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Theorem bnj106 31957
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj106.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj106.2  |-  F  e. 
_V
Assertion
Ref Expression
bnj106  |-  ( [. F  /  f ]. [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
Distinct variable groups:    A, f, n    f, F, i, y    R, f, n    i, n, y
Allowed substitution hints:    ps( y, f, i, n)    A( y,
i)    R( y, i)    F( n)

Proof of Theorem bnj106
StepHypRef Expression
1 bnj106.1 . . . 4  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2 bnj105 31809 . . . 4  |-  1o  e.  _V
31, 2bnj92 31951 . . 3  |-  ( [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
43sbcbii 3267 . 2  |-  ( [. F  /  f ]. [. 1o  /  n ]. ps  <->  [. F  / 
f ]. A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
5 bnj106.2 . . 3  |-  F  e. 
_V
6 fveq1 5711 . . . . . 6  |-  ( f  =  F  ->  (
f `  suc  i )  =  ( F `  suc  i ) )
7 fveq1 5711 . . . . . . 7  |-  ( f  =  F  ->  (
f `  i )  =  ( F `  i ) )
87bnj1113 31875 . . . . . 6  |-  ( f  =  F  ->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) )
96, 8eqeq12d 2457 . . . . 5  |-  ( f  =  F  ->  (
( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R )  <->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
109imbi2d 316 . . . 4  |-  ( f  =  F  ->  (
( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  ( suc  i  e.  1o  ->  ( F `  suc  i
)  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) ) ) )
1110ralbidv 2756 . . 3  |-  ( f  =  F  ->  ( A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) ) )
125, 11sbcie 3242 . 2  |-  ( [. F  /  f ]. A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
134, 12bitri 249 1  |-  ( [. F  /  f ]. [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   A.wral 2736   _Vcvv 2993   [.wsbc 3207   U_ciun 4192   suc csuc 4742   ` cfv 5439   omcom 6497   1oc1o 6934    predc-bnj14 31772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-pw 3883  df-sn 3899  df-uni 4113  df-iun 4194  df-br 4314  df-suc 4746  df-iota 5402  df-fv 5447  df-1o 6941
This theorem is referenced by:  bnj126  31962
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