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Theorem bnj929 28968
Description: Technical lemma for bnj69 29040. 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
bnj929.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj929.4  |-  ( ph'  <->  [. p  /  n ]. ph )
bnj929.7  |-  ( ph"  <->  [. G  / 
f ]. ph' )
bnj929.10  |-  D  =  ( om  \  { (/)
} )
bnj929.13  |-  G  =  ( f  u.  { <. n ,  C >. } )
bnj929.50  |-  C  e. 
_V
Assertion
Ref Expression
bnj929  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph" )
Distinct variable groups:    A, f, n    R, f, n    f, X, n
Allowed substitution hints:    ph( f, n, p)    A( p)    C( f, n, p)    D( f, n, p)    R( p)    G( f, n, p)    X( p)    ph'( f, n, p)   
ph"( f, n, p)

Proof of Theorem bnj929
StepHypRef Expression
1 bnj645 28779 . 2  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph )
2 bnj334 28738 . . . . . . 7  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  p  =  suc  n  /\  ph ) )
3 bnj257 28732 . . . . . . 7  |-  ( ( f  Fn  n  /\  n  e.  D  /\  p  =  suc  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  ph  /\  p  =  suc  n ) )
42, 3bitri 240 . . . . . 6  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  ph  /\  p  =  suc  n ) )
5 bnj345 28739 . . . . . 6  |-  ( ( f  Fn  n  /\  n  e.  D  /\  ph 
/\  p  =  suc  n )  <->  ( p  =  suc  n  /\  f  Fn  n  /\  n  e.  D  /\  ph )
)
6 bnj253 28729 . . . . . 6  |-  ( ( p  =  suc  n  /\  f  Fn  n  /\  n  e.  D  /\  ph )  <->  ( (
p  =  suc  n  /\  f  Fn  n
)  /\  n  e.  D  /\  ph ) )
74, 5, 63bitri 262 . . . . 5  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( p  =  suc  n  /\  f  Fn  n )  /\  n  e.  D  /\  ph ) )
87simp1bi 970 . . . 4  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ( p  =  suc  n  /\  f  Fn  n ) )
9 bnj929.13 . . . . . 6  |-  G  =  ( f  u.  { <. n ,  C >. } )
10 bnj929.50 . . . . . 6  |-  C  e. 
_V
119, 10bnj927 28800 . . . . 5  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p )
1211bnj930 28801 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  Fun  G )
138, 12syl 15 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  Fun  G )
149bnj931 28802 . . . 4  |-  f  C_  G
1514a1i 10 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  f  C_  G )
16 bnj268 28734 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  p  =  suc  n  /\  ph )  <->  ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )
)
17 bnj253 28729 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  p  =  suc  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n )  /\  p  =  suc  n  /\  ph ) )
1816, 17bitr3i 242 . . . . 5  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n )  /\  p  =  suc  n  /\  ph ) )
1918simp1bi 970 . . . 4  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ( n  e.  D  /\  f  Fn  n ) )
20 fndm 5343 . . . . 5  |-  ( f  Fn  n  ->  dom  f  =  n )
21 bnj929.10 . . . . . 6  |-  D  =  ( om  \  { (/)
} )
2221bnj529 28770 . . . . 5  |-  ( n  e.  D  ->  (/)  e.  n
)
23 eleq2 2344 . . . . . 6  |-  ( dom  f  =  n  -> 
( (/)  e.  dom  f  <->  (/)  e.  n ) )
2423biimpar 471 . . . . 5  |-  ( ( dom  f  =  n  /\  (/)  e.  n )  ->  (/)  e.  dom  f
)
2520, 22, 24syl2anr 464 . . . 4  |-  ( ( n  e.  D  /\  f  Fn  n )  -> 
(/)  e.  dom  f )
2619, 25syl 15 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  (/)  e.  dom  f )
2713, 15, 26bnj1502 28880 . 2  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ( G `  (/) )  =  ( f `  (/) ) )
28 bnj929.1 . . 3  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
29 bnj929.4 . . 3  |-  ( ph'  <->  [. p  /  n ]. ph )
30 bnj929.7 . . 3  |-  ( ph"  <->  [. G  / 
f ]. ph' )
319bnj918 28796 . . 3  |-  G  e. 
_V
3228, 29, 30, 31bnj934 28967 . 2  |-  ( (
ph  /\  ( G `  (/) )  =  ( f `  (/) ) )  ->  ph" )
331, 27, 32syl2anc 642 1  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph" )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   suc csuc 4394   omcom 4656   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713
This theorem is referenced by:  bnj944  28970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-reg 7306
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-bnj17 28712
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