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Theorem bnj544 29707
Description: Technical lemma for bnj852 29734. 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
bnj544.1  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj544.2  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj544.3  |-  D  =  ( om  \  { (/)
} )
bnj544.4  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
bnj544.5  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj544.6  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
Assertion
Ref Expression
bnj544  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
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)    G( x, y, f, i, m, n, p)    ph'( x, y, f, i, m, n)    ps'( x, y, f, i, m, n, p)

Proof of Theorem bnj544
StepHypRef Expression
1 bnj544.6 . . 3  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
2 bnj544.3 . . . . 5  |-  D  =  ( om  \  { (/)
} )
32bnj923 29581 . . . 4  |-  ( m  e.  D  ->  m  e.  om )
433anim1i 1192 . . 3  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )  ->  ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m
) )
51, 4sylbi 199 . 2  |-  ( si  ->  ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m
) )
6 bnj544.1 . . 3  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
7 bnj544.2 . . 3  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
8 bnj544.4 . . 3  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
9 bnj544.5 . . 3  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
10 biid 240 . . 3  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m )  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  e.  m )
)
116, 7, 8, 9, 10bnj543 29706 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  ( m  e. 
om  /\  n  =  suc  m  /\  p  e.  m ) )  ->  G  Fn  n )
125, 11syl3an3 1300 1  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ w3a 983    = wceq 1438    e. wcel 1869   A.wral 2776    \ cdif 3434    u. cun 3435   (/)c0 3762   {csn 3997   <.cop 4003   U_ciun 4297   suc csuc 5442    Fn wfn 5594   ` cfv 5599   omcom 6704    predc-bnj14 29495    FrSe w-bnj15 29499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pr 4658  ax-un 6595  ax-reg 8111
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-om 6705  df-bnj17 29494  df-bnj14 29496  df-bnj13 29498  df-bnj15 29500
This theorem is referenced by:  bnj600  29732  bnj908  29744
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