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Theorem bnj929 33290
Description: Technical lemma for bnj69 33362. 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 33103 . 2  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph )
2 bnj334 33062 . . . . . . 7  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  p  =  suc  n  /\  ph ) )
3 bnj257 33056 . . . . . . 7  |-  ( ( f  Fn  n  /\  n  e.  D  /\  p  =  suc  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  ph  /\  p  =  suc  n ) )
42, 3bitri 249 . . . . . 6  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  ph  /\  p  =  suc  n ) )
5 bnj345 33063 . . . . . 6  |-  ( ( f  Fn  n  /\  n  e.  D  /\  ph 
/\  p  =  suc  n )  <->  ( p  =  suc  n  /\  f  Fn  n  /\  n  e.  D  /\  ph )
)
6 bnj253 33053 . . . . . 6  |-  ( ( p  =  suc  n  /\  f  Fn  n  /\  n  e.  D  /\  ph )  <->  ( (
p  =  suc  n  /\  f  Fn  n
)  /\  n  e.  D  /\  ph ) )
74, 5, 63bitri 271 . . . . 5  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( p  =  suc  n  /\  f  Fn  n )  /\  n  e.  D  /\  ph ) )
87simp1bi 1011 . . . 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 33123 . . . . 5  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p )
1211bnj930 33124 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  Fun  G )
138, 12syl 16 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  Fun  G )
149bnj931 33125 . . . 4  |-  f  C_  G
1514a1i 11 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  f  C_  G )
16 bnj268 33058 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  p  =  suc  n  /\  ph )  <->  ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )
)
17 bnj253 33053 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  p  =  suc  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n )  /\  p  =  suc  n  /\  ph ) )
1816, 17bitr3i 251 . . . . 5  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n )  /\  p  =  suc  n  /\  ph ) )
1918simp1bi 1011 . . . 4  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ( n  e.  D  /\  f  Fn  n ) )
20 fndm 5680 . . . . 5  |-  ( f  Fn  n  ->  dom  f  =  n )
21 bnj929.10 . . . . . 6  |-  D  =  ( om  \  { (/)
} )
2221bnj529 33094 . . . . 5  |-  ( n  e.  D  ->  (/)  e.  n
)
23 eleq2 2540 . . . . . 6  |-  ( dom  f  =  n  -> 
( (/)  e.  dom  f  <->  (/)  e.  n ) )
2423biimpar 485 . . . . 5  |-  ( ( dom  f  =  n  /\  (/)  e.  n )  ->  (/)  e.  dom  f
)
2520, 22, 24syl2anr 478 . . . 4  |-  ( ( n  e.  D  /\  f  Fn  n )  -> 
(/)  e.  dom  f )
2619, 25syl 16 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  (/)  e.  dom  f )
2713, 15, 26bnj1502 33202 . 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 33120 . . 3  |-  G  e. 
_V
3228, 29, 30, 31bnj934 33289 . 2  |-  ( (
ph  /\  ( G `  (/) )  =  ( f `  (/) ) )  ->  ph" )
331, 27, 32syl2anc 661 1  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph" )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   [.wsbc 3331    \ cdif 3473    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   suc csuc 4880   dom cdm 4999   Fun wfun 5582    Fn wfn 5583   ` cfv 5588   omcom 6685    /\ w-bnj17 33035    predc-bnj14 33037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577  ax-reg 8019
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-om 6686  df-bnj17 33036
This theorem is referenced by:  bnj944  33292
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