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Theorem bnj929 28647
Description: Technical lemma for bnj69 28719. 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 28458 . 2  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph )
2 bnj334 28417 . . . . . . 7  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  p  =  suc  n  /\  ph ) )
3 bnj257 28411 . . . . . . 7  |-  ( ( f  Fn  n  /\  n  e.  D  /\  p  =  suc  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  ph  /\  p  =  suc  n ) )
42, 3bitri 241 . . . . . 6  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( f  Fn  n  /\  n  e.  D  /\  ph  /\  p  =  suc  n ) )
5 bnj345 28418 . . . . . 6  |-  ( ( f  Fn  n  /\  n  e.  D  /\  ph 
/\  p  =  suc  n )  <->  ( p  =  suc  n  /\  f  Fn  n  /\  n  e.  D  /\  ph )
)
6 bnj253 28408 . . . . . 6  |-  ( ( p  =  suc  n  /\  f  Fn  n  /\  n  e.  D  /\  ph )  <->  ( (
p  =  suc  n  /\  f  Fn  n
)  /\  n  e.  D  /\  ph ) )
74, 5, 63bitri 263 . . . . 5  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( p  =  suc  n  /\  f  Fn  n )  /\  n  e.  D  /\  ph ) )
87simp1bi 972 . . . 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 28479 . . . . 5  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p )
1211bnj930 28480 . . . 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 28481 . . . 4  |-  f  C_  G
1514a1i 11 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  f  C_  G )
16 bnj268 28413 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  p  =  suc  n  /\  ph )  <->  ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )
)
17 bnj253 28408 . . . . . 6  |-  ( ( n  e.  D  /\  f  Fn  n  /\  p  =  suc  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n )  /\  p  =  suc  n  /\  ph ) )
1816, 17bitr3i 243 . . . . 5  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  <->  ( ( n  e.  D  /\  f  Fn  n )  /\  p  =  suc  n  /\  ph ) )
1918simp1bi 972 . . . 4  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ( n  e.  D  /\  f  Fn  n ) )
20 fndm 5486 . . . . 5  |-  ( f  Fn  n  ->  dom  f  =  n )
21 bnj929.10 . . . . . 6  |-  D  =  ( om  \  { (/)
} )
2221bnj529 28449 . . . . 5  |-  ( n  e.  D  ->  (/)  e.  n
)
23 eleq2 2450 . . . . . 6  |-  ( dom  f  =  n  -> 
( (/)  e.  dom  f  <->  (/)  e.  n ) )
2423biimpar 472 . . . . 5  |-  ( ( dom  f  =  n  /\  (/)  e.  n )  ->  (/)  e.  dom  f
)
2520, 22, 24syl2anr 465 . . . 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 28559 . 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 28475 . . 3  |-  G  e. 
_V
3228, 29, 30, 31bnj934 28646 . 2  |-  ( (
ph  /\  ( G `  (/) )  =  ( f `  (/) ) )  ->  ph" )
331, 27, 32syl2anc 643 1  |-  ( ( n  e.  D  /\  p  =  suc  n  /\  f  Fn  n  /\  ph )  ->  ph" )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2901   [.wsbc 3106    \ cdif 3262    u. cun 3263    C_ wss 3265   (/)c0 3573   {csn 3759   <.cop 3762   suc csuc 4526   omcom 4787   dom cdm 4820   Fun wfun 5390    Fn wfn 5391   ` cfv 5396    /\ w-bnj17 28390    predc-bnj14 28392
This theorem is referenced by:  bnj944  28649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643  ax-reg 7495
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-res 4832  df-iota 5360  df-fun 5398  df-fn 5399  df-fv 5404  df-bnj17 28391
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