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Theorem bnj106 34273
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 34124 . . . 4  |-  1o  e.  _V
31, 2bnj92 34267 . . 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 3308 . 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 5773 . . . . . 6  |-  ( f  =  F  ->  (
f `  suc  i )  =  ( F `  suc  i ) )
7 fveq1 5773 . . . . . . 7  |-  ( f  =  F  ->  (
f `  i )  =  ( F `  i ) )
87bnj1113 34191 . . . . . 6  |-  ( f  =  F  ->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) )
96, 8eqeq12d 2404 . . . . 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 314 . . . 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 2821 . . 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 3287 . 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 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034   [.wsbc 3252   U_ciun 4243   suc csuc 4794   ` cfv 5496   omcom 6599   1oc1o 7041    predc-bnj14 34087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-pw 3929  df-sn 3945  df-uni 4164  df-iun 4245  df-br 4368  df-suc 4798  df-iota 5460  df-fv 5504  df-1o 7048
This theorem is referenced by:  bnj126  34278
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