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Theorem bnj121 31860
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj121.1  |-  ( ze  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )
bnj121.2  |-  ( ze'  <->  [. 1o  /  n ]. ze )
bnj121.3  |-  ( ph'  <->  [. 1o  /  n ]. ph )
bnj121.4  |-  ( ps'  <->  [. 1o  /  n ]. ps )
Assertion
Ref Expression
bnj121  |-  ( ze'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
Distinct variable groups:    A, n    R, n    f, n    x, n
Allowed substitution hints:    ph( x, f, n)    ps( x, f, n)    ze( x, f, n)    A( x, f)    R( x, f)    ph'( x, f, n)    ps'( x, f, n)    ze'( x, f, n)

Proof of Theorem bnj121
StepHypRef Expression
1 bnj121.1 . . 3  |-  ( ze  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )
21sbcbii 3244 . 2  |-  ( [. 1o  /  n ]. ze  <->  [. 1o  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )
3 bnj121.2 . 2  |-  ( ze'  <->  [. 1o  /  n ]. ze )
4 bnj105 31710 . . . . . . . 8  |-  1o  e.  _V
54bnj90 31708 . . . . . . 7  |-  ( [. 1o  /  n ]. f  Fn  n  <->  f  Fn  1o )
65bicomi 202 . . . . . 6  |-  ( f  Fn  1o  <->  [. 1o  /  n ]. f  Fn  n
)
7 bnj121.3 . . . . . 6  |-  ( ph'  <->  [. 1o  /  n ]. ph )
8 bnj121.4 . . . . . 6  |-  ( ps'  <->  [. 1o  /  n ]. ps )
96, 7, 83anbi123i 1176 . . . . 5  |-  ( ( f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( [. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
10 sbc3an 3247 . . . . 5  |-  ( [. 1o  /  n ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  ( [. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
119, 10bitr4i 252 . . . 4  |-  ( ( f  Fn  1o  /\  ph' 
/\  ps' )  <->  [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\  ps ) )
1211imbi2i 312 . . 3  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  ph'  /\  ps' ) )  <-> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\  ps ) ) )
13 nfv 1673 . . . . 5  |-  F/ n
( R  FrSe  A  /\  x  e.  A
)
1413sbc19.21g 3257 . . . 4  |-  ( 1o  e.  _V  ->  ( [. 1o  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\ 
ps ) ) ) )
154, 14ax-mp 5 . . 3  |-  ( [. 1o  /  n ]. (
( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\ 
ps ) ) )
1612, 15bitr4i 252 . 2  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  ph'  /\  ps' ) )  <->  [. 1o  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )
172, 3, 163bitr4i 277 1  |-  ( ze'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1756   _Vcvv 2970   [.wsbc 3184    Fn wfn 5411   1oc1o 6911    FrSe w-bnj15 31677
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-pw 3860  df-sn 3876  df-suc 4723  df-fn 5419  df-1o 6918
This theorem is referenced by:  bnj150  31866  bnj153  31870
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