Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj945 Structured version   Unicode version

Theorem bnj945 33954
Description: Technical lemma for bnj69 34188. 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 5686 . . . . . . 7  |-  ( f  Fn  n  ->  dom  f  =  n )
21ad2antll 728 . . . . . 6  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  dom  f  =  n )
32eleq2d 2527 . . . . 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 33953 . . . . . . . 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 33950 . . . . . 6  |-  ( ( C  e.  _V  /\  ( p  =  suc  n  /\  f  Fn  n
) )  ->  Fun  G )
95bnj931 33951 . . . . . 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 33861 . . . 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 5887 . 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 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469    C_ wss 3471   {csn 4032   <.cop 4038   suc csuc 4889   dom cdm 5008   Fun wfun 5588    Fn wfn 5589   ` cfv 5594    /\ w-bnj17 33860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-reg 8036
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-bnj17 33861
This theorem is referenced by:  bnj966  34124  bnj967  34125  bnj1006  34139
  Copyright terms: Public domain W3C validator