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Theorem bnj590 28999
Description: Technical lemma for bnj852 29010. 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 2734 . . . 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 188 . . 3  |-  ( ps 
->  ( i  e.  om  ->  ( suc  i  e.  n  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )
4 eleq1 2472 . . . . 5  |-  ( B  =  suc  i  -> 
( B  e.  n  <->  suc  i  e.  n ) )
5 fveq2 5695 . . . . . 6  |-  ( B  =  suc  i  -> 
( f `  B
)  =  ( f `
 suc  i )
)
65eqeq1d 2420 . . . . 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 312 . . . 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 308 . . 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 213 . 2  |-  ( B  =  suc  i  -> 
( ps  ->  (
i  e.  om  ->  ( B  e.  n  -> 
( f `  B
)  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) ) ) ) )
109imp 419 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   U_ciun 4061   suc csuc 4551   omcom 4812   ` cfv 5421    predc-bnj14 28770
This theorem is referenced by:  bnj594  29001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429
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