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Theorem bnj222 29254
Description: Technical lemma for bnj229 29255. 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 4646 . . . . 5  |-  ( i  =  m  ->  suc  i  =  suc  m )
32eleq1d 2502 . . . 4  |-  ( i  =  m  ->  ( suc  i  e.  N  <->  suc  m  e.  N ) )
42fveq2d 5732 . . . . 5  |-  ( i  =  m  ->  ( F `  suc  i )  =  ( F `  suc  m ) )
5 fveq2 5728 . . . . . 6  |-  ( i  =  m  ->  ( F `  i )  =  ( F `  m ) )
65bnj1113 29156 . . . . 5  |-  ( i  =  m  ->  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) )
74, 6eqeq12d 2450 . . . 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 312 . . 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 2932 . 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 241 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 177    = 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:  bnj229  29255  bnj589  29280
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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