Metamath Proof Explorer

Theorem bnj1491

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 bnj1491.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1491.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1491.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1491.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1491.5 
|- D = { x e. A | -. E. f ta }
bnj1491.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1491.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1491.8 
|- ( ta' <-> [. y / x ]. ta )
bnj1491.9 
|- H = { f | E. y e. _pred ( x , A , R ) ta' }
bnj1491.10 
|- P = U. H
bnj1491.11 
|- Z = <. x , ( P |` _pred ( x , A , R ) ) >.
bnj1491.12 
|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1491.13 
|- ( ch -> ( Q e. C /\ dom Q = ( { x } u. _trCl ( x , A , R ) ) ) )
Assertion bnj1491 
|- ( ( ch /\ Q e. _V ) -> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )

Proof

Step Hyp Ref Expression
1 bnj1491.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1491.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1491.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1491.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1491.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1491.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1491.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1491.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1491.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1491.10 
 |-  P = U. H
11 bnj1491.11 
 |-  Z = <. x , ( P |` _pred ( x , A , R ) ) >.
12 bnj1491.12 
 |-  Q = ( P u. { <. x , ( G ` Z ) >. } )
13 bnj1491.13 
 |-  ( ch -> ( Q e. C /\ dom Q = ( { x } u. _trCl ( x , A , R ) ) ) )
14 1 2 3 4 5 6 7 8 9 10 11 12 bnj1466 
 |-  ( w e. Q -> A. f w e. Q )
15 14 nfcii 
 |-  F/_ f Q
16 3 bnj1317 
 |-  ( w e. C -> A. f w e. C )
17 16 nfcii 
 |-  F/_ f C
18 15 17 nfel 
 |-  F/ f Q e. C
19 15 nfdm 
 |-  F/_ f dom Q
20 19 nfeq1 
 |-  F/ f dom Q = ( { x } u. _trCl ( x , A , R ) )
21 18 20 nfan 
 |-  F/ f ( Q e. C /\ dom Q = ( { x } u. _trCl ( x , A , R ) ) )
22 eleq1 
 |-  ( f = Q -> ( f e. C <-> Q e. C ) )
23 dmeq 
 |-  ( f = Q -> dom f = dom Q )
24 23 eqeq1d 
 |-  ( f = Q -> ( dom f = ( { x } u. _trCl ( x , A , R ) ) <-> dom Q = ( { x } u. _trCl ( x , A , R ) ) ) )
25 22 24 anbi12d 
 |-  ( f = Q -> ( ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) <-> ( Q e. C /\ dom Q = ( { x } u. _trCl ( x , A , R ) ) ) ) )
26 15 21 25 spcegf 
 |-  ( Q e. _V -> ( ( Q e. C /\ dom Q = ( { x } u. _trCl ( x , A , R ) ) ) -> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) )
27 13 26 mpan9 
 |-  ( ( ch /\ Q e. _V ) -> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )