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Theorem bnj126 29181
Description: Technical lemma for bnj150 29184. 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 3208 . 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 29172 . . 3  |-  F  e. 
_V
74, 6bnj106 29176 . 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 1652    e. wcel 1725   A.wral 2697   [.wsbc 3153   (/)c0 3620   {csn 3806   <.cop 3809   U_ciun 4085   suc csuc 4575   omcom 4837   ` cfv 5446   1oc1o 6709    predc-bnj14 28989
This theorem is referenced by:  bnj150  29184  bnj153  29188
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  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-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-pw 3793  df-sn 3812  df-pr 3813  df-uni 4008  df-iun 4087  df-br 4205  df-suc 4579  df-iota 5410  df-fv 5454  df-1o 6716
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