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Theorem bnj222 29268
Description: Technical lemma for bnj229 29269. 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
bnj222.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj222  |-  ( ps  <->  A. m  e.  om  ( suc  m  e.  N  -> 
( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) ) )
Distinct variable groups:    A, i, m    i, F, m, y   
i, N, m    R, i, m
Allowed substitution hints:    ps( y, i, m)    A( y)    R( y)    N( y)

Proof of Theorem bnj222
StepHypRef Expression
1 bnj222.1 . 2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
2 suceq 5475 . . . . 5  |-  ( i  =  m  ->  suc  i  =  suc  m )
32eleq1d 2471 . . . 4  |-  ( i  =  m  ->  ( suc  i  e.  N  <->  suc  m  e.  N ) )
42fveq2d 5853 . . . . 5  |-  ( i  =  m  ->  ( F `  suc  i )  =  ( F `  suc  m ) )
5 fveq2 5849 . . . . . 6  |-  ( i  =  m  ->  ( F `  i )  =  ( F `  m ) )
65bnj1113 29171 . . . . 5  |-  ( i  =  m  ->  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) )
74, 6eqeq12d 2424 . . . 4  |-  ( i  =  m  ->  (
( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R )  <->  ( F `  suc  m )  = 
U_ y  e.  ( F `  m ) 
pred ( y ,  A ,  R ) ) )
83, 7imbi12d 318 . . 3  |-  ( i  =  m  ->  (
( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )  <->  ( suc  m  e.  N  ->  ( F `  suc  m
)  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) ) ) )
98cbvralv 3034 . 2  |-  ( A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )  <->  A. m  e.  om  ( suc  m  e.  N  ->  ( F `
 suc  m )  =  U_ y  e.  ( F `  m ) 
pred ( y ,  A ,  R ) ) )
101, 9bitri 249 1  |-  ( ps  <->  A. m  e.  om  ( suc  m  e.  N  -> 
( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405    e. wcel 1842   A.wral 2754   U_ciun 4271   suc csuc 5412   ` cfv 5569   omcom 6683    predc-bnj14 29067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-suc 5416  df-iota 5533  df-fv 5577
This theorem is referenced by:  bnj229  29269  bnj589  29294
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