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 ) ) |