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Theorem bnj999 29305
Description: Technical lemma for bnj69 29356. 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 2804 . . . . . . 7  |-  p  e. 
_V
61, 2, 3, 4, 5bnj919 29113 . . . . . 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 29112 . . . . . 6  |-  G  e. 
_V
126, 7, 8, 9, 11bnj976 29125 . . . . 5  |-  ( ch"  <->  ( p  e.  D  /\  G  Fn  p  /\  ph"  /\  ps" ) )
1312bnj1254 29158 . . . 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 29044 . . 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 29044 . . 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 3961 . . . 4  |-  ( y  e.  ( G `  i )  ->  pred (
y ,  A ,  R )  C_  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
1918bnj708 29101 . . 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 29239 . . . . . . 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 29290 . . . . . 6  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  p  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
2726bnj228 29079 . . . . 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 29102 . . 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 3228 . 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 1632    e. wcel 1696   A.wral 2556   [.wsbc 3004    u. cun 3163    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   U_ciun 3921   suc csuc 4410   omcom 4672    Fn wfn 5266   ` cfv 5271    /\ w-bnj17 29027    predc-bnj14 29029
This theorem is referenced by:  bnj1006  29307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-bnj17 29028
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