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Theorem bnj929 29576
Description: Technical lemma for bnj69 29648. 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 29389 . 2  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph )
2 bnj334 29347 . . . . . . 7  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  p  =  suc  n  /\  ph ) )
3 bnj257 29341 . . . . . . 7  |-  ( ( f  Fn  n  /\  n  e.  D  /\  p  =  suc  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  ph  /\  p  =  suc  n ) )
42, 3bitri 252 . . . . . 6  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  ph  /\  p  =  suc  n ) )
5 bnj345 29348 . . . . . 6  |-  ( ( f  Fn  n  /\  n  e.  D  /\  ph 
/\  p  =  suc  n )  <->  ( p  =  suc  n  /\  f  Fn  n  /\  n  e.  D  /\  ph )
)
6 bnj253 29338 . . . . . 6  |-  ( ( p  =  suc  n  /\  f  Fn  n  /\  n  e.  D  /\  ph )  <->  ( (
p  =  suc  n  /\  f  Fn  n
)  /\  n  e.  D  /\  ph ) )
74, 5, 63bitri 274 . . . . 5  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( p  =  suc  n  /\  f  Fn  n )  /\  n  e.  D  /\  ph ) )
87simp1bi 1020 . . . 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 29409 . . . . 5  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p )
1211bnj930 29410 . . . 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 29411 . . . 4  |-  f  C_  G
1514a1i 11 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  f  C_  G )
16 bnj268 29343 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  p  =  suc  n  /\  ph )  <->  ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )
)
17 bnj253 29338 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  p  =  suc  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n )  /\  p  =  suc  n  /\  ph ) )
1816, 17bitr3i 254 . . . . 5  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n )  /\  p  =  suc  n  /\  ph ) )
1918simp1bi 1020 . . . 4  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ( n  e.  D  /\  f  Fn  n ) )
20 fndm 5684 . . . . 5  |-  ( f  Fn  n  ->  dom  f  =  n )
21 bnj929.10 . . . . . 6  |-  D  =  ( om  \  { (/)
} )
2221bnj529 29380 . . . . 5  |-  ( n  e.  D  ->  (/)  e.  n
)
23 eleq2 2493 . . . . . 6  |-  ( dom  f  =  n  -> 
( (/)  e.  dom  f  <->  (/)  e.  n ) )
2423biimpar 487 . . . . 5  |-  ( ( dom  f  =  n  /\  (/)  e.  n )  ->  (/)  e.  dom  f
)
2520, 22, 24syl2anr 480 . . . 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 29488 . 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 29406 . . 3  |-  G  e. 
_V
3228, 29, 30, 31bnj934 29575 . 2  |-  ( (
ph  /\  ( G `  (/) )  =  ( f `  (/) ) )  ->  ph" )
331, 27, 32syl2anc 665 1  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph" )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   _Vcvv 3078   [.wsbc 3296    \ cdif 3430    u. cun 3431    C_ wss 3433   (/)c0 3758   {csn 3993   <.cop 3999   dom cdm 4845   suc csuc 5435   Fun wfun 5586    Fn wfn 5587   ` cfv 5592   omcom 6697    /\ w-bnj17 29320    predc-bnj14 29322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652  ax-un 6588  ax-reg 8098
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-fv 5600  df-om 6698  df-bnj17 29321
This theorem is referenced by:  bnj944  29578
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