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Theorem bnj126 32169
Description: Technical lemma for bnj150 32172. 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 3347 . 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 32160 . . 3  |-  F  e. 
_V
74, 6bnj106 32164 . 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 1370    e. wcel 1758   A.wral 2795   [.wsbc 3287   (/)c0 3738   {csn 3978   <.cop 3984   U_ciun 4272   suc csuc 4822   ` cfv 5519   omcom 6579   1oc1o 7016    predc-bnj14 31979
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-pw 3963  df-sn 3979  df-pr 3981  df-uni 4193  df-iun 4274  df-br 4394  df-suc 4826  df-iota 5482  df-fv 5527  df-1o 7023
This theorem is referenced by:  bnj150  32172  bnj153  32176
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