Metamath Proof Explorer

Theorem bnj852

Description: Technical lemma for bnj69 . 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 bnj852.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj852.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj852.3 
|- D = ( _om \ { (/) } )
Assertion bnj852 
|- ( ( R _FrSe A /\ X e. A ) -> A. n e. D E! f ( f Fn n /\ ph /\ ps ) )

Proof

Step Hyp Ref Expression
1 bnj852.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj852.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj852.3 
 |-  D = ( _om \ { (/) } )
4 elisset 
 |-  ( X e. A -> E. x x = X )
5 4 adantl 
 |-  ( ( R _FrSe A /\ X e. A ) -> E. x x = X )
6 5 ancri 
 |-  ( ( R _FrSe A /\ X e. A ) -> ( E. x x = X /\ ( R _FrSe A /\ X e. A ) ) )
7 6 bnj534 
 |-  ( ( R _FrSe A /\ X e. A ) -> E. x ( x = X /\ ( R _FrSe A /\ X e. A ) ) )
8 eleq1 
 |-  ( x = X -> ( x e. A <-> X e. A ) )
9 8 anbi2d 
 |-  ( x = X -> ( ( R _FrSe A /\ x e. A ) <-> ( R _FrSe A /\ X e. A ) ) )
10 9 biimpar 
 |-  ( ( x = X /\ ( R _FrSe A /\ X e. A ) ) -> ( R _FrSe A /\ x e. A ) )
11 biid 
 |-  ( A. z e. D ( z _E n -> [. z / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) ) <-> A. z e. D ( z _E n -> [. z / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) ) )
12 omex 
 |-  _om e. _V
13 difexg 
 |-  ( _om e. _V -> ( _om \ { (/) } ) e. _V )
14 12 13 ax-mp 
 |-  ( _om \ { (/) } ) e. _V
15 3 14 eqeltri 
 |-  D e. _V
16 zfregfr 
 |-  _E Fr D
17 11 15 16 bnj157 
 |-  ( A. n e. D ( A. z e. D ( z _E n -> [. z / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) ) -> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) ) -> A. n e. D ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) )
18 biid 
 |-  ( ( f ` (/) ) = _pred ( x , A , R ) <-> ( f ` (/) ) = _pred ( x , A , R ) )
19 biid 
 |-  ( ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) )
20 18 2 3 19 11 bnj153 
 |-  ( n = 1o -> ( ( n e. D /\ A. z e. D ( z _E n -> [. z / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) ) ) -> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) ) )
21 18 2 3 19 11 bnj601 
 |-  ( n =/= 1o -> ( ( n e. D /\ A. z e. D ( z _E n -> [. z / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) ) ) -> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) ) )
22 20 21 pm2.61ine 
 |-  ( ( n e. D /\ A. z e. D ( z _E n -> [. z / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) ) ) -> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) )
23 22 ex 
 |-  ( n e. D -> ( A. z e. D ( z _E n -> [. z / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) ) -> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) ) )
24 17 23 mprg 
 |-  A. n e. D ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) )
25 r19.21v 
 |-  ( A. n e. D ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) <-> ( ( R _FrSe A /\ x e. A ) -> A. n e. D E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) ) )
26 24 25 mpbi 
 |-  ( ( R _FrSe A /\ x e. A ) -> A. n e. D E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) )
27 10 26 syl 
 |-  ( ( x = X /\ ( R _FrSe A /\ X e. A ) ) -> A. n e. D E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) )
28 bnj602 
 |-  ( x = X -> _pred ( x , A , R ) = _pred ( X , A , R ) )
29 28 eqeq2d 
 |-  ( x = X -> ( ( f ` (/) ) = _pred ( x , A , R ) <-> ( f ` (/) ) = _pred ( X , A , R ) ) )
30 29 1 bitr4di 
 |-  ( x = X -> ( ( f ` (/) ) = _pred ( x , A , R ) <-> ph ) )
31 30 3anbi2d 
 |-  ( x = X -> ( ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) <-> ( f Fn n /\ ph /\ ps ) ) )
32 31 eubidv 
 |-  ( x = X -> ( E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) <-> E! f ( f Fn n /\ ph /\ ps ) ) )
33 32 ralbidv 
 |-  ( x = X -> ( A. n e. D E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) <-> A. n e. D E! f ( f Fn n /\ ph /\ ps ) ) )
34 33 adantr 
 |-  ( ( x = X /\ ( R _FrSe A /\ X e. A ) ) -> ( A. n e. D E! f ( f Fn n /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps ) <-> A. n e. D E! f ( f Fn n /\ ph /\ ps ) ) )
35 27 34 mpbid 
 |-  ( ( x = X /\ ( R _FrSe A /\ X e. A ) ) -> A. n e. D E! f ( f Fn n /\ ph /\ ps ) )
36 7 35 bnj593 
 |-  ( ( R _FrSe A /\ X e. A ) -> E. x A. n e. D E! f ( f Fn n /\ ph /\ ps ) )
37 36 bnj937 
 |-  ( ( R _FrSe A /\ X e. A ) -> A. n e. D E! f ( f Fn n /\ ph /\ ps ) )