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Theorem bnj539 28296
Description: Technical lemma for bnj852 28326. 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
bnj539.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
bnj539.2  |-  ( ps'  <->  [. M  /  n ]. ps )
bnj539.3  |-  M  e. 
_V
Assertion
Ref Expression
bnj539  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  M  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
Distinct variable groups:    A, n    n, F    i, M    R, n    i, n    y, n
Allowed substitution hints:    ps( y, i, n)    A( y, i)    R( y, i)    F( y, i)    M( y, n)    ps'( y, i, n)

Proof of Theorem bnj539
StepHypRef Expression
1 bnj539.2 . 2  |-  ( ps'  <->  [. M  /  n ]. ps )
2 bnj539.1 . . . 4  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
3 bnj539.3 . . . 4  |-  M  e. 
_V
42, 3bnj524 28139 . . 3  |-  ( [. M  /  n ]. ps  <->  [. M  /  n ]. A. i  e.  om  ( suc  i  e.  n  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
53bnj538 28142 . . . 4  |-  ( [. M  /  n ]. A. i  e.  om  ( suc  i  e.  n  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  [. M  /  n ]. ( suc  i  e.  n  ->  ( F `
 suc  i )  =  U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
6 sbcimg 3032 . . . . . . 7  |-  ( M  e.  _V  ->  ( [. M  /  n ]. ( suc  i  e.  n  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) )  <->  ( [. M  /  n ]. suc  i  e.  n  ->  [. M  /  n ]. ( F `
 suc  i )  =  U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) ) )
73, 6ax-mp 8 . . . . . 6  |-  ( [. M  /  n ]. ( suc  i  e.  n  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )  <->  ( [. M  /  n ]. suc  i  e.  n  ->  [. M  /  n ]. ( F `  suc  i
)  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) ) )
8 sbcel2gv 3051 . . . . . . . 8  |-  ( M  e.  _V  ->  ( [. M  /  n ]. suc  i  e.  n  <->  suc  i  e.  M ) )
93, 8ax-mp 8 . . . . . . 7  |-  ( [. M  /  n ]. suc  i  e.  n  <->  suc  i  e.  M )
103bnj525 28140 . . . . . . 7  |-  ( [. M  /  n ]. ( F `  suc  i )  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R )  <-> 
( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )
119, 10imbi12i 316 . . . . . 6  |-  ( (
[. M  /  n ]. suc  i  e.  n  ->  [. M  /  n ]. ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )  <->  ( suc  i  e.  M  ->  ( F `  suc  i
)  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) ) )
127, 11bitri 240 . . . . 5  |-  ( [. M  /  n ]. ( suc  i  e.  n  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )  <->  ( suc  i  e.  M  ->  ( F `  suc  i
)  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) ) )
1312ralbii 2567 . . . 4  |-  ( A. i  e.  om  [. M  /  n ]. ( suc  i  e.  n  -> 
( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  ( suc  i  e.  M  ->  ( F `
 suc  i )  =  U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
145, 13bitri 240 . . 3  |-  ( [. M  /  n ]. A. i  e.  om  ( suc  i  e.  n  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  ( suc  i  e.  M  ->  ( F `
 suc  i )  =  U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
154, 14bitri 240 . 2  |-  ( [. M  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  M  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
161, 15bitri 240 1  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  M  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   [.wsbc 2991   U_ciun 3905   suc csuc 4394   omcom 4656   ` cfv 5255    predc-bnj14 28086
This theorem is referenced by:  bnj600  28324  bnj908  28336  bnj964  28348  bnj999  28362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-sbc 2992
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