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Theorem bnj945 32920
Description: Technical lemma for bnj69 33154. 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 5679 . . . . . . 7  |-  ( f  Fn  n  ->  dom  f  =  n )
21ad2antll 728 . . . . . 6  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  dom  f  =  n )
32eleq2d 2537 . . . . 5  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  ( A  e.  dom  f  <->  A  e.  n ) )
43pm5.32i 637 . . . 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 32919 . . . . . . . 8  |-  ( C  e.  _V  ->  (
( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p ) )
76imp 429 . . . . . . 7  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  G  Fn  p )
87bnj930 32916 . . . . . 6  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  Fun  G )
95bnj931 32917 . . . . . 6  |-  f  C_  G
108, 9jctir 538 . . . . 5  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  ( Fun  G  /\  f  C_  G ) )
1110anim1i 568 . . . 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 213 . . 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 32828 . . . 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 982 . . . . . 6  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  <-> 
( C  e.  _V  /\  p  =  suc  n  /\  f  Fn  n
) )
15 3anass 977 . . . . . 6  |-  ( ( C  e.  _V  /\  p  =  suc  n  /\  f  Fn  n )  <->  ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) ) )
1614, 15bitri 249 . . . . 5  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  <-> 
( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) ) )
1716anbi1i 695 . . . 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 249 . . 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 975 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  A  e. 
dom  f )  <->  ( ( Fun  G  /\  f  C_  G )  /\  A  e.  dom  f ) )
2012, 18, 193imtr4i 266 . 2  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  A  e.  n )  ->  ( Fun  G  /\  f  C_  G  /\  A  e.  dom  f ) )
21 funssfv 5880 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474    C_ wss 3476   {csn 4027   <.cop 4033   suc csuc 4880   dom cdm 4999   Fun wfun 5581    Fn wfn 5582   ` cfv 5587    /\ w-bnj17 32827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-reg 8017
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5550  df-fun 5589  df-fn 5590  df-fv 5595  df-bnj17 32828
This theorem is referenced by:  bnj966  33090  bnj967  33091  bnj1006  33105
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