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Theorem bnj864 33077
Description: Technical lemma for bnj69 33163. 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 33076 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) )
5 df-ral 2819 . . . . . 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 312 . . . . 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 1930 . . . . 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 446 . . . . . . 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 975 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  <->  ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D ) )
109bicomi 202 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D )  <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D ) )
1110imbi1i 325 . . . . . . 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 251 . . . . . 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 1620 . . . . 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 273 . . . 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 208 . . 3  |-  A. n
( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
)
1615spi 1813 . 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 2300 . 2  |-  ( E! f th  <->  E! f
( f  Fn  n  /\  ph  /\  ps )
)
2016, 17, 193imtr4i 266 1  |-  ( ch 
->  E! f th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767   E!weu 2275   A.wral 2814    \ cdif 3473   (/)c0 3785   {csn 4027   U_ciun 4325   suc csuc 4880    Fn wfn 5583   ` cfv 5588   omcom 6684    predc-bnj14 32838    FrSe w-bnj15 32842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-reg 8018  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-1o 7130  df-bnj17 32837  df-bnj14 32839  df-bnj13 32841  df-bnj15 32843
This theorem is referenced by:  bnj849  33080
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