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Theorem bnj1112 34459
Description: Technical lemma for bnj69 34486. 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 34198 . 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 2526 . . . . 5  |-  ( i  =  j  ->  (
i  e.  om  <->  j  e.  om ) )
4 suceq 4932 . . . . . 6  |-  ( i  =  j  ->  suc  i  =  suc  j )
54eleq1d 2523 . . . . 5  |-  ( i  =  j  ->  ( suc  i  e.  n  <->  suc  j  e.  n ) )
63, 5anbi12d 708 . . . 4  |-  ( i  =  j  ->  (
( i  e.  om  /\ 
suc  i  e.  n
)  <->  ( j  e. 
om  /\  suc  j  e.  n ) ) )
74fveq2d 5852 . . . . 5  |-  ( i  =  j  ->  (
f `  suc  i )  =  ( f `  suc  j ) )
8 fveq2 5848 . . . . . 6  |-  ( i  =  j  ->  (
f `  i )  =  ( f `  j ) )
98bnj1113 34264 . . . . 5  |-  ( i  =  j  ->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) )
107, 9eqeq12d 2476 . . . 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 318 . . 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 2028 . 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 249 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398    e. wcel 1823   A.wral 2804   U_ciun 4315   suc csuc 4869   ` cfv 5570   omcom 6673    predc-bnj14 34160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-suc 4873  df-iota 5534  df-fv 5578
This theorem is referenced by:  bnj1118  34460
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