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Theorem bnj945 28805
Description: Technical lemma for bnj69 29040. 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 5343 . . . . . . 7  |-  ( f  Fn  n  ->  dom  f  =  n )
21ad2antll 709 . . . . . 6  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  dom  f  =  n )
32eleq2d 2350 . . . . 5  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  ( A  e.  dom  f  <->  A  e.  n ) )
43pm5.32i 618 . . . 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 28804 . . . . . . . 8  |-  ( C  e.  _V  ->  (
( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p ) )
76imp 418 . . . . . . 7  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  G  Fn  p )
87bnj930 28801 . . . . . 6  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  Fun  G )
95bnj931 28802 . . . . . 6  |-  f  C_  G
108, 9jctir 524 . . . . 5  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  ( Fun  G  /\  f  C_  G ) )
1110anim1i 551 . . . 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 204 . . 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 28712 . . . 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 943 . . . . . 6  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  <-> 
( C  e.  _V  /\  p  =  suc  n  /\  f  Fn  n
) )
15 3anass 938 . . . . . 6  |-  ( ( C  e.  _V  /\  p  =  suc  n  /\  f  Fn  n )  <->  ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) ) )
1614, 15bitri 240 . . . . 5  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  <-> 
( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) ) )
1716anbi1i 676 . . . 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 240 . . 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 936 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  A  e. 
dom  f )  <->  ( ( Fun  G  /\  f  C_  G )  /\  A  e.  dom  f ) )
2012, 18, 193imtr4i 257 . 2  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  A  e.  n )  ->  ( Fun  G  /\  f  C_  G  /\  A  e.  dom  f ) )
21 funssfv 5543 . 2  |-  ( ( Fun  G  /\  f  C_  G  /\  A  e. 
dom  f )  -> 
( G `  A
)  =  ( f `
 A ) )
2220, 21syl 15 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    C_ wss 3152   {csn 3640   <.cop 3643   suc csuc 4394   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711
This theorem is referenced by:  bnj966  28976  bnj967  28977  bnj1006  28991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-bnj17 28712
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