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Theorem bnj865 28700
Description: Technical lemma for bnj69 28785. 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 28698 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) )
5 omex 7431 . . . . . . . . 9  |-  om  e.  _V
6 difexg 4241 . . . . . . . . 9  |-  ( om  e.  _V  ->  ( om  \  { (/) } )  e.  _V )
75, 6ax-mp 8 . . . . . . . 8  |-  ( om 
\  { (/) } )  e.  _V
83, 7eqeltri 2428 . . . . . . 7  |-  D  e. 
_V
9 raleq 2812 . . . . . . . 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 2812 . . . . . . . . 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 1626 . . . . . . . 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 311 . . . . . . 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 5831 . . . . . . 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 2914 . . . . . 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 15 . . . . 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 1904 . . . . 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 200 . . . 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 2624 . . . . . . . 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 303 . . . . . . 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 1895 . . . . . . 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 243 . . . . . 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 1582 . . . . 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 433 . . . . . . . 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 936 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  <->  ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D ) )
2524bicomi 193 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D )  <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D ) )
2625imbi1i 315 . . . . . . . 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 242 . . . . . . 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 1566 . . . . . 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 1582 . . . . 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 240 . . . 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 199 . . 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 193 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  <->  ch )
3433imbi1i 315 . . . . 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 1566 . . . 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 1582 . . 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 199 . 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 2644 . . . . 5  |-  ( E. f  e.  w  th  <->  E. f  e.  w  ( f  Fn  n  /\  ph 
/\  ps ) )
4039imbi2i 303 . . . 4  |-  ( ( ch  ->  E. f  e.  w  th )  <->  ( ch  ->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps ) ) )
4140albii 1566 . . 3  |-  ( A. n ( ch  ->  E. f  e.  w  th ) 
<-> 
A. n ( ch 
->  E. f  e.  w  ( f  Fn  n  /\  ph  /\  ps )
) )
4241exbii 1582 . 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 200 1  |-  E. w A. n ( ch  ->  E. f  e.  w  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1540   E.wex 1541    = wceq 1642    e. wcel 1710   E!weu 2209   A.wral 2619   E.wrex 2620   _Vcvv 2864    \ cdif 3225   (/)c0 3531   {csn 3716   U_ciun 3984   suc csuc 4473   omcom 4735    Fn wfn 5329   ` cfv 5334    predc-bnj14 28458    FrSe w-bnj15 28462
This theorem is referenced by:  bnj849  28702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-reg 7393  ax-inf2 7429
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-1o 6563  df-bnj17 28457  df-bnj14 28459  df-bnj13 28461  df-bnj15 28463
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