Metamath Proof Explorer

Theorem bnj1421

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 bnj1421.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1421.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1421.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1421.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1421.5 
|- D = { x e. A | -. E. f ta }
bnj1421.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1421.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1421.8 
|- ( ta' <-> [. y / x ]. ta )
bnj1421.9 
|- H = { f | E. y e. _pred ( x , A , R ) ta' }
bnj1421.10 
|- P = U. H
bnj1421.11 
|- Z = <. x , ( P |` _pred ( x , A , R ) ) >.
bnj1421.12 
|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1421.13 
|- ( ch -> Fun P )
bnj1421.14 
|- ( ch -> dom Q = ( { x } u. _trCl ( x , A , R ) ) )
bnj1421.15 
|- ( ch -> dom P = _trCl ( x , A , R ) )
Assertion bnj1421 
|- ( ch -> Fun Q )

Proof

Step Hyp Ref Expression
1 bnj1421.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1421.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1421.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1421.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1421.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1421.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1421.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1421.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1421.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1421.10 
 |-  P = U. H
11 bnj1421.11 
 |-  Z = <. x , ( P |` _pred ( x , A , R ) ) >.
12 bnj1421.12 
 |-  Q = ( P u. { <. x , ( G ` Z ) >. } )
13 bnj1421.13 
 |-  ( ch -> Fun P )
14 bnj1421.14 
 |-  ( ch -> dom Q = ( { x } u. _trCl ( x , A , R ) ) )
15 bnj1421.15 
 |-  ( ch -> dom P = _trCl ( x , A , R ) )
16 vex 
 |-  x e. _V
17 fvex 
 |-  ( G ` Z ) e. _V
18 16 17 funsn 
 |-  Fun { <. x , ( G ` Z ) >. }
19 13 18 jctir 
 |-  ( ch -> ( Fun P /\ Fun { <. x , ( G ` Z ) >. } ) )
20 17 dmsnop 
 |-  dom { <. x , ( G ` Z ) >. } = { x }
21 20 a1i 
 |-  ( ch -> dom { <. x , ( G ` Z ) >. } = { x } )
22 15 21 ineq12d 
 |-  ( ch -> ( dom P i^i dom { <. x , ( G ` Z ) >. } ) = ( _trCl ( x , A , R ) i^i { x } ) )
23 6 simplbi 
 |-  ( ps -> R _FrSe A )
24 7 23 bnj835 
 |-  ( ch -> R _FrSe A )
25 biid 
 |-  ( R _FrSe A <-> R _FrSe A )
26 biid 
 |-  ( -. x e. _trCl ( x , A , R ) <-> -. x e. _trCl ( x , A , R ) )
27 biid 
 |-  ( A. z e. A ( z R x -> [. z / x ]. -. x e. _trCl ( x , A , R ) ) <-> A. z e. A ( z R x -> [. z / x ]. -. x e. _trCl ( x , A , R ) ) )
28 biid 
 |-  ( ( R _FrSe A /\ x e. A /\ A. z e. A ( z R x -> [. z / x ]. -. x e. _trCl ( x , A , R ) ) ) <-> ( R _FrSe A /\ x e. A /\ A. z e. A ( z R x -> [. z / x ]. -. x e. _trCl ( x , A , R ) ) ) )
29 eqid 
 |-  ( _pred ( x , A , R ) u. U_ z e. _pred ( x , A , R ) _trCl ( z , A , R ) ) = ( _pred ( x , A , R ) u. U_ z e. _pred ( x , A , R ) _trCl ( z , A , R ) )
30 25 26 27 28 29 bnj1417 
 |-  ( R _FrSe A -> A. x e. A -. x e. _trCl ( x , A , R ) )
31 disjsn 
 |-  ( ( _trCl ( x , A , R ) i^i { x } ) = (/) <-> -. x e. _trCl ( x , A , R ) )
32 31 ralbii 
 |-  ( A. x e. A ( _trCl ( x , A , R ) i^i { x } ) = (/) <-> A. x e. A -. x e. _trCl ( x , A , R ) )
33 30 32 sylibr 
 |-  ( R _FrSe A -> A. x e. A ( _trCl ( x , A , R ) i^i { x } ) = (/) )
34 24 33 syl 
 |-  ( ch -> A. x e. A ( _trCl ( x , A , R ) i^i { x } ) = (/) )
35 5 7 bnj1212 
 |-  ( ch -> x e. A )
36 34 35 bnj1294 
 |-  ( ch -> ( _trCl ( x , A , R ) i^i { x } ) = (/) )
37 22 36 eqtrd 
 |-  ( ch -> ( dom P i^i dom { <. x , ( G ` Z ) >. } ) = (/) )
38 funun 
 |-  ( ( ( Fun P /\ Fun { <. x , ( G ` Z ) >. } ) /\ ( dom P i^i dom { <. x , ( G ` Z ) >. } ) = (/) ) -> Fun ( P u. { <. x , ( G ` Z ) >. } ) )
39 19 37 38 syl2anc 
 |-  ( ch -> Fun ( P u. { <. x , ( G ` Z ) >. } ) )
40 12 funeqi 
 |-  ( Fun Q <-> Fun ( P u. { <. x , ( G ` Z ) >. } ) )
41 39 40 sylibr 
 |-  ( ch -> Fun Q )