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Theorem bnj965 29290
Description: Technical lemma for bnj852 29269. 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
bnj965.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj965.2  |-  ( ps"  <->  [. G  / 
f ]. ps )
bnj965.12000  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj965.13000  |-  G  =  ( f  u.  { <. n ,  C >. } )
Assertion
Ref Expression
bnj965  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
Distinct variable groups:    A, f    i, G    f, N    R, f    f, i, y    y, n
Allowed substitution hints:    ps( y, f, i, m, n)    A( y, i, m, n)    C( y, f, i, m, n)    R( y, i, m, n)    G( y, f, m, n)    N( y, i, m, n)    ps"( y, f, i, m, n)

Proof of Theorem bnj965
StepHypRef Expression
1 bnj965.1 . 2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2 bnj965.2 . 2  |-  ( ps"  <->  [. G  / 
f ]. ps )
3 bnj965.13000 . . 3  |-  G  =  ( f  u.  { <. n ,  C >. } )
43bnj918 29112 . 2  |-  G  e. 
_V
5 bnj965.12000 . 2  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
61, 2, 4, 5, 3bnj1000 29289 1  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   A.wral 2556   [.wsbc 3004    u. cun 3163   {csn 3653   <.cop 3656   U_ciun 3921   suc csuc 4410   omcom 4672   ` cfv 5271    predc-bnj14 29029
This theorem is referenced by:  bnj964  29291  bnj999  29305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-iota 5235  df-fv 5279
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