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Theorem bnj865 29294
Description: Technical lemma for bnj69 29379. 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
bnj865.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj865.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj865.3  |-  D  =  ( om  \  { (/)
} )
bnj865.5  |-  ( ch  <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
)
bnj865.6  |-  ( th  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
Assertion
Ref Expression
bnj865  |-  E. w A. n ( ch  ->  E. f  e.  w  th )
Distinct variable groups:    A, f,
i, n, y    w, A, f, n    D, f, i, n    w, D    R, f, i, n, y   
w, R    f, X, n, w    ph, w    ps, w
Allowed substitution hints:    ph( y, f, i, n)    ps( y,
f, i, n)    ch( y, w, f, i, n)    th( y, w, f, i, n)    D( y)    X( y, i)

Proof of Theorem bnj865
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bnj865.1 . . . . . . 7  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj865.2 . . . . . . 7  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj865.3 . . . . . . 7  |-  D  =  ( om  \  { (/)
} )
41, 2, 3bnj852 29292 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) )
5 omex 7598 . . . . . . . . 9  |-  om  e.  _V
6 difexg 4351 . . . . . . . . 9  |-  ( om  e.  _V  ->  ( om  \  { (/) } )  e.  _V )
75, 6ax-mp 8 . . . . . . . 8  |-  ( om 
\  { (/) } )  e.  _V
83, 7eqeltri 2506 . . . . . . 7  |-  D  e. 
_V
9 raleq 2904 . . . . . . . 8  |-  ( z  =  D  ->  ( A. n  e.  z  E! f ( f  Fn  n  /\  ph  /\  ps )  <->  A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) ) )
10 raleq 2904 . . . . . . . . 9  |-  ( z  =  D  ->  ( A. n  e.  z  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )  <->  A. n  e.  D  E. f  e.  w  (
f  Fn  n  /\  ph 
/\  ps ) ) )
1110exbidv 1636 . . . . . . . 8  |-  ( z  =  D  ->  ( E. w A. n  e.  z  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )  <->  E. w A. n  e.  D  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) )
129, 11imbi12d 312 . . . . . . 7  |-  ( z  =  D  ->  (
( A. n  e.  z  E! f ( f  Fn  n  /\  ph 
/\  ps )  ->  E. w A. n  e.  z  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
)  <->  ( A. n  e.  D  E! f
( f  Fn  n  /\  ph  /\  ps )  ->  E. w A. n  e.  D  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) ) )
13 zfrep6 5968 . . . . . . 7  |-  ( A. n  e.  z  E! f ( f  Fn  n  /\  ph  /\  ps )  ->  E. w A. n  e.  z  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
)
148, 12, 13vtocl 3006 . . . . . 6  |-  ( A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps )  ->  E. w A. n  e.  D  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
)
154, 14syl 16 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. w A. n  e.  D  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )
16 19.37v 1922 . . . . 5  |-  ( E. w ( ( R 
FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )  <->  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. w A. n  e.  D  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
) )
1715, 16mpbir 201 . . . 4  |-  E. w
( ( R  FrSe  A  /\  X  e.  A
)  ->  A. n  e.  D  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )
18 df-ral 2710 . . . . . . . 8  |-  ( A. n  e.  D  E. f  e.  w  (
f  Fn  n  /\  ph 
/\  ps )  <->  A. n
( n  e.  D  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
) )
1918imbi2i 304 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  ->  A. n  e.  D  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )  <->  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. n
( n  e.  D  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
) ) )
20 19.21v 1913 . . . . . . 7  |-  ( A. n ( ( R 
FrSe  A  /\  X  e.  A )  ->  (
n  e.  D  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
) )  <->  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. n
( n  e.  D  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
) ) )
2119, 20bitr4i 244 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  ->  A. n  e.  D  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )  <->  A. n
( ( R  FrSe  A  /\  X  e.  A
)  ->  ( n  e.  D  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) ) )
2221exbii 1592 . . . . 5  |-  ( E. w ( ( R 
FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )  <->  E. w A. n ( ( R 
FrSe  A  /\  X  e.  A )  ->  (
n  e.  D  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
) ) )
23 impexp 434 . . . . . . . 8  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D )  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )  <->  ( ( R  FrSe  A  /\  X  e.  A )  ->  (
n  e.  D  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
) ) )
24 df-3an 938 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  <->  ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D ) )
2524bicomi 194 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D )  <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D ) )
2625imbi1i 316 . . . . . . . 8  |-  ( ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D )  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )  <->  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) )
2723, 26bitr3i 243 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  ->  ( n  e.  D  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) )  <->  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) )
2827albii 1575 . . . . . 6  |-  ( A. n ( ( R 
FrSe  A  /\  X  e.  A )  ->  (
n  e.  D  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
) )  <->  A. n
( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D
)  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) )
2928exbii 1592 . . . . 5  |-  ( E. w A. n ( ( R  FrSe  A  /\  X  e.  A
)  ->  ( n  e.  D  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) )  <->  E. w A. n ( ( R 
FrSe  A  /\  X  e.  A  /\  n  e.  D )  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) )
3022, 29bitri 241 . . . 4  |-  ( E. w ( ( R 
FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )  <->  E. w A. n ( ( R 
FrSe  A  /\  X  e.  A  /\  n  e.  D )  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) )
3117, 30mpbi 200 . . 3  |-  E. w A. n ( ( R 
FrSe  A  /\  X  e.  A  /\  n  e.  D )  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )
32 bnj865.5 . . . . . . 7  |-  ( ch  <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
)
3332bicomi 194 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  <->  ch )
3433imbi1i 316 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D
)  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )  <->  ( ch  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
) )
3534albii 1575 . . . 4  |-  ( A. n ( ( R 
FrSe  A  /\  X  e.  A  /\  n  e.  D )  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )  <->  A. n
( ch  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) )
3635exbii 1592 . . 3  |-  ( E. w A. n ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D
)  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) )  <->  E. w A. n ( ch  ->  E. f  e.  w  ( f  Fn  n  /\  ph 
/\  ps ) ) )
3731, 36mpbi 200 . 2  |-  E. w A. n ( ch  ->  E. f  e.  w  ( f  Fn  n  /\  ph 
/\  ps ) )
38 bnj865.6 . . . . . 6  |-  ( th  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
3938rexbii 2730 . . . . 5  |-  ( E. f  e.  w  th  <->  E. f  e.  w  ( f  Fn  n  /\  ph 
/\  ps ) )
4039imbi2i 304 . . . 4  |-  ( ( ch  ->  E. f  e.  w  th )  <->  ( ch  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) )
4140albii 1575 . . 3  |-  ( A. n ( ch  ->  E. f  e.  w  th ) 
<-> 
A. n ( ch 
->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
) )
4241exbii 1592 . 2  |-  ( E. w A. n ( ch  ->  E. f  e.  w  th )  <->  E. w A. n ( ch  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) )
4337, 42mpbir 201 1  |-  E. w A. n ( ch  ->  E. f  e.  w  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2281   A.wral 2705   E.wrex 2706   _Vcvv 2956    \ cdif 3317   (/)c0 3628   {csn 3814   U_ciun 4093   suc csuc 4583   omcom 4845    Fn wfn 5449   ` cfv 5454    predc-bnj14 29052    FrSe w-bnj15 29056
This theorem is referenced by:  bnj849  29296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-reg 7560  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-bnj17 29051  df-bnj14 29053  df-bnj13 29055  df-bnj15 29057
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