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Theorem bnj121 29218
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 ) ) )
2 bnj105 29066 . . 3  |-  1o  e.  _V
31, 2bnj524 29082 . 2  |-  ( [. 1o  /  n ]. ze  <->  [. 1o  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )
4 bnj121.2 . 2  |-  ( ze'  <->  [. 1o  /  n ]. ze )
52bnj90 29064 . . . . . . 7  |-  ( [. 1o  /  n ]. f  Fn  n  <->  f  Fn  1o )
65bicomi 193 . . . . . 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 1140 . . . . 5  |-  ( ( f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( [. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
10 sbc3ang 3062 . . . . . 6  |-  ( 1o  e.  _V  ->  ( [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\  ps )  <->  (
[. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps )
) )
112, 10ax-mp 8 . . . . 5  |-  ( [. 1o  /  n ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  ( [. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
129, 11bitr4i 243 . . . 4  |-  ( ( f  Fn  1o  /\  ph' 
/\  ps' )  <->  [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\  ps ) )
1312imbi2i 303 . . 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 ) ) )
14 nfv 1609 . . . . 5  |-  F/ n
( R  FrSe  A  /\  x  e.  A
)
1514sbc19.21g 3068 . . . 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 ) ) ) )
162, 15ax-mp 8 . . 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 ) ) )
1713, 16bitr4i 243 . 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 ) ) )
183, 4, 173bitr4i 268 1  |-  ( ze'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696   _Vcvv 2801   [.wsbc 3004    Fn wfn 5266   1oc1o 6488    FrSe w-bnj15 29033
This theorem is referenced by:  bnj150  29224  bnj153  29228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-suc 4414  df-fn 5274  df-1o 6495
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