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Theorem bnj92 29170
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj92.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj92.2  |-  Z  e. 
_V
Assertion
Ref Expression
bnj92  |-  ( [. Z  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  Z  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
Distinct variable groups:    A, n    R, n    i, Z    f, n    i, n    y, n
Allowed substitution hints:    ps( y, f, i, n)    A( y,
f, i)    R( y,
f, i)    Z( y,
f, n)

Proof of Theorem bnj92
StepHypRef Expression
1 bnj92.1 . . 3  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
21sbcbii 3208 . 2  |-  ( [. Z  /  n ]. ps  <->  [. Z  /  n ]. A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj92.2 . . 3  |-  Z  e. 
_V
43bnj538 29045 . 2  |-  ( [. Z  /  n ]. A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  [. Z  /  n ]. ( suc  i  e.  n  ->  ( f `
 suc  i )  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
5 sbcimg 3194 . . . . 5  |-  ( Z  e.  _V  ->  ( [. Z  /  n ]. ( suc  i  e.  n  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) )  <->  ( [. Z  /  n ]. suc  i  e.  n  ->  [. Z  /  n ]. ( f `
 suc  i )  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )
63, 5ax-mp 8 . . . 4  |-  ( [. Z  /  n ]. ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  ( [. Z  /  n ]. suc  i  e.  n  ->  [. Z  /  n ]. ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
7 sbcel2gv 3213 . . . . . 6  |-  ( Z  e.  _V  ->  ( [. Z  /  n ]. suc  i  e.  n  <->  suc  i  e.  Z ) )
83, 7ax-mp 8 . . . . 5  |-  ( [. Z  /  n ]. suc  i  e.  n  <->  suc  i  e.  Z )
93bnj525 29043 . . . . 5  |-  ( [. Z  /  n ]. (
f `  suc  i )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  <-> 
( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
108, 9imbi12i 317 . . . 4  |-  ( (
[. Z  /  n ]. suc  i  e.  n  ->  [. Z  /  n ]. ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  ( suc  i  e.  Z  ->  ( f `  suc  i
)  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) ) )
116, 10bitri 241 . . 3  |-  ( [. Z  /  n ]. ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  ( suc  i  e.  Z  ->  ( f `  suc  i
)  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) ) )
1211ralbii 2721 . 2  |-  ( A. i  e.  om  [. Z  /  n ]. ( suc  i  e.  n  -> 
( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  ( suc  i  e.  Z  ->  ( f `
 suc  i )  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
132, 4, 123bitri 263 1  |-  ( [. Z  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  Z  ->  ( 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 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   [.wsbc 3153   U_ciun 4085   suc csuc 4575   omcom 4837   ` cfv 5446    predc-bnj14 28989
This theorem is referenced by:  bnj106  29176  bnj153  29188
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 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-sbc 3154
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