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Metamath Proof Explorer

Theorem bnj1423

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 bnj1423.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1423.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1423.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1423.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1423.5 
|- D = { x e. A | -. E. f ta }
bnj1423.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1423.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1423.8 
|- ( ta' <-> [. y / x ]. ta )
bnj1423.9 
|- H = { f | E. y e. _pred ( x , A , R ) ta' }
bnj1423.10 
|- P = U. H
bnj1423.11 
|- Z = <. x , ( P |` _pred ( x , A , R ) ) >.
bnj1423.12 
|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1423.13 
|- W = <. z , ( Q |` _pred ( z , A , R ) ) >.
bnj1423.14 
|- E = ( { x } u. _trCl ( x , A , R ) )
bnj1423.15 
|- ( ch -> P Fn _trCl ( x , A , R ) )
bnj1423.16 
|- ( ch -> Q Fn ( { x } u. _trCl ( x , A , R ) ) )
Assertion bnj1423 
|- ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) )

Proof

Step Hyp Ref Expression
1 bnj1423.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1423.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1423.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1423.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1423.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1423.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1423.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1423.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1423.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1423.10 
 |-  P = U. H
11 bnj1423.11 
 |-  Z = <. x , ( P |` _pred ( x , A , R ) ) >.
12 bnj1423.12 
 |-  Q = ( P u. { <. x , ( G ` Z ) >. } )
13 bnj1423.13 
 |-  W = <. z , ( Q |` _pred ( z , A , R ) ) >.
14 bnj1423.14 
 |-  E = ( { x } u. _trCl ( x , A , R ) )
15 bnj1423.15 
 |-  ( ch -> P Fn _trCl ( x , A , R ) )
16 bnj1423.16 
 |-  ( ch -> Q Fn ( { x } u. _trCl ( x , A , R ) ) )
17 biid 
 |-  ( ( ch /\ z e. E ) <-> ( ch /\ z e. E ) )
18 biid 
 |-  ( ( ( ch /\ z e. E ) /\ z e. { x } ) <-> ( ( ch /\ z e. E ) /\ z e. { x } ) )
19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 bnj1442 
 |-  ( ( ( ch /\ z e. E ) /\ z e. { x } ) -> ( Q ` z ) = ( G ` W ) )
20 biid 
 |-  ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) <-> ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) )
21 biid 
 |-  ( ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) <-> ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) )
22 biid 
 |-  ( ( ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) /\ y e. _pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) <-> ( ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) /\ y e. _pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
23 biid 
 |-  ( ( ( ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) /\ y e. _pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) /\ d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> ( ( ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) /\ y e. _pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) /\ d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
24 eqid 
 |-  <. z , ( f |` _pred ( z , A , R ) ) >. = <. z , ( f |` _pred ( z , A , R ) ) >.
25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 23 24 bnj1450 
 |-  ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) -> ( Q ` z ) = ( G ` W ) )
26 14 bnj1424 
 |-  ( z e. E -> ( z e. { x } \/ z e. _trCl ( x , A , R ) ) )
27 26 adantl 
 |-  ( ( ch /\ z e. E ) -> ( z e. { x } \/ z e. _trCl ( x , A , R ) ) )
28 19 25 27 mpjaodan 
 |-  ( ( ch /\ z e. E ) -> ( Q ` z ) = ( G ` W ) )
29 28 ralrimiva 
 |-  ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) )