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Theorem bnj92 28572
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 3160 . 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 28447 . 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 3146 . . . . 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 3165 . . . . . 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 28445 . . . . 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 2674 . 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 1649    e. wcel 1717   A.wral 2650   _Vcvv 2900   [.wsbc 3105   U_ciun 4036   suc csuc 4525   omcom 4786   ` cfv 5395    predc-bnj14 28391
This theorem is referenced by:  bnj106  28578  bnj153  28590
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-v 2902  df-sbc 3106
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