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Theorem bnj1039 28671
Description: Technical lemma for bnj69 28710. 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 2895 . . 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 2692 . . . . 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 1576 . . . 4  |-  F/ i ps
65sbcgf 3160 . . 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 1649    e. wcel 1717   A.wral 2642   _Vcvv 2892   [.wsbc 3097   U_ciun 4028   suc csuc 4517   omcom 4778   ` cfv 5387    predc-bnj14 28383
This theorem is referenced by:  bnj1128  28690
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-v 2894  df-sbc 3098
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