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Theorem bnj999 28989
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
bnj999.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj999.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj999.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj999.7  |-  ( ph'  <->  [. p  /  n ]. ph )
bnj999.8  |-  ( ps'  <->  [. p  /  n ]. ps )
bnj999.9  |-  ( ch'  <->  [. p  /  n ]. ch )
bnj999.10  |-  ( ph"  <->  [. G  / 
f ]. ph' )
bnj999.11  |-  ( ps"  <->  [. G  / 
f ]. ps' )
bnj999.12  |-  ( ch"  <->  [. G  / 
f ]. ch' )
bnj999.15  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj999.16  |-  G  =  ( f  u.  { <. n ,  C >. } )
Assertion
Ref Expression
bnj999  |-  ( ( ch" 
/\  i  e.  om  /\ 
suc  i  e.  p  /\  y  e.  ( G `  i )
)  ->  pred ( y ,  A ,  R
)  C_  ( G `  suc  i ) )
Distinct variable groups:    f, i, n, y    A, f, n    D, f, n    i, G    R, f, n    f, X, n    f, p, i, n
Allowed substitution hints:    ph( y, f, i, m, n, p)    ps( y, f, i, m, n, p)    ch( y,
f, i, m, n, p)    A( y, i, m, p)    C( y, f, i, m, n, p)    D( y, i, m, p)    R( y, i, m, p)    G( y, f, m, n, p)    X( y, i, m, p)    ph'( y, f, i, m, n, p)    ps'( y, f, i, m, n, p)    ch'( y, f, i, m, n, p)    ph"( y, f, i, m, n, p)    ps"( y, f, i, m, n, p)    ch"( y, f, i, m, n, p)

Proof of Theorem bnj999
StepHypRef Expression
1 bnj999.3 . . . . . . 7  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
2 bnj999.7 . . . . . . 7  |-  ( ph'  <->  [. p  /  n ]. ph )
3 bnj999.8 . . . . . . 7  |-  ( ps'  <->  [. p  /  n ]. ps )
4 bnj999.9 . . . . . . 7  |-  ( ch'  <->  [. p  /  n ]. ch )
5 vex 2791 . . . . . . 7  |-  p  e. 
_V
61, 2, 3, 4, 5bnj919 28797 . . . . . 6  |-  ( ch'  <->  (
p  e.  D  /\  f  Fn  p  /\  ph' 
/\  ps' ) )
7 bnj999.10 . . . . . 6  |-  ( ph"  <->  [. G  / 
f ]. ph' )
8 bnj999.11 . . . . . 6  |-  ( ps"  <->  [. G  / 
f ]. ps' )
9 bnj999.12 . . . . . 6  |-  ( ch"  <->  [. G  / 
f ]. ch' )
10 bnj999.16 . . . . . . 7  |-  G  =  ( f  u.  { <. n ,  C >. } )
1110bnj918 28796 . . . . . 6  |-  G  e. 
_V
126, 7, 8, 9, 11bnj976 28809 . . . . 5  |-  ( ch"  <->  ( p  e.  D  /\  G  Fn  p  /\  ph"  /\  ps" ) )
1312bnj1254 28842 . . . 4  |-  ( ch"  ->  ps" )
1413anim1i 551 . . 3  |-  ( ( ch" 
/\  ( i  e. 
om  /\  suc  i  e.  p  /\  y  e.  ( G `  i
) ) )  -> 
( ps"  /\  ( i  e.  om  /\  suc  i  e.  p  /\  y  e.  ( G `  i ) ) ) )
15 bnj252 28728 . . 3  |-  ( ( ch" 
/\  i  e.  om  /\ 
suc  i  e.  p  /\  y  e.  ( G `  i )
)  <->  ( ch"  /\  (
i  e.  om  /\  suc  i  e.  p  /\  y  e.  ( G `  i )
) ) )
16 bnj252 28728 . . 3  |-  ( ( ps" 
/\  i  e.  om  /\ 
suc  i  e.  p  /\  y  e.  ( G `  i )
)  <->  ( ps"  /\  (
i  e.  om  /\  suc  i  e.  p  /\  y  e.  ( G `  i )
) ) )
1714, 15, 163imtr4i 257 . 2  |-  ( ( ch" 
/\  i  e.  om  /\ 
suc  i  e.  p  /\  y  e.  ( G `  i )
)  ->  ( ps"  /\  i  e.  om  /\  suc  i  e.  p  /\  y  e.  ( G `  i
) ) )
18 ssiun2 3945 . . . 4  |-  ( y  e.  ( G `  i )  ->  pred (
y ,  A ,  R )  C_  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
1918bnj708 28785 . . 3  |-  ( ( ps" 
/\  i  e.  om  /\ 
suc  i  e.  p  /\  y  e.  ( G `  i )
)  ->  pred ( y ,  A ,  R
)  C_  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
20 3simpa 952 . . . . . 6  |-  ( ( ps" 
/\  i  e.  om  /\ 
suc  i  e.  p
)  ->  ( ps"  /\  i  e.  om ) )
2120ancomd 438 . . . . 5  |-  ( ( ps" 
/\  i  e.  om  /\ 
suc  i  e.  p
)  ->  ( i  e.  om  /\  ps" ) )
22 simp3 957 . . . . 5  |-  ( ( ps" 
/\  i  e.  om  /\ 
suc  i  e.  p
)  ->  suc  i  e.  p )
23 bnj999.2 . . . . . . . 8  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2423, 3, 5bnj539 28923 . . . . . . 7  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  p  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
25 bnj999.15 . . . . . . 7  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
2624, 8, 25, 10bnj965 28974 . . . . . 6  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  p  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
2726bnj228 28763 . . . . 5  |-  ( ( i  e.  om  /\  ps" )  ->  ( suc  i  e.  p  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
2821, 22, 27sylc 56 . . . 4  |-  ( ( ps" 
/\  i  e.  om  /\ 
suc  i  e.  p
)  ->  ( G `  suc  i )  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) )
2928bnj721 28786 . . 3  |-  ( ( ps" 
/\  i  e.  om  /\ 
suc  i  e.  p  /\  y  e.  ( G `  i )
)  ->  ( G `  suc  i )  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) )
3019, 29sseqtr4d 3215 . 2  |-  ( ( ps" 
/\  i  e.  om  /\ 
suc  i  e.  p  /\  y  e.  ( G `  i )
)  ->  pred ( y ,  A ,  R
)  C_  ( G `  suc  i ) )
3117, 30syl 15 1  |-  ( ( ch" 
/\  i  e.  om  /\ 
suc  i  e.  p  /\  y  e.  ( G `  i )
)  ->  pred ( y ,  A ,  R
)  C_  ( G `  suc  i ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   [.wsbc 2991    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   U_ciun 3905   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713
This theorem is referenced by:  bnj1006  28991
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-iun 3907  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-bnj17 28712
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