Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj864 Unicode version

Theorem bnj864 29270
Description: Technical lemma for bnj69 29356. 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
bnj864.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj864.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj864.3  |-  D  =  ( om  \  { (/)
} )
bnj864.4  |-  ( ch  <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
)
bnj864.5  |-  ( th  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
Assertion
Ref Expression
bnj864  |-  ( ch 
->  E! f th )
Distinct variable groups:    A, f,
i, n, y    D, f, i, n    R, f, i, n, y    f, X, n
Allowed substitution hints:    ph( y, f, i, n)    ps( y,
f, i, n)    ch( y, f, i, n)    th( y,
f, i, n)    D( y)    X( y, i)

Proof of Theorem bnj864
StepHypRef Expression
1 bnj864.1 . . . . 5  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj864.2 . . . . 5  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj864.3 . . . . 5  |-  D  =  ( om  \  { (/)
} )
41, 2, 3bnj852 29269 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) )
5 df-ral 2561 . . . . . 6  |-  ( A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps )  <->  A. n ( n  e.  D  ->  E! f ( f  Fn  n  /\  ph  /\  ps ) ) )
65imbi2i 303 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  ->  A. n  e.  D  E! f
( f  Fn  n  /\  ph  /\  ps )
)  <->  ( ( R 
FrSe  A  /\  X  e.  A )  ->  A. n
( n  e.  D  ->  E! f ( f  Fn  n  /\  ph  /\ 
ps ) ) ) )
7 19.21v 1843 . . . . 5  |-  ( A. n ( ( R 
FrSe  A  /\  X  e.  A )  ->  (
n  e.  D  ->  E! f ( f  Fn  n  /\  ph  /\  ps ) ) )  <->  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. n
( n  e.  D  ->  E! f ( f  Fn  n  /\  ph  /\ 
ps ) ) ) )
8 impexp 433 . . . . . . 7  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D )  ->  E! f ( f  Fn  n  /\  ph  /\  ps ) )  <->  ( ( R  FrSe  A  /\  X  e.  A )  ->  (
n  e.  D  ->  E! f ( f  Fn  n  /\  ph  /\  ps ) ) ) )
9 df-3an 936 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  <->  ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D ) )
109bicomi 193 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D )  <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D ) )
1110imbi1i 315 . . . . . . 7  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D )  ->  E! f ( f  Fn  n  /\  ph  /\  ps ) )  <->  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  ->  E! f ( f  Fn  n  /\  ph  /\  ps ) ) )
128, 11bitr3i 242 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  ->  ( n  e.  D  ->  E! f ( f  Fn  n  /\  ph  /\  ps )
) )  <->  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  ->  E! f ( f  Fn  n  /\  ph  /\  ps ) ) )
1312albii 1556 . . . . 5  |-  ( A. n ( ( R 
FrSe  A  /\  X  e.  A )  ->  (
n  e.  D  ->  E! f ( f  Fn  n  /\  ph  /\  ps ) ) )  <->  A. n
( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
146, 7, 133bitr2i 264 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  ->  A. n  e.  D  E! f
( f  Fn  n  /\  ph  /\  ps )
)  <->  A. n ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  ->  E! f ( f  Fn  n  /\  ph  /\ 
ps ) ) )
154, 14mpbi 199 . . 3  |-  A. n
( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
)
1615spi 1750 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  ->  E! f ( f  Fn  n  /\  ph  /\ 
ps ) )
17 bnj864.4 . 2  |-  ( ch  <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
)
18 bnj864.5 . . 3  |-  ( th  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
1918eubii 2165 . 2  |-  ( E! f th  <->  E! f
( f  Fn  n  /\  ph  /\  ps )
)
2016, 17, 193imtr4i 257 1  |-  ( ch 
->  E! f th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530    = wceq 1632    e. wcel 1696   E!weu 2156   A.wral 2556    \ cdif 3162   (/)c0 3468   {csn 3653   U_ciun 3921   suc csuc 4410   omcom 4672    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    FrSe w-bnj15 29033
This theorem is referenced by:  bnj849  29273
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034
  Copyright terms: Public domain W3C validator