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Metamath Proof Explorer

Theorem bnj1415

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 bnj1415.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1415.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1415.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1415.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1415.5 
|- D = { x e. A | -. E. f ta }
bnj1415.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1415.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1415.8 
|- ( ta' <-> [. y / x ]. ta )
bnj1415.9 
|- H = { f | E. y e. _pred ( x , A , R ) ta' }
bnj1415.10 
|- P = U. H
Assertion bnj1415 
|- ( ch -> dom P = _trCl ( x , A , R ) )

Proof

Step Hyp Ref Expression
1 bnj1415.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1415.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1415.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1415.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1415.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1415.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1415.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1415.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1415.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1415.10 
 |-  P = U. H
11 6 simplbi 
 |-  ( ps -> R _FrSe A )
12 7 11 bnj835 
 |-  ( ch -> R _FrSe A )
13 5 7 bnj1212 
 |-  ( ch -> x e. A )
14 eqid 
 |-  ( _pred ( x , A , R ) u. U_ y e. _pred ( x , A , R ) _trCl ( y , A , R ) ) = ( _pred ( x , A , R ) u. U_ y e. _pred ( x , A , R ) _trCl ( y , A , R ) )
15 14 bnj1414 
 |-  ( ( R _FrSe A /\ x e. A ) -> _trCl ( x , A , R ) = ( _pred ( x , A , R ) u. U_ y e. _pred ( x , A , R ) _trCl ( y , A , R ) ) )
16 12 13 15 syl2anc 
 |-  ( ch -> _trCl ( x , A , R ) = ( _pred ( x , A , R ) u. U_ y e. _pred ( x , A , R ) _trCl ( y , A , R ) ) )
17 iunun 
 |-  U_ y e. _pred ( x , A , R ) ( { y } u. _trCl ( y , A , R ) ) = ( U_ y e. _pred ( x , A , R ) { y } u. U_ y e. _pred ( x , A , R ) _trCl ( y , A , R ) )
18 iunid 
 |-  U_ y e. _pred ( x , A , R ) { y } = _pred ( x , A , R )
19 18 uneq1i 
 |-  ( U_ y e. _pred ( x , A , R ) { y } u. U_ y e. _pred ( x , A , R ) _trCl ( y , A , R ) ) = ( _pred ( x , A , R ) u. U_ y e. _pred ( x , A , R ) _trCl ( y , A , R ) )
20 17 19 eqtri 
 |-  U_ y e. _pred ( x , A , R ) ( { y } u. _trCl ( y , A , R ) ) = ( _pred ( x , A , R ) u. U_ y e. _pred ( x , A , R ) _trCl ( y , A , R ) )
21 biid 
 |-  ( ( ch /\ z e. U_ y e. _pred ( x , A , R ) ( { y } u. _trCl ( y , A , R ) ) ) <-> ( ch /\ z e. U_ y e. _pred ( x , A , R ) ( { y } u. _trCl ( y , A , R ) ) ) )
22 biid 
 |-  ( ( ( ch /\ z e. U_ y e. _pred ( x , A , R ) ( { y } u. _trCl ( y , A , R ) ) ) /\ y e. _pred ( x , A , R ) /\ z e. ( { y } u. _trCl ( y , A , R ) ) ) <-> ( ( ch /\ z e. U_ y e. _pred ( x , A , R ) ( { y } u. _trCl ( y , A , R ) ) ) /\ y e. _pred ( x , A , R ) /\ z e. ( { y } u. _trCl ( y , A , R ) ) ) )
23 1 2 3 4 5 6 7 8 9 10 21 22 bnj1398 
 |-  ( ch -> U_ y e. _pred ( x , A , R ) ( { y } u. _trCl ( y , A , R ) ) = dom P )
24 20 23 eqtr3id 
 |-  ( ch -> ( _pred ( x , A , R ) u. U_ y e. _pred ( x , A , R ) _trCl ( y , A , R ) ) = dom P )
25 16 24 eqtr2d 
 |-  ( ch -> dom P = _trCl ( x , A , R ) )