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Theorem bnj546 29267
Description: Technical lemma for bnj852 29292. 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
bnj546.1  |-  D  =  ( om  \  { (/)
} )
bnj546.2  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj546.3  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
bnj546.4  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj546.5  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj546  |-  ( ( R  FrSe  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   
p, ph'
Allowed substitution hints:    ta( x, y, f, i, m, n, p)    si( x, y, f, i, m, n, p)    A( x, f, m, n)    D( x, y, f, i, m, n, p)    R( x, f, m, n)    ph'( x, y, f, i, m, n)    ps'( x, y, f, i, m, n, p)

Proof of Theorem bnj546
StepHypRef Expression
1 bnj546.2 . . . . . . 7  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
2 3simpc 956 . . . . . . 7  |-  ( ( f  Fn  m  /\  ph' 
/\  ps' )  ->  ( ph' 
/\  ps' ) )
31, 2sylbi 188 . . . . . 6  |-  ( ta 
->  ( ph'  /\  ps' ) )
4 bnj546.3 . . . . . . 7  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
5 bnj546.1 . . . . . . . . . 10  |-  D  =  ( om  \  { (/)
} )
65bnj923 29137 . . . . . . . . 9  |-  ( m  e.  D  ->  m  e.  om )
763ad2ant1 978 . . . . . . . 8  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )  ->  m  e.  om )
8 simp3 959 . . . . . . . 8  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )  ->  p  e.  m )
97, 8jca 519 . . . . . . 7  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )  ->  ( m  e.  om  /\  p  e.  m ) )
104, 9sylbi 188 . . . . . 6  |-  ( si  ->  ( m  e.  om  /\  p  e.  m ) )
113, 10anim12i 550 . . . . 5  |-  ( ( ta  /\  si )  ->  ( ( ph'  /\  ps' )  /\  ( m  e.  om  /\  p  e.  m ) ) )
12 bnj256 29070 . . . . 5  |-  ( ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m )  <->  ( ( ph' 
/\  ps' )  /\  (
m  e.  om  /\  p  e.  m )
) )
1311, 12sylibr 204 . . . 4  |-  ( ( ta  /\  si )  ->  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m ) )
1413anim2i 553 . . 3  |-  ( ( R  FrSe  A  /\  ( ta  /\  si )
)  ->  ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m ) ) )
15143impb 1149 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m ) ) )
16 bnj546.4 . . 3  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
17 bnj546.5 . . 3  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
18 biid 228 . . 3  |-  ( ( ph'  /\  ps'  /\  m  e. 
om  /\  p  e.  m )  <->  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m ) )
1916, 17, 18bnj518 29257 . 2  |-  ( ( R  FrSe  A  /\  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m ) )  ->  A. y  e.  (
f `  p )  pred ( y ,  A ,  R )  e.  _V )
20 fvex 5742 . . 3  |-  ( f `
 p )  e. 
_V
21 iunexg 5987 . . 3  |-  ( ( ( f `  p
)  e.  _V  /\  A. y  e.  ( f `
 p )  pred ( y ,  A ,  R )  e.  _V )  ->  U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )  e.  _V )
2220, 21mpan 652 . 2  |-  ( A. y  e.  ( f `  p )  pred (
y ,  A ,  R )  e.  _V  ->  U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )  e.  _V )
2315, 19, 223syl 19 1  |-  ( ( R  FrSe  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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    \ cdif 3317   (/)c0 3628   {csn 3814   U_ciun 4093   suc csuc 4583   omcom 4845    Fn wfn 5449   ` cfv 5454    /\ w-bnj17 29050    predc-bnj14 29052    FrSe w-bnj15 29056
This theorem is referenced by:  bnj938  29308
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-pr 4403  ax-un 4701
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-bnj17 29051  df-bnj14 29053  df-bnj13 29055  df-bnj15 29057
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