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Theorem bnj1112 29352
Description: Technical lemma for bnj69 29379. 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 29090 . 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 2496 . . . . 5  |-  ( i  =  j  ->  (
i  e.  om  <->  j  e.  om ) )
4 suceq 4646 . . . . . 6  |-  ( i  =  j  ->  suc  i  =  suc  j )
54eleq1d 2502 . . . . 5  |-  ( i  =  j  ->  ( suc  i  e.  n  <->  suc  j  e.  n ) )
63, 5anbi12d 692 . . . 4  |-  ( i  =  j  ->  (
( i  e.  om  /\ 
suc  i  e.  n
)  <->  ( j  e. 
om  /\  suc  j  e.  n ) ) )
74fveq2d 5732 . . . . 5  |-  ( i  =  j  ->  (
f `  suc  i )  =  ( f `  suc  j ) )
8 fveq2 5728 . . . . . 6  |-  ( i  =  j  ->  (
f `  i )  =  ( f `  j ) )
98bnj1113 29156 . . . . 5  |-  ( i  =  j  ->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) )
107, 9eqeq12d 2450 . . . 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 312 . . 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 1984 . 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 241 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 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   A.wral 2705   U_ciun 4093   suc csuc 4583   omcom 4845   ` cfv 5454    predc-bnj14 29052
This theorem is referenced by:  bnj1118  29353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-suc 4587  df-iota 5418  df-fv 5462
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