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Theorem bnj1112 29013
Description: Technical lemma for bnj69 29040. 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
bnj1112.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj1112  |-  ( ps  <->  A. j ( ( j  e.  om  /\  suc  j  e.  n )  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
Distinct variable groups:    A, i,
j    R, i, j    f,
i, j, y    i, n, j
Allowed substitution hints:    ps( y, f, i, j, n)    A( y, f, n)    R( y,
f, n)

Proof of Theorem bnj1112
StepHypRef Expression
1 bnj1112.1 . . 3  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
21bnj115 28751 . 2  |-  ( ps  <->  A. i ( ( i  e.  om  /\  suc  i  e.  n )  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 eleq1 2343 . . . . 5  |-  ( i  =  j  ->  (
i  e.  om  <->  j  e.  om ) )
4 suceq 4457 . . . . . 6  |-  ( i  =  j  ->  suc  i  =  suc  j )
54eleq1d 2349 . . . . 5  |-  ( i  =  j  ->  ( suc  i  e.  n  <->  suc  j  e.  n ) )
63, 5anbi12d 691 . . . 4  |-  ( i  =  j  ->  (
( i  e.  om  /\ 
suc  i  e.  n
)  <->  ( j  e. 
om  /\  suc  j  e.  n ) ) )
74fveq2d 5529 . . . . 5  |-  ( i  =  j  ->  (
f `  suc  i )  =  ( f `  suc  j ) )
8 fveq2 5525 . . . . . 6  |-  ( i  =  j  ->  (
f `  i )  =  ( f `  j ) )
98bnj1113 28817 . . . . 5  |-  ( i  =  j  ->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) )
107, 9eqeq12d 2297 . . . 4  |-  ( i  =  j  ->  (
( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R )  <->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) )
116, 10imbi12d 311 . . 3  |-  ( i  =  j  ->  (
( ( i  e. 
om  /\  suc  i  e.  n )  ->  (
f `  suc  i )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )  <->  ( ( j  e.  om  /\  suc  j  e.  n )  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) ) )
1211cbvalv 1942 . 2  |-  ( A. i ( ( i  e.  om  /\  suc  i  e.  n )  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  A. j
( ( j  e. 
om  /\  suc  j  e.  n )  ->  (
f `  suc  j )  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) ) )
132, 12bitri 240 1  |-  ( ps  <->  A. j ( ( j  e.  om  /\  suc  j  e.  n )  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = 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:  bnj1118  29014
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-iun 3907  df-br 4024  df-suc 4398  df-iota 5219  df-fv 5263
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