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Metamath Proof Explorer

Theorem bnj1093

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 bnj1093.1 
|- E. j ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` i ) C_ B )
bnj1093.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj1093.3 
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
Assertion bnj1093 
|- ( ( th /\ ta /\ ch /\ ze ) -> A. i E. j ( ph0 -> ( f ` i ) C_ B ) )

Proof

Step Hyp Ref Expression
1 bnj1093.1 
 |-  E. j ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` i ) C_ B )
2 bnj1093.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj1093.3 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4 2 bnj1095 
 |-  ( ps -> A. i ps )
5 4 3 bnj1096 
 |-  ( ch -> A. i ch )
6 5 bnj1350 
 |-  ( ( th /\ ta /\ ch ) -> A. i ( th /\ ta /\ ch ) )
7 impexp 
 |-  ( ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` i ) C_ B ) <-> ( ( th /\ ta /\ ch ) -> ( ph0 -> ( f ` i ) C_ B ) ) )
8 7 exbii 
 |-  ( E. j ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` i ) C_ B ) <-> E. j ( ( th /\ ta /\ ch ) -> ( ph0 -> ( f ` i ) C_ B ) ) )
9 1 8 mpbi 
 |-  E. j ( ( th /\ ta /\ ch ) -> ( ph0 -> ( f ` i ) C_ B ) )
10 9 19.37iv 
 |-  ( ( th /\ ta /\ ch ) -> E. j ( ph0 -> ( f ` i ) C_ B ) )
11 6 10 alrimih 
 |-  ( ( th /\ ta /\ ch ) -> A. i E. j ( ph0 -> ( f ` i ) C_ B ) )
12 11 bnj721 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> A. i E. j ( ph0 -> ( f ` i ) C_ B ) )