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Theorem bnj938 29370
Description: Technical lemma for bnj69 29441. 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 2968 . . 3  |-  ( X  e.  A  ->  E. x  x  =  X )
21bnj706 29184 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  E. x  x  =  X )
3 bnj291 29137 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  <->  ( ( R  FrSe  A  /\  ta  /\ 
si )  /\  X  e.  A ) )
43simplbi 448 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  -> 
( R  FrSe  A  /\  ta  /\  si )
)
5 bnj602 29348 . . . . . . . . . 10  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
65eqeq2d 2449 . . . . . . . . 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 256 . . . . . . . 8  |-  ( x  =  X  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <->  ph' ) )
983anbi2d 1260 . . . . . . 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 256 . . . . . 6  |-  ( x  =  X  ->  (
( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  <->  ta )
)
12113anbi2d 1260 . . . . 5  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  ( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  /\  si ) 
<->  ( R  FrSe  A  /\  ta  /\  si )
) )
134, 12syl5ibr 214 . . . 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 229 . . . . 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 229 . . . . 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 29329 . . . 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 32 . . 3  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )  e.  _V ) )
2120exlimiv 1645 . 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 35 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 178    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    \ cdif 3319   (/)c0 3630   {csn 3816   U_ciun 4095   suc csuc 4585   omcom 4847    Fn wfn 5451   ` cfv 5456    /\ w-bnj17 29112    predc-bnj14 29114    FrSe w-bnj15 29118
This theorem is referenced by:  bnj944  29371  bnj969  29379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-bnj17 29113  df-bnj14 29115  df-bnj13 29117  df-bnj15 29119
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