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Theorem bnj106 28570
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 28420 . . . 4  |-  1o  e.  _V
31, 2bnj92 28564 . . 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 3152 . 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 5660 . . . . . 6  |-  ( f  =  F  ->  (
f `  suc  i )  =  ( F `  suc  i ) )
7 fveq1 5660 . . . . . . 7  |-  ( f  =  F  ->  (
f `  i )  =  ( F `  i ) )
87bnj1113 28487 . . . . . 6  |-  ( f  =  F  ->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) )
96, 8eqeq12d 2394 . . . . 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 308 . . . 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 2662 . . 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 3131 . 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 241 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 set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   A.wral 2642   _Vcvv 2892   [.wsbc 3097   U_ciun 4028   suc csuc 4517   omcom 4778   ` cfv 5387   1oc1o 6646    predc-bnj14 28383
This theorem is referenced by:  bnj126  28575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-pw 3737  df-sn 3756  df-uni 3951  df-iun 4030  df-br 4147  df-suc 4521  df-iota 5351  df-fv 5395  df-1o 6653
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