This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem bnj1416

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 bnj1416.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1416.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1416.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1416.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1416.5 
|- D = { x e. A | -. E. f ta }
bnj1416.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1416.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1416.8 
|- ( ta' <-> [. y / x ]. ta )
bnj1416.9 
|- H = { f | E. y e. _pred ( x , A , R ) ta' }
bnj1416.10 
|- P = U. H
bnj1416.11 
|- Z = <. x , ( P |` _pred ( x , A , R ) ) >.
bnj1416.12 
|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1416.28 
|- ( ch -> dom P = _trCl ( x , A , R ) )
Assertion bnj1416 
|- ( ch -> dom Q = ( { x } u. _trCl ( x , A , R ) ) )

Proof

Step Hyp Ref Expression
1 bnj1416.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1416.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1416.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1416.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1416.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1416.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1416.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1416.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1416.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1416.10 
 |-  P = U. H
11 bnj1416.11 
 |-  Z = <. x , ( P |` _pred ( x , A , R ) ) >.
12 bnj1416.12 
 |-  Q = ( P u. { <. x , ( G ` Z ) >. } )
13 bnj1416.28 
 |-  ( ch -> dom P = _trCl ( x , A , R ) )
14 12 dmeqi 
 |-  dom Q = dom ( P u. { <. x , ( G ` Z ) >. } )
15 dmun 
 |-  dom ( P u. { <. x , ( G ` Z ) >. } ) = ( dom P u. dom { <. x , ( G ` Z ) >. } )
16 fvex 
 |-  ( G ` Z ) e. _V
17 16 dmsnop 
 |-  dom { <. x , ( G ` Z ) >. } = { x }
18 17 uneq2i 
 |-  ( dom P u. dom { <. x , ( G ` Z ) >. } ) = ( dom P u. { x } )
19 14 15 18 3eqtri 
 |-  dom Q = ( dom P u. { x } )
20 13 uneq1d 
 |-  ( ch -> ( dom P u. { x } ) = ( _trCl ( x , A , R ) u. { x } ) )
21 uncom 
 |-  ( _trCl ( x , A , R ) u. { x } ) = ( { x } u. _trCl ( x , A , R ) )
22 20 21 eqtrdi 
 |-  ( ch -> ( dom P u. { x } ) = ( { x } u. _trCl ( x , A , R ) ) )
23 19 22 eqtrid 
 |-  ( ch -> dom Q = ( { x } u. _trCl ( x , A , R ) ) )