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Theorem bnj222 33020
Description: Technical lemma for bnj229 33021. 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 4943 . . . . 5  |-  ( i  =  m  ->  suc  i  =  suc  m )
32eleq1d 2536 . . . 4  |-  ( i  =  m  ->  ( suc  i  e.  N  <->  suc  m  e.  N ) )
42fveq2d 5868 . . . . 5  |-  ( i  =  m  ->  ( F `  suc  i )  =  ( F `  suc  m ) )
5 fveq2 5864 . . . . . 6  |-  ( i  =  m  ->  ( F `  i )  =  ( F `  m ) )
65bnj1113 32923 . . . . 5  |-  ( i  =  m  ->  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) )
74, 6eqeq12d 2489 . . . 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 320 . . 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 3088 . 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 1379    e. wcel 1767   A.wral 2814   U_ciun 4325   suc csuc 4880   ` cfv 5586   omcom 6678    predc-bnj14 32820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-suc 4884  df-iota 5549  df-fv 5594
This theorem is referenced by:  bnj229  33021  bnj589  33046
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