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Theorem bnj590 31903
Description: Technical lemma for bnj852 31914. 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 2776 . . . 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 195 . . 3  |-  ( ps 
->  ( i  e.  om  ->  ( suc  i  e.  n  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )
4 eleq1 2503 . . . . 5  |-  ( B  =  suc  i  -> 
( B  e.  n  <->  suc  i  e.  n ) )
5 fveq2 5691 . . . . . 6  |-  ( B  =  suc  i  -> 
( f `  B
)  =  ( f `
 suc  i )
)
65eqeq1d 2451 . . . . 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 320 . . . 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 316 . . 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 221 . 2  |-  ( B  =  suc  i  -> 
( ps  ->  (
i  e.  om  ->  ( B  e.  n  -> 
( f `  B
)  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) ) ) ) )
109imp 429 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   U_ciun 4171   suc csuc 4721   ` cfv 5418   omcom 6476    predc-bnj14 31676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426
This theorem is referenced by:  bnj594  31905
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