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Theorem bnj965 29761
Description: Technical lemma for bnj852 29740. 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 29585 . 2  |-  G  e. 
_V
5 bnj965.12000 . 2  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
61, 2, 4, 5, 3bnj1000 29760 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1872   A.wral 2771   [.wsbc 3299    u. cun 3434   {csn 3998   <.cop 4004   U_ciun 4299   suc csuc 5444   ` cfv 5601   omcom 6706    predc-bnj14 29501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-iota 5565  df-fv 5609
This theorem is referenced by:  bnj964  29762  bnj999  29776
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