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Theorem bnj965 29375
Description: Technical lemma for bnj852 29354. 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 29197 . 2  |-  G  e. 
_V
5 bnj965.12000 . 2  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
61, 2, 4, 5, 3bnj1000 29374 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 178    = wceq 1653    e. wcel 1726   A.wral 2707   [.wsbc 3163    u. cun 3320   {csn 3816   <.cop 3819   U_ciun 4095   suc csuc 4585   omcom 4847   ` cfv 5456    predc-bnj14 29114
This theorem is referenced by:  bnj964  29376  bnj999  29390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-iota 5420  df-fv 5464
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