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Theorem bnj130 28583
Description: Technical lemma for bnj151 28586. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj130.1  |-  ( th  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
bnj130.2  |-  ( ph'  <->  [. 1o  /  n ]. ph )
bnj130.3  |-  ( ps'  <->  [. 1o  /  n ]. ps )
bnj130.4  |-  ( th'  <->  [. 1o  /  n ]. th )
Assertion
Ref Expression
bnj130  |-  ( th'  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( 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)    th( x, f, n)    A( x, f)    R( x, f)    ph'( x, f, n)    ps'( x, f, n)    th'( x, f, n)

Proof of Theorem bnj130
StepHypRef Expression
1 bnj130.1 . . 3  |-  ( th  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
21sbcbii 3159 . 2  |-  ( [. 1o  /  n ]. th  <->  [. 1o  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
3 bnj130.4 . 2  |-  ( th'  <->  [. 1o  /  n ]. th )
4 bnj105 28427 . . . . . . . . . 10  |-  1o  e.  _V
54bnj90 28425 . . . . . . . . 9  |-  ( [. 1o  /  n ]. f  Fn  n  <->  f  Fn  1o )
65bicomi 194 . . . . . . . 8  |-  ( f  Fn  1o  <->  [. 1o  /  n ]. f  Fn  n
)
7 bnj130.2 . . . . . . . 8  |-  ( ph'  <->  [. 1o  /  n ]. ph )
8 bnj130.3 . . . . . . . 8  |-  ( ps'  <->  [. 1o  /  n ]. ps )
96, 7, 83anbi123i 1142 . . . . . . 7  |-  ( ( f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( [. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
10 sbc3ang 3162 . . . . . . . 8  |-  ( 1o  e.  _V  ->  ( [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\  ps )  <->  (
[. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps )
) )
114, 10ax-mp 8 . . . . . . 7  |-  ( [. 1o  /  n ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  ( [. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
129, 11bitr4i 244 . . . . . 6  |-  ( ( f  Fn  1o  /\  ph' 
/\  ps' )  <->  [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\  ps ) )
1312eubii 2247 . . . . 5  |-  ( E! f ( f  Fn  1o  /\  ph'  /\  ps' )  <->  E! f [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\  ps )
)
144bnj89 28424 . . . . 5  |-  ( [. 1o  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )  <->  E! f [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\ 
ps ) )
1513, 14bitr4i 244 . . . 4  |-  ( E! f ( f  Fn  1o  /\  ph'  /\  ps' )  <->  [. 1o  /  n ]. E! f ( f  Fn  n  /\  ph 
/\  ps ) )
1615imbi2i 304 . . 3  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  1o  /\  ph'  /\  ps' ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  [. 1o  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )
) )
17 nfv 1626 . . . . 5  |-  F/ n
( R  FrSe  A  /\  x  e.  A
)
1817sbc19.21g 3168 . . . 4  |-  ( 1o  e.  _V  ->  ( [. 1o  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
)  <->  ( ( R 
FrSe  A  /\  x  e.  A )  ->  [. 1o  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )
) ) )
194, 18ax-mp 8 . . 3  |-  ( [. 1o  /  n ]. (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
)  <->  ( ( R 
FrSe  A  /\  x  e.  A )  ->  [. 1o  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )
) )
2016, 19bitr4i 244 . 2  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  1o  /\  ph'  /\  ps' ) )  <->  [. 1o  /  n ]. ( ( R 
FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph  /\  ps ) ) )
212, 3, 203bitr4i 269 1  |-  ( th'  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1717   E!weu 2238   _Vcvv 2899   [.wsbc 3104    Fn wfn 5389   1oc1o 6653    FrSe w-bnj15 28394
This theorem is referenced by:  bnj151  28586
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318
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-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-pw 3744  df-sn 3763  df-suc 4528  df-fn 5397  df-1o 6660
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