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Theorem bnj938 34096
Description: Technical lemma for bnj69 34167. 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 3120 . . 3  |-  ( X  e.  A  ->  E. x  x  =  X )
21bnj706 33912 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  E. x  x  =  X )
3 bnj291 33864 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  <->  ( ( R  FrSe  A  /\  ta  /\ 
si )  /\  X  e.  A ) )
43simplbi 460 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  -> 
( R  FrSe  A  /\  ta  /\  si )
)
5 bnj602 34074 . . . . . . . . . 10  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
65eqeq2d 2471 . . . . . . . . 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 263 . . . . . . . 8  |-  ( x  =  X  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <->  ph' ) )
983anbi2d 1304 . . . . . . 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 263 . . . . . 6  |-  ( x  =  X  ->  (
( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  <->  ta )
)
12113anbi2d 1304 . . . . 5  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  ( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  /\  si ) 
<->  ( R  FrSe  A  /\  ta  /\  si )
) )
134, 12syl5ibr 221 . . . 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 236 . . . . 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 236 . . . . 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 34055 . . . 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 33 . . 3  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )  e.  _V ) )
2120exlimiv 1723 . 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 36 1  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  (
f `  p )  pred ( y ,  A ,  R )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819   A.wral 2807   _Vcvv 3109    \ cdif 3468   (/)c0 3793   {csn 4032   U_ciun 4332   suc csuc 4889    Fn wfn 5589   ` cfv 5594   omcom 6699    /\ w-bnj17 33839    predc-bnj14 33841    FrSe w-bnj15 33845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-bnj17 33840  df-bnj14 33842  df-bnj13 33844  df-bnj15 33846
This theorem is referenced by:  bnj944  34097  bnj969  34105
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