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Theorem bnj1039 32275
Description: Technical lemma for bnj69 32314. 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 3075 . . 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 2807 . . . . 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 1616 . . . 4  |-  F/ i ps
65sbcgf 3360 . . 3  |-  ( j  e.  _V  ->  ( [. j  /  i ]. ps  <->  ps ) )
72, 6ax-mp 5 . 2  |-  ( [. j  /  i ]. ps  <->  ps )
81, 7, 33bitri 271 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   A.wral 2796   _Vcvv 3072   [.wsbc 3288   U_ciun 4274   suc csuc 4824   ` cfv 5521   omcom 6581    predc-bnj14 31989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-ral 2801  df-v 3074  df-sbc 3289
This theorem is referenced by:  bnj1128  32294
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