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Theorem bnj126 28582
Description: Technical lemma for bnj150 28585. 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
bnj126.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj126.2  |-  ( ps'  <->  [. 1o  /  n ]. ps )
bnj126.3  |-  ( ps"  <->  [. F  / 
f ]. ps' )
bnj126.4  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
Assertion
Ref Expression
bnj126  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
Distinct variable groups:    A, f, n    f, F, i, y    R, f, n    i, n, y
Allowed substitution hints:    ps( x, y, f, i, n)    A( x, y, i)    R( x, y, i)    F( x, n)    ps'( x, y, f, i, n)    ps"( x, y, f, i, n)

Proof of Theorem bnj126
StepHypRef Expression
1 bnj126.3 . 2  |-  ( ps"  <->  [. F  / 
f ]. ps' )
2 bnj126.2 . . 3  |-  ( ps'  <->  [. 1o  /  n ]. ps )
32sbcbii 3159 . 2  |-  ( [. F  /  f ]. ps'  <->  [. F  / 
f ]. [. 1o  /  n ]. ps )
4 bnj126.1 . . 3  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
5 bnj126.4 . . . 4  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
65bnj95 28573 . . 3  |-  F  e. 
_V
74, 6bnj106 28577 . 2  |-  ( [. F  /  f ]. [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
81, 3, 73bitri 263 1  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   A.wral 2649   [.wsbc 3104   (/)c0 3571   {csn 3757   <.cop 3760   U_ciun 4035   suc csuc 4524   omcom 4785   ` cfv 5394   1oc1o 6653    predc-bnj14 28390
This theorem is referenced by:  bnj150  28585  bnj153  28589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-pw 3744  df-sn 3763  df-pr 3764  df-uni 3958  df-iun 4037  df-br 4154  df-suc 4528  df-iota 5358  df-fv 5402  df-1o 6660
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