Metamath Proof Explorer

Theorem bnj1447

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 bnj1447.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1447.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1447.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1447.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1447.5 
|- D = { x e. A | -. E. f ta }
bnj1447.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1447.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1447.8 
|- ( ta' <-> [. y / x ]. ta )
bnj1447.9 
|- H = { f | E. y e. _pred ( x , A , R ) ta' }
bnj1447.10 
|- P = U. H
bnj1447.11 
|- Z = <. x , ( P |` _pred ( x , A , R ) ) >.
bnj1447.12 
|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1447.13 
|- W = <. z , ( Q |` _pred ( z , A , R ) ) >.
Assertion bnj1447 
|- ( ( Q ` z ) = ( G ` W ) -> A. y ( Q ` z ) = ( G ` W ) )

Proof

Step Hyp Ref Expression
1 bnj1447.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1447.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1447.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1447.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1447.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1447.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1447.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1447.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1447.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1447.10 
 |-  P = U. H
11 bnj1447.11 
 |-  Z = <. x , ( P |` _pred ( x , A , R ) ) >.
12 bnj1447.12 
 |-  Q = ( P u. { <. x , ( G ` Z ) >. } )
13 bnj1447.13 
 |-  W = <. z , ( Q |` _pred ( z , A , R ) ) >.
14 nfre1 
 |-  F/ y E. y e. _pred ( x , A , R ) ta'
15 14 nfab 
 |-  F/_ y { f | E. y e. _pred ( x , A , R ) ta' }
16 9 15 nfcxfr 
 |-  F/_ y H
17 16 nfuni 
 |-  F/_ y U. H
18 10 17 nfcxfr 
 |-  F/_ y P
19 nfcv 
 |-  F/_ y x
20 nfcv 
 |-  F/_ y G
21 nfcv 
 |-  F/_ y _pred ( x , A , R )
22 18 21 nfres 
 |-  F/_ y ( P |` _pred ( x , A , R ) )
23 19 22 nfop 
 |-  F/_ y <. x , ( P |` _pred ( x , A , R ) ) >.
24 11 23 nfcxfr 
 |-  F/_ y Z
25 20 24 nffv 
 |-  F/_ y ( G ` Z )
26 19 25 nfop 
 |-  F/_ y <. x , ( G ` Z ) >.
27 26 nfsn 
 |-  F/_ y { <. x , ( G ` Z ) >. }
28 18 27 nfun 
 |-  F/_ y ( P u. { <. x , ( G ` Z ) >. } )
29 12 28 nfcxfr 
 |-  F/_ y Q
30 nfcv 
 |-  F/_ y z
31 29 30 nffv 
 |-  F/_ y ( Q ` z )
32 nfcv 
 |-  F/_ y _pred ( z , A , R )
33 29 32 nfres 
 |-  F/_ y ( Q |` _pred ( z , A , R ) )
34 30 33 nfop 
 |-  F/_ y <. z , ( Q |` _pred ( z , A , R ) ) >.
35 13 34 nfcxfr 
 |-  F/_ y W
36 20 35 nffv 
 |-  F/_ y ( G ` W )
37 31 36 nfeq 
 |-  F/ y ( Q ` z ) = ( G ` W )
38 37 nf5ri 
 |-  ( ( Q ` z ) = ( G ` W ) -> A. y ( Q ` z ) = ( G ` W ) )