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Theorem bnj929 29759
Description: Technical lemma for bnj69 29831. 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 29572 . 2  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph )
2 bnj334 29530 . . . . . . 7  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  p  =  suc  n  /\  ph ) )
3 bnj257 29524 . . . . . . 7  |-  ( ( f  Fn  n  /\  n  e.  D  /\  p  =  suc  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  ph  /\  p  =  suc  n ) )
42, 3bitri 253 . . . . . 6  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  ph  /\  p  =  suc  n ) )
5 bnj345 29531 . . . . . 6  |-  ( ( f  Fn  n  /\  n  e.  D  /\  ph 
/\  p  =  suc  n )  <->  ( p  =  suc  n  /\  f  Fn  n  /\  n  e.  D  /\  ph )
)
6 bnj253 29521 . . . . . 6  |-  ( ( p  =  suc  n  /\  f  Fn  n  /\  n  e.  D  /\  ph )  <->  ( (
p  =  suc  n  /\  f  Fn  n
)  /\  n  e.  D  /\  ph ) )
74, 5, 63bitri 275 . . . . 5  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( p  =  suc  n  /\  f  Fn  n )  /\  n  e.  D  /\  ph ) )
87simp1bi 1024 . . . 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 29592 . . . . 5  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p )
1211bnj930 29593 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  Fun  G )
138, 12syl 17 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  Fun  G )
149bnj931 29594 . . . 4  |-  f  C_  G
1514a1i 11 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  f  C_  G )
16 bnj268 29526 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  p  =  suc  n  /\  ph )  <->  ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )
)
17 bnj253 29521 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  p  =  suc  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n )  /\  p  =  suc  n  /\  ph ) )
1816, 17bitr3i 255 . . . . 5  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n )  /\  p  =  suc  n  /\  ph ) )
1918simp1bi 1024 . . . 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 29563 . . . . 5  |-  ( n  e.  D  ->  (/)  e.  n
)
23 eleq2 2520 . . . . . 6  |-  ( dom  f  =  n  -> 
( (/)  e.  dom  f  <->  (/)  e.  n ) )
2423biimpar 488 . . . . 5  |-  ( ( dom  f  =  n  /\  (/)  e.  n )  ->  (/)  e.  dom  f
)
2520, 22, 24syl2anr 481 . . . 4  |-  ( ( n  e.  D  /\  f  Fn  n )  -> 
(/)  e.  dom  f )
2619, 25syl 17 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  (/)  e.  dom  f )
2713, 15, 26bnj1502 29671 . 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 29589 . . 3  |-  G  e. 
_V
3228, 29, 30, 31bnj934 29758 . 2  |-  ( (
ph  /\  ( G `  (/) )  =  ( f `  (/) ) )  ->  ph" )
331, 27, 32syl2anc 667 1  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph" )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   _Vcvv 3047   [.wsbc 3269    \ cdif 3403    u. cun 3404    C_ wss 3406   (/)c0 3733   {csn 3970   <.cop 3976   dom cdm 4837   suc csuc 5428   Fun wfun 5579    Fn wfn 5580   ` cfv 5585   omcom 6697    /\ w-bnj17 29503    predc-bnj14 29505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642  ax-un 6588  ax-reg 8112
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-res 4849  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-fv 5593  df-om 6698  df-bnj17 29504
This theorem is referenced by:  bnj944  29761
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