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Theorem bnj1039 29267
Description: Technical lemma for bnj69 29306. 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
bnj1039.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1039.2  |-  ( ps'  <->  [. j  /  i ]. ps )
Assertion
Ref Expression
bnj1039  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )

Proof of Theorem bnj1039
StepHypRef Expression
1 bnj1039.2 . 2  |-  ( ps'  <->  [. j  /  i ]. ps )
2 vex 2951 . . 3  |-  j  e. 
_V
3 bnj1039.1 . . . . 5  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
4 nfra1 2748 . . . . 5  |-  F/ i A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
53, 4nfxfr 1579 . . . 4  |-  F/ i ps
65sbcgf 3216 . . 3  |-  ( j  e.  _V  ->  ( [. j  /  i ]. ps  <->  ps ) )
72, 6ax-mp 8 . 2  |-  ( [. j  /  i ]. ps  <->  ps )
81, 7, 33bitri 263 1  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( 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 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   [.wsbc 3153   U_ciun 4085   suc csuc 4575   omcom 4837   ` cfv 5446    predc-bnj14 28979
This theorem is referenced by:  bnj1128  29286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-sbc 3154
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