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Theorem bnj590 29355
Description: Technical lemma for bnj852 29366. 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 2768 . . . 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 189 . . 3  |-  ( ps 
->  ( i  e.  om  ->  ( suc  i  e.  n  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )
4 eleq1 2498 . . . . 5  |-  ( B  =  suc  i  -> 
( B  e.  n  <->  suc  i  e.  n ) )
5 fveq2 5731 . . . . . 6  |-  ( B  =  suc  i  -> 
( f `  B
)  =  ( f `
 suc  i )
)
65eqeq1d 2446 . . . . 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 313 . . . 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 309 . . 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 214 . 2  |-  ( B  =  suc  i  -> 
( ps  ->  (
i  e.  om  ->  ( B  e.  n  -> 
( f `  B
)  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) ) ) ) )
109imp 420 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   U_ciun 4095   suc csuc 4586   omcom 4848   ` cfv 5457    predc-bnj14 29126
This theorem is referenced by:  bnj594  29357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465
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