Metamath Proof Explorer

Theorem bnj1448

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 bnj1448.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1448.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1448.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1448.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1448.5 
|- D = { x e. A | -. E. f ta }
bnj1448.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1448.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1448.8 
|- ( ta' <-> [. y / x ]. ta )
bnj1448.9 
|- H = { f | E. y e. _pred ( x , A , R ) ta' }
bnj1448.10 
|- P = U. H
bnj1448.11 
|- Z = <. x , ( P |` _pred ( x , A , R ) ) >.
bnj1448.12 
|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1448.13 
|- W = <. z , ( Q |` _pred ( z , A , R ) ) >.
Assertion bnj1448 
|- ( ( Q ` z ) = ( G ` W ) -> A. f ( Q ` z ) = ( G ` W ) )

Proof

Step Hyp Ref Expression
1 bnj1448.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1448.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1448.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1448.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1448.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1448.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1448.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1448.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1448.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1448.10 
 |-  P = U. H
11 bnj1448.11 
 |-  Z = <. x , ( P |` _pred ( x , A , R ) ) >.
12 bnj1448.12 
 |-  Q = ( P u. { <. x , ( G ` Z ) >. } )
13 bnj1448.13 
 |-  W = <. z , ( Q |` _pred ( z , A , R ) ) >.
14 9 bnj1317 
 |-  ( w e. H -> A. f w e. H )
15 14 nfcii 
 |-  F/_ f H
16 15 nfuni 
 |-  F/_ f U. H
17 10 16 nfcxfr 
 |-  F/_ f P
18 nfcv 
 |-  F/_ f x
19 nfcv 
 |-  F/_ f G
20 nfcv 
 |-  F/_ f _pred ( x , A , R )
21 17 20 nfres 
 |-  F/_ f ( P |` _pred ( x , A , R ) )
22 18 21 nfop 
 |-  F/_ f <. x , ( P |` _pred ( x , A , R ) ) >.
23 11 22 nfcxfr 
 |-  F/_ f Z
24 19 23 nffv 
 |-  F/_ f ( G ` Z )
25 18 24 nfop 
 |-  F/_ f <. x , ( G ` Z ) >.
26 25 nfsn 
 |-  F/_ f { <. x , ( G ` Z ) >. }
27 17 26 nfun 
 |-  F/_ f ( P u. { <. x , ( G ` Z ) >. } )
28 12 27 nfcxfr 
 |-  F/_ f Q
29 nfcv 
 |-  F/_ f z
30 28 29 nffv 
 |-  F/_ f ( Q ` z )
31 nfcv 
 |-  F/_ f _pred ( z , A , R )
32 28 31 nfres 
 |-  F/_ f ( Q |` _pred ( z , A , R ) )
33 29 32 nfop 
 |-  F/_ f <. z , ( Q |` _pred ( z , A , R ) ) >.
34 13 33 nfcxfr 
 |-  F/_ f W
35 19 34 nffv 
 |-  F/_ f ( G ` W )
36 30 35 nfeq 
 |-  F/ f ( Q ` z ) = ( G ` W )
37 36 nf5ri 
 |-  ( ( Q ` z ) = ( G ` W ) -> A. f ( Q ` z ) = ( G ` W ) )