Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1423.1 |
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) } |
2 |
|
bnj1423.2 |
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >. |
3 |
|
bnj1423.3 |
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) } |
4 |
|
bnj1423.4 |
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) |
5 |
|
bnj1423.5 |
|- D = { x e. A | -. E. f ta } |
6 |
|
bnj1423.6 |
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) ) |
7 |
|
bnj1423.7 |
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) ) |
8 |
|
bnj1423.8 |
|- ( ta' <-> [. y / x ]. ta ) |
9 |
|
bnj1423.9 |
|- H = { f | E. y e. _pred ( x , A , R ) ta' } |
10 |
|
bnj1423.10 |
|- P = U. H |
11 |
|
bnj1423.11 |
|- Z = <. x , ( P |` _pred ( x , A , R ) ) >. |
12 |
|
bnj1423.12 |
|- Q = ( P u. { <. x , ( G ` Z ) >. } ) |
13 |
|
bnj1423.13 |
|- W = <. z , ( Q |` _pred ( z , A , R ) ) >. |
14 |
|
bnj1423.14 |
|- E = ( { x } u. _trCl ( x , A , R ) ) |
15 |
|
bnj1423.15 |
|- ( ch -> P Fn _trCl ( x , A , R ) ) |
16 |
|
bnj1423.16 |
|- ( ch -> Q Fn ( { x } u. _trCl ( x , A , R ) ) ) |
17 |
|
biid |
|- ( ( ch /\ z e. E ) <-> ( ch /\ z e. E ) ) |
18 |
|
biid |
|- ( ( ( ch /\ z e. E ) /\ z e. { x } ) <-> ( ( ch /\ z e. E ) /\ z e. { x } ) ) |
19 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
|
bnj1442 |
|- ( ( ( ch /\ z e. E ) /\ z e. { x } ) -> ( Q ` z ) = ( G ` W ) ) |
20 |
|
biid |
|- ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) <-> ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) ) |
21 |
|
biid |
|- ( ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) <-> ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) ) |
22 |
|
biid |
|- ( ( ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) /\ y e. _pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) <-> ( ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) /\ y e. _pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) |
23 |
|
biid |
|- ( ( ( ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) /\ y e. _pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) /\ d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> ( ( ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) /\ f e. H /\ z e. dom f ) /\ y e. _pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) /\ d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) |
24 |
|
eqid |
|- <. z , ( f |` _pred ( z , A , R ) ) >. = <. z , ( f |` _pred ( z , A , R ) ) >. |
25 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 23 24
|
bnj1450 |
|- ( ( ( ch /\ z e. E ) /\ z e. _trCl ( x , A , R ) ) -> ( Q ` z ) = ( G ` W ) ) |
26 |
14
|
bnj1424 |
|- ( z e. E -> ( z e. { x } \/ z e. _trCl ( x , A , R ) ) ) |
27 |
26
|
adantl |
|- ( ( ch /\ z e. E ) -> ( z e. { x } \/ z e. _trCl ( x , A , R ) ) ) |
28 |
19 25 27
|
mpjaodan |
|- ( ( ch /\ z e. E ) -> ( Q ` z ) = ( G ` W ) ) |
29 |
28
|
ralrimiva |
|- ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) ) |