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Theorem bnj1093 29376
Description: Technical lemma for bnj69 29406. 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.)
Hypotheses
Ref Expression
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
Ref Expression
bnj1093  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i E. j (
ph0  ->  ( f `  i )  C_  B
) )
Distinct variable groups:    ch, j    ta, i    th, i    ta, j    th, j    D, i    f, i   
i, n    ph, i
Allowed substitution hints:    ph( y, f, j, n)    ps( y,
f, i, j, n)    ch( y, f, i, n)    th( y, f, n)    ta( y, f, n)    ze( y,
f, i, j, n)    A( y, f, i, j, n)    B( y, f, i, j, n)    D( y,
f, j, n)    R( y, f, i, j, n)    ph0( y, f, i, j, n)

Proof of Theorem bnj1093
StepHypRef Expression
1 bnj1093.2 . . . . . 6  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
21bnj1095 29180 . . . . 5  |-  ( ps 
->  A. i ps )
3 bnj1093.3 . . . . 5  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
42, 3bnj1096 29181 . . . 4  |-  ( ch 
->  A. i ch )
54bnj1350 29224 . . 3  |-  ( ( th  /\  ta  /\  ch )  ->  A. i
( th  /\  ta  /\ 
ch ) )
6 bnj1093.1 . . . . 5  |-  E. j
( ( ( th 
/\  ta  /\  ch )  /\  ph0 )  ->  (
f `  i )  C_  B )
7 impexp 446 . . . . . 6  |-  ( ( ( ( th  /\  ta  /\  ch )  /\  ph0 )  ->  ( f `  i )  C_  B
)  <->  ( ( th 
/\  ta  /\  ch )  ->  ( ph0  ->  ( f `
 i )  C_  B ) ) )
87exbii 1690 . . . . 5  |-  ( E. j ( ( ( th  /\  ta  /\  ch )  /\  ph0 )  ->  ( f `  i
)  C_  B )  <->  E. j ( ( th 
/\  ta  /\  ch )  ->  ( ph0  ->  ( f `
 i )  C_  B ) ) )
96, 8mpbi 210 . . . 4  |-  E. j
( ( th  /\  ta  /\  ch )  -> 
( ph0  ->  ( f `  i )  C_  B
) )
10919.37iv 1795 . . 3  |-  ( ( th  /\  ta  /\  ch )  ->  E. j
( ph0  ->  ( f `  i )  C_  B
) )
115, 10alrimih 1665 . 2  |-  ( ( th  /\  ta  /\  ch )  ->  A. i E. j ( ph0  ->  (
f `  i )  C_  B ) )
1211bnj721 29154 1  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i E. j (
ph0  ->  ( f `  i )  C_  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976   A.wal 1405    = wceq 1407   E.wex 1635    e. wcel 1844   A.wral 2756    C_ wss 3416   U_ciun 4273   suc csuc 5414    Fn wfn 5566   ` cfv 5571   omcom 6685    /\ w-bnj17 29078    predc-bnj14 29080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-12 1880
This theorem depends on definitions:  df-bi 187  df-an 371  df-3an 978  df-ex 1636  df-nf 1640  df-ral 2761  df-bnj17 29079
This theorem is referenced by:  bnj1030  29383
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