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Theorem bnj590 28942
Description: Technical lemma for bnj852 28953. 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.)
Hypothesis
Ref Expression
bnj590.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj590  |-  ( ( B  =  suc  i  /\  ps )  ->  (
i  e.  om  ->  ( B  e.  n  -> 
( f `  B
)  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) ) ) )

Proof of Theorem bnj590
StepHypRef Expression
1 bnj590.1 . . . 4  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2 rsp 2603 . . . 4  |-  ( A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  -> 
( i  e.  om  ->  ( suc  i  e.  n  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )
31, 2sylbi 187 . . 3  |-  ( ps 
->  ( i  e.  om  ->  ( suc  i  e.  n  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )
4 eleq1 2343 . . . . 5  |-  ( B  =  suc  i  -> 
( B  e.  n  <->  suc  i  e.  n ) )
5 fveq2 5525 . . . . . 6  |-  ( B  =  suc  i  -> 
( f `  B
)  =  ( f `
 suc  i )
)
65eqeq1d 2291 . . . . 5  |-  ( B  =  suc  i  -> 
( ( f `  B )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R )  <->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
74, 6imbi12d 311 . . . 4  |-  ( B  =  suc  i  -> 
( ( B  e.  n  ->  ( f `  B )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  ( suc  i  e.  n  ->  ( f `  suc  i
)  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) ) ) )
87imbi2d 307 . . 3  |-  ( B  =  suc  i  -> 
( ( i  e. 
om  ->  ( B  e.  n  ->  ( f `  B )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )  <-> 
( i  e.  om  ->  ( suc  i  e.  n  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) ) )
93, 8syl5ibr 212 . 2  |-  ( B  =  suc  i  -> 
( ps  ->  (
i  e.  om  ->  ( B  e.  n  -> 
( f `  B
)  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) ) ) ) )
109imp 418 1  |-  ( ( B  =  suc  i  /\  ps )  ->  (
i  e.  om  ->  ( B  e.  n  -> 
( f `  B
)  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   U_ciun 3905   suc csuc 4394   omcom 4656   ` cfv 5255    predc-bnj14 28713
This theorem is referenced by:  bnj594  28944
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-3an 936  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-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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