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Theorem bnj938 29285
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
bnj938.1  |-  D  =  ( om  \  { (/)
} )
bnj938.2  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj938.3  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
bnj938.4  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj938.5  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj938  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  (
f `  p )  pred ( y ,  A ,  R )  e.  _V )
Distinct variable groups:    A, i, p, y    R, i, p, y    f, i, p, y    i, m, p
Allowed substitution hints:    ta( y, f, i, m, n, p)    si( y, f, i, m, n, p)    A( f, m, n)    D( y, f, i, m, n, p)    R( f, m, n)    X( y, f, i, m, n, p)    ph'( y, f, i, m, n, p)    ps'( y, f, i, m, n, p)

Proof of Theorem bnj938
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elisset 2811 . . 3  |-  ( X  e.  A  ->  E. x  x  =  X )
21bnj706 29099 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  E. x  x  =  X )
3 bnj291 29052 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  <->  ( ( R  FrSe  A  /\  ta  /\ 
si )  /\  X  e.  A ) )
43simplbi 446 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  -> 
( R  FrSe  A  /\  ta  /\  si )
)
5 bnj602 29263 . . . . . . . . . 10  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
65eqeq2d 2307 . . . . . . . . 9  |-  ( x  =  X  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( f `  (/) )  = 
pred ( X ,  A ,  R )
) )
7 bnj938.4 . . . . . . . . 9  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)
86, 7syl6bbr 254 . . . . . . . 8  |-  ( x  =  X  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <->  ph' ) )
983anbi2d 1257 . . . . . . 7  |-  ( x  =  X  ->  (
( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  <->  ( f  Fn  m  /\  ph'  /\  ps' ) ) )
10 bnj938.2 . . . . . . 7  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
119, 10syl6bbr 254 . . . . . 6  |-  ( x  =  X  ->  (
( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  <->  ta )
)
12113anbi2d 1257 . . . . 5  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  ( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  /\  si ) 
<->  ( R  FrSe  A  /\  ta  /\  si )
) )
134, 12syl5ibr 212 . . . 4  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  ( R  FrSe  A  /\  ( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  /\  si ) ) )
14 bnj938.1 . . . . 5  |-  D  =  ( om  \  { (/)
} )
15 biid 227 . . . . 5  |-  ( ( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  <->  ( f  Fn  m  /\  (
f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' ) )
16 bnj938.3 . . . . 5  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
17 biid 227 . . . . 5  |-  ( ( f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( f `  (/) )  = 
pred ( x ,  A ,  R ) )
18 bnj938.5 . . . . 5  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
1914, 15, 16, 17, 18bnj546 29244 . . . 4  |-  ( ( R  FrSe  A  /\  ( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  /\  si )  ->  U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )  e.  _V )
2013, 19syl6 29 . . 3  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )  e.  _V ) )
2120exlimiv 1624 . 2  |-  ( E. x  x  =  X  ->  ( ( R 
FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )  e.  _V ) )
222, 21mpcom 32 1  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  (
f `  p )  pred ( y ,  A ,  R )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162   (/)c0 3468   {csn 3653   U_ciun 3921   suc csuc 4410   omcom 4672    Fn wfn 5266   ` cfv 5271    /\ w-bnj17 29027    predc-bnj14 29029    FrSe w-bnj15 29033
This theorem is referenced by:  bnj944  29286  bnj969  29294
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-pr 4230  ax-un 4528
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-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034
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