. x , ( G ` Z ) >. } )
13 bnj1445.13 
 |-  W = <. z , ( Q |` _pred ( z , A , R ) ) >.
14 bnj1445.14 
 |-  E = ( { x } u. _trCl ( x , A , R ) )
15 bnj1445.15 
 |-  ( ch -> P Fn _trCl ( x , A , R ) )
16 bnj1445.16 
 |-  ( ch -> Q Fn ( { x } u. _trCl ( x , A , R ) ) )
17 bnj1445.17 
 |-  ( th <-> ( ch /\ z e. E ) )
18 bnj1445.18 
 |-  ( et <-> ( th /\ z e. { x } ) )
19 bnj1445.19 
 |-  ( ze <-> ( th /\ z e. _trCl ( x , A , R ) ) )
20 bnj1445.20 
 |-  ( rh <-> ( ze /\ f e. H /\ z e. dom f ) )
21 bnj1445.21 
 |-  ( si <-> ( rh /\ y e. _pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
22 bnj1445.22 
 |-  ( ph <-> ( si /\ d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
23 bnj1445.23 
 |-  X = <. z , ( f |` _pred ( z , A , R ) ) >.
24 nfv 
 |-  F/ d R _FrSe A
25 nfre1 
 |-  F/ d E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) )
26 25 nfab 
 |-  F/_ d { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
27 3 26 nfcxfr 
 |-  F/_ d C
28 27 nfcri 
 |-  F/ d f e. C
29 nfv 
 |-  F/ d dom f = ( { x } u. _trCl ( x , A , R ) )
30 28 29 nfan 
 |-  F/ d ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) )
31 4 30 nfxfr 
 |-  F/ d ta
32 31 nfex 
 |-  F/ d E. f ta
33 32 nfn 
 |-  F/ d -. E. f ta
34 nfcv 
 |-  F/_ d A
35 33 34 nfrabw 
 |-  F/_ d { x e. A | -. E. f ta }
36 5 35 nfcxfr 
 |-  F/_ d D
37 nfcv 
 |-  F/_ d (/)
38 36 37 nfne 
 |-  F/ d D =/= (/)
39 24 38 nfan 
 |-  F/ d ( R _FrSe A /\ D =/= (/) )
40 6 39 nfxfr 
 |-  F/ d ps
41 36 nfcri 
 |-  F/ d x e. D
42 nfv 
 |-  F/ d -. y R x
43 36 42 nfralw 
 |-  F/ d A. y e. D -. y R x
44 40 41 43 nf3an 
 |-  F/ d ( ps /\ x e. D /\ A. y e. D -. y R x )
45 7 44 nfxfr 
 |-  F/ d ch
46 45 nf5ri 
 |-  ( ch -> A. d ch )
47 46 bnj1351 
 |-  ( ( ch /\ z e. E ) -> A. d ( ch /\ z e. E ) )
48 47 nf5i 
 |-  F/ d ( ch /\ z e. E )
49 17 48 nfxfr 
 |-  F/ d th
50 nfv 
 |-  F/ d z e. _trCl ( x , A , R )
51 49 50 nfan 
 |-  F/ d ( th /\ z e. _trCl ( x , A , R ) )
52 19 51 nfxfr 
 |-  F/ d ze
53 nfcv 
 |-  F/_ d _pred ( x , A , R )
54 nfcv 
 |-  F/_ d y
55 54 31 nfsbcw 
 |-  F/ d [. y / x ]. ta
56 8 55 nfxfr 
 |-  F/ d ta'
57 53 56 nfrexw 
 |-  F/ d E. y e. _pred ( x , A , R ) ta'
58 57 nfab 
 |-  F/_ d { f | E. y e. _pred ( x , A , R ) ta' }
59 9 58 nfcxfr 
 |-  F/_ d H
60 59 nfcri 
 |-  F/ d f e. H
61 nfv 
 |-  F/ d z e. dom f
62 52 60 61 nf3an 
 |-  F/ d ( ze /\ f e. H /\ z e. dom f )
63 20 62 nfxfr 
 |-  F/ d rh
64 63 nf5ri 
 |-  ( rh -> A. d rh )
65 ax-5 
 |-  ( y e. _pred ( x , A , R ) -> A. d y e. _pred ( x , A , R ) )
66 28 nf5ri 
 |-  ( f e. C -> A. d f e. C )
67 ax-5 
 |-  ( dom f = ( { y } u. _trCl ( y , A , R ) ) -> A. d dom f = ( { y } u. _trCl ( y , A , R ) ) )
68 64 65 66 67 bnj982 
 |-  ( ( rh /\ y e. _pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) -> A. d ( rh /\ y e. _pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
69 21 68 hbxfrbi 
 |-  ( si -> A. d si )