Metamath Proof Explorer

Theorem bnj1446

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 bnj1446.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1446.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1446.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1446.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1446.5 
|- D = { x e. A | -. E. f ta }
bnj1446.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1446.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1446.8 
|- ( ta' <-> [. y / x ]. ta )
bnj1446.9 
|- H = { f | E. y e. _pred ( x , A , R ) ta' }
bnj1446.10 
|- P = U. H
bnj1446.11 
|- Z = <. x , ( P |` _pred ( x , A , R ) ) >.
bnj1446.12 
|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1446.13 
|- W = <. z , ( Q |` _pred ( z , A , R ) ) >.
Assertion bnj1446 
|- ( ( Q ` z ) = ( G ` W ) -> A. d ( Q ` z ) = ( G ` W ) )

Proof

Step Hyp Ref Expression
1 bnj1446.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1446.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1446.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1446.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1446.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1446.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1446.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1446.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1446.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1446.10 
 |-  P = U. H
11 bnj1446.11 
 |-  Z = <. x , ( P |` _pred ( x , A , R ) ) >.
12 bnj1446.12 
 |-  Q = ( P u. { <. x , ( G ` Z ) >. } )
13 bnj1446.13 
 |-  W = <. z , ( Q |` _pred ( z , A , R ) ) >.
14 nfcv 
 |-  F/_ d _pred ( x , A , R )
15 nfcv 
 |-  F/_ d y
16 nfre1 
 |-  F/ d E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) )
17 16 nfab 
 |-  F/_ d { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
18 3 17 nfcxfr 
 |-  F/_ d C
19 18 nfcri 
 |-  F/ d f e. C
20 nfv 
 |-  F/ d dom f = ( { x } u. _trCl ( x , A , R ) )
21 19 20 nfan 
 |-  F/ d ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) )
22 4 21 nfxfr 
 |-  F/ d ta
23 15 22 nfsbcw 
 |-  F/ d [. y / x ]. ta
24 8 23 nfxfr 
 |-  F/ d ta'
25 14 24 nfrex 
 |-  F/ d E. y e. _pred ( x , A , R ) ta'
26 25 nfab 
 |-  F/_ d { f | E. y e. _pred ( x , A , R ) ta' }
27 9 26 nfcxfr 
 |-  F/_ d H
28 27 nfuni 
 |-  F/_ d U. H
29 10 28 nfcxfr 
 |-  F/_ d P
30 nfcv 
 |-  F/_ d x
31 nfcv 
 |-  F/_ d G
32 29 14 nfres 
 |-  F/_ d ( P |` _pred ( x , A , R ) )
33 30 32 nfop 
 |-  F/_ d <. x , ( P |` _pred ( x , A , R ) ) >.
34 11 33 nfcxfr 
 |-  F/_ d Z
35 31 34 nffv 
 |-  F/_ d ( G ` Z )
36 30 35 nfop 
 |-  F/_ d <. x , ( G ` Z ) >.
37 36 nfsn 
 |-  F/_ d { <. x , ( G ` Z ) >. }
38 29 37 nfun 
 |-  F/_ d ( P u. { <. x , ( G ` Z ) >. } )
39 12 38 nfcxfr 
 |-  F/_ d Q
40 nfcv 
 |-  F/_ d z
41 39 40 nffv 
 |-  F/_ d ( Q ` z )
42 nfcv 
 |-  F/_ d _pred ( z , A , R )
43 39 42 nfres 
 |-  F/_ d ( Q |` _pred ( z , A , R ) )
44 40 43 nfop 
 |-  F/_ d <. z , ( Q |` _pred ( z , A , R ) ) >.
45 13 44 nfcxfr 
 |-  F/_ d W
46 31 45 nffv 
 |-  F/_ d ( G ` W )
47 41 46 nfeq 
 |-  F/ d ( Q ` z ) = ( G ` W )
48 47 nf5ri 
 |-  ( ( Q ` z ) = ( G ` W ) -> A. d ( Q ` z ) = ( G ` W ) )