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Theorem bnj945 28862
Description: Technical lemma for bnj69 29097. 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.)
Hypothesis
Ref Expression
bnj945.1  |-  G  =  ( f  u.  { <. n ,  C >. } )
Assertion
Ref Expression
bnj945  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  A  e.  n )  ->  ( G `  A
)  =  ( f `
 A ) )

Proof of Theorem bnj945
StepHypRef Expression
1 fndm 5511 . . . . . . 7  |-  ( f  Fn  n  ->  dom  f  =  n )
21ad2antll 710 . . . . . 6  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  dom  f  =  n )
32eleq2d 2479 . . . . 5  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  ( A  e.  dom  f  <->  A  e.  n ) )
43pm5.32i 619 . . . 4  |-  ( ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  /\  A  e.  dom  f )  <->  ( ( C  e.  _V  /\  (
p  =  suc  n  /\  f  Fn  n
) )  /\  A  e.  n ) )
5 bnj945.1 . . . . . . . . 9  |-  G  =  ( f  u.  { <. n ,  C >. } )
65bnj941 28861 . . . . . . . 8  |-  ( C  e.  _V  ->  (
( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p ) )
76imp 419 . . . . . . 7  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  G  Fn  p )
87bnj930 28858 . . . . . 6  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  Fun  G )
95bnj931 28859 . . . . . 6  |-  f  C_  G
108, 9jctir 525 . . . . 5  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  ( Fun  G  /\  f  C_  G ) )
1110anim1i 552 . . . 4  |-  ( ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  /\  A  e.  dom  f )  -> 
( ( Fun  G  /\  f  C_  G )  /\  A  e.  dom  f ) )
124, 11sylbir 205 . . 3  |-  ( ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  /\  A  e.  n )  ->  (
( Fun  G  /\  f  C_  G )  /\  A  e.  dom  f ) )
13 df-bnj17 28769 . . . 4  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  A  e.  n )  <->  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  /\  A  e.  n
) )
14 3ancomb 945 . . . . . 6  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  <-> 
( C  e.  _V  /\  p  =  suc  n  /\  f  Fn  n
) )
15 3anass 940 . . . . . 6  |-  ( ( C  e.  _V  /\  p  =  suc  n  /\  f  Fn  n )  <->  ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) ) )
1614, 15bitri 241 . . . . 5  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  <-> 
( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) ) )
1716anbi1i 677 . . . 4  |-  ( ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  /\  A  e.  n
)  <->  ( ( C  e.  _V  /\  (
p  =  suc  n  /\  f  Fn  n
) )  /\  A  e.  n ) )
1813, 17bitri 241 . . 3  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  A  e.  n )  <->  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  /\  A  e.  n ) )
19 df-3an 938 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  A  e. 
dom  f )  <->  ( ( Fun  G  /\  f  C_  G )  /\  A  e.  dom  f ) )
2012, 18, 193imtr4i 258 . 2  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  A  e.  n )  ->  ( Fun  G  /\  f  C_  G  /\  A  e.  dom  f ) )
21 funssfv 5713 . 2  |-  ( ( Fun  G  /\  f  C_  G  /\  A  e. 
dom  f )  -> 
( G `  A
)  =  ( f `
 A ) )
2220, 21syl 16 1  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  A  e.  n )  ->  ( G `  A
)  =  ( f `
 A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2924    u. cun 3286    C_ wss 3288   {csn 3782   <.cop 3785   suc csuc 4551   dom cdm 4845   Fun wfun 5415    Fn wfn 5416   ` cfv 5421    /\ w-bnj17 28768
This theorem is referenced by:  bnj966  29033  bnj967  29034  bnj1006  29048
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-reg 7524
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-suc 4555  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5385  df-fun 5423  df-fn 5424  df-fv 5429  df-bnj17 28769
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