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Theorem bnj945 29657
Description: Technical lemma for bnj69 29891. 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 5685 . . . . . . 7  |-  ( f  Fn  n  ->  dom  f  =  n )
21ad2antll 743 . . . . . 6  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  dom  f  =  n )
32eleq2d 2534 . . . . 5  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  ( A  e.  dom  f  <->  A  e.  n ) )
43pm5.32i 649 . . . 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 29656 . . . . . . . 8  |-  ( C  e.  _V  ->  (
( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p ) )
76imp 436 . . . . . . 7  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  G  Fn  p )
87bnj930 29653 . . . . . 6  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  Fun  G )
95bnj931 29654 . . . . . 6  |-  f  C_  G
108, 9jctir 547 . . . . 5  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  ( Fun  G  /\  f  C_  G ) )
1110anim1i 578 . . . 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 218 . . 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 29564 . . . 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 1016 . . . . . 6  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  <-> 
( C  e.  _V  /\  p  =  suc  n  /\  f  Fn  n
) )
15 3anass 1011 . . . . . 6  |-  ( ( C  e.  _V  /\  p  =  suc  n  /\  f  Fn  n )  <->  ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) ) )
1614, 15bitri 257 . . . . 5  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  <-> 
( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) ) )
1716anbi1i 709 . . . 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 257 . . 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 1009 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  A  e. 
dom  f )  <->  ( ( Fun  G  /\  f  C_  G )  /\  A  e.  dom  f ) )
2012, 18, 193imtr4i 274 . 2  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  A  e.  n )  ->  ( Fun  G  /\  f  C_  G  /\  A  e.  dom  f ) )
21 funssfv 5894 . 2  |-  ( ( Fun  G  /\  f  C_  G  /\  A  e. 
dom  f )  -> 
( G `  A
)  =  ( f `
 A ) )
2220, 21syl 17 1  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  A  e.  n )  ->  ( G `  A
)  =  ( f `
 A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031    u. cun 3388    C_ wss 3390   {csn 3959   <.cop 3965   dom cdm 4839   suc csuc 5432   Fun wfun 5583    Fn wfn 5584   ` cfv 5589    /\ w-bnj17 29563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-reg 8125
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-res 4851  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597  df-bnj17 29564
This theorem is referenced by:  bnj966  29827  bnj967  29828  bnj1006  29842
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