Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1097 Unicode version

Theorem bnj1097 29011
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
bnj1097.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1097.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1097.5  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
Assertion
Ref Expression
bnj1097  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( f `  i )  C_  B
)

Proof of Theorem bnj1097
StepHypRef Expression
1 bnj1097.3 . . . . . . . 8  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
2 bnj1097.1 . . . . . . . . 9  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
32biimpi 186 . . . . . . . 8  |-  ( ph  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
41, 3bnj771 28794 . . . . . . 7  |-  ( ch 
->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
543ad2ant3 978 . . . . . 6  |-  ( ( th  /\  ta  /\  ch )  ->  ( f `
 (/) )  =  pred ( X ,  A ,  R ) )
65adantr 451 . . . . 5  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
7 bnj1097.5 . . . . . . . 8  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
87simp3bi 972 . . . . . . 7  |-  ( ta 
->  pred ( X ,  A ,  R )  C_  B )
983ad2ant2 977 . . . . . 6  |-  ( ( th  /\  ta  /\  ch )  ->  pred ( X ,  A ,  R )  C_  B
)
109adantr 451 . . . . 5  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  pred ( X ,  A ,  R )  C_  B )
116, 10jca 518 . . . 4  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  ( ( f `  (/) )  =  pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
1211anim2i 552 . . 3  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( i  =  (/)  /\  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) ) )
13 3anass 938 . . 3  |-  ( ( i  =  (/)  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B )  <->  ( i  =  (/)  /\  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) ) )
1412, 13sylibr 203 . 2  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( i  =  (/)  /\  ( f `
 (/) )  =  pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
15 fveq2 5525 . . . . . . 7  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
1615eqeq1d 2291 . . . . . 6  |-  ( i  =  (/)  ->  ( ( f `  i )  =  pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
) )
1716biimpar 471 . . . . 5  |-  ( ( i  =  (/)  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( f `  i )  =  pred ( X ,  A ,  R ) )
1817adantr 451 . . . 4  |-  ( ( ( i  =  (/)  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  /\  pred ( X ,  A ,  R
)  C_  B )  ->  ( f `  i
)  =  pred ( X ,  A ,  R ) )
19 simpr 447 . . . 4  |-  ( ( ( i  =  (/)  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  /\  pred ( X ,  A ,  R
)  C_  B )  ->  pred ( X ,  A ,  R )  C_  B )
2018, 19eqsstrd 3212 . . 3  |-  ( ( ( i  =  (/)  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  /\  pred ( X ,  A ,  R
)  C_  B )  ->  ( f `  i
)  C_  B )
21203impa 1146 . 2  |-  ( ( i  =  (/)  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B )  ->  (
f `  i )  C_  B )
2214, 21syl 15 1  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( f `  i )  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    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-or 359  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-nfc 2408  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-iota 5219  df-fv 5263  df-bnj17 28712
  Copyright terms: Public domain W3C validator