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Theorem bnj126 34332
Description: Technical lemma for bnj150 34335. 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 3380 . 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 34323 . . 3  |-  F  e. 
_V
74, 6bnj106 34327 . 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 271 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823   A.wral 2804   [.wsbc 3324   (/)c0 3783   {csn 4016   <.cop 4022   U_ciun 4315   suc csuc 4869   ` cfv 5570   omcom 6673   1oc1o 7115    predc-bnj14 34141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-pw 4001  df-sn 4017  df-pr 4019  df-uni 4236  df-iun 4317  df-br 4440  df-suc 4873  df-iota 5534  df-fv 5578  df-1o 7122
This theorem is referenced by:  bnj150  34335  bnj153  34339
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