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Theorem bnj1121 29015
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.)
Hypotheses
Ref Expression
bnj1121.1  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
bnj1121.2  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
bnj1121.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1121.4  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
bnj1121.5  |-  ( et  <->  ( ( f  e.  K  /\  i  e.  dom  f )  ->  (
f `  i )  C_  B ) )
bnj1121.6  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i  e.  n  et )
bnj1121.7  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
Assertion
Ref Expression
bnj1121  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B )

Proof of Theorem bnj1121
StepHypRef Expression
1 19.8a 1718 . . . . 5  |-  ( ch 
->  E. n ch )
21bnj707 28784 . . . 4  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  E. n ch )
3 bnj1121.3 . . . . 5  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
4 bnj1121.7 . . . . 5  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
53, 4bnj1083 29008 . . . 4  |-  ( f  e.  K  <->  E. n ch )
62, 5sylibr 203 . . 3  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
f  e.  K )
7 bnj1121.4 . . . . . 6  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
87simplbi 446 . . . . 5  |-  ( ze 
->  i  e.  n
)
98bnj708 28785 . . . 4  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
i  e.  n )
103bnj1235 28837 . . . . . 6  |-  ( ch 
->  f  Fn  n
)
1110bnj707 28784 . . . . 5  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
f  Fn  n )
12 fndm 5343 . . . . 5  |-  ( f  Fn  n  ->  dom  f  =  n )
1311, 12syl 15 . . . 4  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  dom  f  =  n
)
149, 13eleqtrrd 2360 . . 3  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
i  e.  dom  f
)
15 bnj1121.6 . . . . 5  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i  e.  n  et )
1615, 9bnj1294 28850 . . . 4  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  et )
17 bnj1121.5 . . . 4  |-  ( et  <->  ( ( f  e.  K  /\  i  e.  dom  f )  ->  (
f `  i )  C_  B ) )
1816, 17sylib 188 . . 3  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
( ( f  e.  K  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )
)
196, 14, 18mp2and 660 . 2  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
( f `  i
)  C_  B )
207simprbi 450 . . 3  |-  ( ze 
->  z  e.  (
f `  i )
)
2120bnj708 28785 . 2  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  ( f `
 i ) )
2219, 21sseldd 3181 1  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   dom cdm 4689    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717    TrFow-bnj19 28721
This theorem is referenced by:  bnj1030  29017
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-rex 2549  df-in 3159  df-ss 3166  df-fn 5258  df-bnj17 28712
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