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Theorem bnj222 29231
Description: Technical lemma for bnj229 29232. 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 4473 . . . . 5  |-  ( i  =  m  ->  suc  i  =  suc  m )
32eleq1d 2362 . . . 4  |-  ( i  =  m  ->  ( suc  i  e.  N  <->  suc  m  e.  N ) )
42fveq2d 5545 . . . . 5  |-  ( i  =  m  ->  ( F `  suc  i )  =  ( F `  suc  m ) )
5 fveq2 5541 . . . . . 6  |-  ( i  =  m  ->  ( F `  i )  =  ( F `  m ) )
65bnj1113 29133 . . . . 5  |-  ( i  =  m  ->  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) )
74, 6eqeq12d 2310 . . . 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 311 . . 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 2777 . 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 240 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 set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   A.wral 2556   U_ciun 3921   suc csuc 4410   omcom 4672   ` cfv 5271    predc-bnj14 29029
This theorem is referenced by:  bnj229  29232  bnj589  29257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-suc 4414  df-iota 5235  df-fv 5279
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