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Metamath Proof Explorer

Theorem bnj1463

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 bnj1463.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1463.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1463.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1463.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1463.5 
|- D = { x e. A | -. E. f ta }
bnj1463.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1463.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1463.8 
|- ( ta' <-> [. y / x ]. ta )
bnj1463.9 
|- H = { f | E. y e. _pred ( x , A , R ) ta' }
bnj1463.10 
|- P = U. H
bnj1463.11 
|- Z = <. x , ( P |` _pred ( x , A , R ) ) >.
bnj1463.12 
|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1463.13 
|- W = <. z , ( Q |` _pred ( z , A , R ) ) >.
bnj1463.14 
|- E = ( { x } u. _trCl ( x , A , R ) )
bnj1463.15 
|- ( ch -> Q e. _V )
bnj1463.16 
|- ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) )
bnj1463.17 
|- ( ch -> Q Fn E )
bnj1463.18 
|- ( ch -> E e. B )
Assertion bnj1463 
|- ( ch -> Q e. C )

Proof

Step Hyp Ref Expression
1 bnj1463.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1463.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1463.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1463.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1463.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1463.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1463.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1463.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1463.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1463.10 
 |-  P = U. H
11 bnj1463.11 
 |-  Z = <. x , ( P |` _pred ( x , A , R ) ) >.
12 bnj1463.12 
 |-  Q = ( P u. { <. x , ( G ` Z ) >. } )
13 bnj1463.13 
 |-  W = <. z , ( Q |` _pred ( z , A , R ) ) >.
14 bnj1463.14 
 |-  E = ( { x } u. _trCl ( x , A , R ) )
15 bnj1463.15 
 |-  ( ch -> Q e. _V )
16 bnj1463.16 
 |-  ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) )
17 bnj1463.17 
 |-  ( ch -> Q Fn E )
18 bnj1463.18 
 |-  ( ch -> E e. B )
19 18 elexd 
 |-  ( ch -> E e. _V )
20 eleq1 
 |-  ( d = E -> ( d e. B <-> E e. B ) )
21 fneq2 
 |-  ( d = E -> ( Q Fn d <-> Q Fn E ) )
22 raleq 
 |-  ( d = E -> ( A. z e. d ( Q ` z ) = ( G ` W ) <-> A. z e. E ( Q ` z ) = ( G ` W ) ) )
23 21 22 anbi12d 
 |-  ( d = E -> ( ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) <-> ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) ) )
24 20 23 anbi12d 
 |-  ( d = E -> ( ( d e. B /\ ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) <-> ( E e. B /\ ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) ) ) )
25 1 bnj1317 
 |-  ( w e. B -> A. d w e. B )
26 25 nfcii 
 |-  F/_ d B
27 26 nfel2 
 |-  F/ d E e. B
28 1 2 3 4 5 6 7 8 9 10 11 12 bnj1467 
 |-  ( w e. Q -> A. d w e. Q )
29 28 nfcii 
 |-  F/_ d Q
30 nfcv 
 |-  F/_ d E
31 29 30 nffn 
 |-  F/ d Q Fn E
32 1 2 3 4 5 6 7 8 9 10 11 12 13 bnj1446 
 |-  ( ( Q ` z ) = ( G ` W ) -> A. d ( Q ` z ) = ( G ` W ) )
33 32 nf5i 
 |-  F/ d ( Q ` z ) = ( G ` W )
34 30 33 nfralw 
 |-  F/ d A. z e. E ( Q ` z ) = ( G ` W )
35 31 34 nfan 
 |-  F/ d ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) )
36 27 35 nfan 
 |-  F/ d ( E e. B /\ ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) )
37 36 nf5ri 
 |-  ( ( E e. B /\ ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) ) -> A. d ( E e. B /\ ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) ) )
38 18 17 16 jca32 
 |-  ( ch -> ( E e. B /\ ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) ) )
39 24 37 38 bnj1465 
 |-  ( ( ch /\ E e. _V ) -> E. d ( d e. B /\ ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
40 19 39 mpdan 
 |-  ( ch -> E. d ( d e. B /\ ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
41 df-rex 
 |-  ( E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) <-> E. d ( d e. B /\ ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
42 40 41 sylibr 
 |-  ( ch -> E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) )
43 nfcv 
 |-  F/_ f B
44 1 2 3 4 5 6 7 8 9 10 11 12 bnj1466 
 |-  ( w e. Q -> A. f w e. Q )
45 44 nfcii 
 |-  F/_ f Q
46 nfcv 
 |-  F/_ f d
47 45 46 nffn 
 |-  F/ f Q Fn d
48 1 2 3 4 5 6 7 8 9 10 11 12 13 bnj1448 
 |-  ( ( Q ` z ) = ( G ` W ) -> A. f ( Q ` z ) = ( G ` W ) )
49 48 nf5i 
 |-  F/ f ( Q ` z ) = ( G ` W )
50 46 49 nfralw 
 |-  F/ f A. z e. d ( Q ` z ) = ( G ` W )
51 47 50 nfan 
 |-  F/ f ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) )
52 43 51 nfrexw 
 |-  F/ f E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) )
53 52 nf5ri 
 |-  ( E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) -> A. f E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) )
54 29 nfeq2 
 |-  F/ d f = Q
55 fneq1 
 |-  ( f = Q -> ( f Fn d <-> Q Fn d ) )
56 fveq1 
 |-  ( f = Q -> ( f ` z ) = ( Q ` z ) )
57 reseq1 
 |-  ( f = Q -> ( f |` _pred ( z , A , R ) ) = ( Q |` _pred ( z , A , R ) ) )
58 57 opeq2d 
 |-  ( f = Q -> <. z , ( f |` _pred ( z , A , R ) ) >. = <. z , ( Q |` _pred ( z , A , R ) ) >. )
59 58 13 eqtr4di 
 |-  ( f = Q -> <. z , ( f |` _pred ( z , A , R ) ) >. = W )
60 59 fveq2d 
 |-  ( f = Q -> ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) = ( G ` W ) )
61 56 60 eqeq12d 
 |-  ( f = Q -> ( ( f ` z ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) <-> ( Q ` z ) = ( G ` W ) ) )
62 61 ralbidv 
 |-  ( f = Q -> ( A. z e. d ( f ` z ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) <-> A. z e. d ( Q ` z ) = ( G ` W ) ) )
63 55 62 anbi12d 
 |-  ( f = Q -> ( ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) ) <-> ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
64 54 63 rexbid 
 |-  ( f = Q -> ( E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) ) <-> E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
65 53 64 44 bnj1468 
 |-  ( Q e. _V -> ( [. Q / f ]. E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) ) <-> E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
66 15 65 syl 
 |-  ( ch -> ( [. Q / f ]. E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) ) <-> E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
67 42 66 mpbird 
 |-  ( ch -> [. Q / f ]. E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) ) )
68 fveq2 
 |-  ( x = z -> ( f ` x ) = ( f ` z ) )
69 id 
 |-  ( x = z -> x = z )
70 bnj602 
 |-  ( x = z -> _pred ( x , A , R ) = _pred ( z , A , R ) )
71 70 reseq2d 
 |-  ( x = z -> ( f |` _pred ( x , A , R ) ) = ( f |` _pred ( z , A , R ) ) )
72 69 71 opeq12d 
 |-  ( x = z -> <. x , ( f |` _pred ( x , A , R ) ) >. = <. z , ( f |` _pred ( z , A , R ) ) >. )
73 2 72 eqtrid 
 |-  ( x = z -> Y = <. z , ( f |` _pred ( z , A , R ) ) >. )
74 73 fveq2d 
 |-  ( x = z -> ( G ` Y ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) )
75 68 74 eqeq12d 
 |-  ( x = z -> ( ( f ` x ) = ( G ` Y ) <-> ( f ` z ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) ) )
76 75 cbvralvw 
 |-  ( A. x e. d ( f ` x ) = ( G ` Y ) <-> A. z e. d ( f ` z ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) )
77 76 anbi2i 
 |-  ( ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) ) )
78 77 rexbii 
 |-  ( E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) ) )
79 78 sbcbii 
 |-  ( [. Q / f ]. E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> [. Q / f ]. E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` _pred ( z , A , R ) ) >. ) ) )
80 67 79 sylibr 
 |-  ( ch -> [. Q / f ]. E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
81 3 bnj1454 
 |-  ( Q e. _V -> ( Q e. C <-> [. Q / f ]. E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
82 15 81 syl 
 |-  ( ch -> ( Q e. C <-> [. Q / f ]. E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
83 80 82 mpbird 
 |-  ( ch -> Q e. C )