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Theorem bnj864 29521
Description: Technical lemma for bnj69 29607. 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 29520 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) )
5 df-ral 2787 . . . . . 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 313 . . . . 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 1778 . . . . 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 447 . . . . . . 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 984 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  <->  ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D ) )
109bicomi 205 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D )  <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D ) )
1110imbi1i 326 . . . . . . 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 254 . . . . . 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 1687 . . . . 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 276 . . . 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 211 . . 3  |-  A. n
( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
)
1615spi 1917 . 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 2290 . 2  |-  ( E! f th  <->  E! f
( f  Fn  n  /\  ph  /\  ps )
)
2016, 17, 193imtr4i 269 1  |-  ( ch 
->  E! f th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1870   E!weu 2266   A.wral 2782    \ cdif 3439   (/)c0 3767   {csn 4002   U_ciun 4302   suc csuc 5444    Fn wfn 5596   ` cfv 5601   omcom 6706    predc-bnj14 29281    FrSe w-bnj15 29285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-reg 8107  ax-inf2 8146
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-1o 7190  df-bnj17 29280  df-bnj14 29282  df-bnj13 29284  df-bnj15 29286
This theorem is referenced by:  bnj849  29524
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