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Theorem bnj518 32212
Description: Technical lemma for bnj852 32247. 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
bnj518.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj518.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj518.3  |-  ( ta  <->  (
ph  /\  ps  /\  n  e.  om  /\  p  e.  n ) )
Assertion
Ref Expression
bnj518  |-  ( ( R  FrSe  A  /\  ta )  ->  A. y  e.  ( f `  p
)  pred ( y ,  A ,  R )  e.  _V )
Distinct variable groups:    f, i, p, y    i, n, p    A, i, p, y    y, R
Allowed substitution hints:    ph( x, y, f, i, n, p)    ps( x, y, f, i, n, p)    ta( x, y, f, i, n, p)    A( x, f, n)    R( x, f, i, n, p)

Proof of Theorem bnj518
StepHypRef Expression
1 bnj518.3 . . . 4  |-  ( ta  <->  (
ph  /\  ps  /\  n  e.  om  /\  p  e.  n ) )
2 bnj334 32034 . . . 4  |-  ( (
ph  /\  ps  /\  n  e.  om  /\  p  e.  n )  <->  ( n  e.  om  /\  ph  /\  ps  /\  p  e.  n
) )
31, 2bitri 249 . . 3  |-  ( ta  <->  ( n  e.  om  /\  ph 
/\  ps  /\  p  e.  n ) )
4 df-bnj17 32008 . . . 4  |-  ( ( n  e.  om  /\  ph 
/\  ps  /\  p  e.  n )  <->  ( (
n  e.  om  /\  ph 
/\  ps )  /\  p  e.  n ) )
5 bnj518.1 . . . . . 6  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
6 bnj518.2 . . . . . 6  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
75, 6bnj517 32211 . . . . 5  |-  ( ( n  e.  om  /\  ph 
/\  ps )  ->  A. p  e.  n  ( f `  p )  C_  A
)
87r19.21bi 2920 . . . 4  |-  ( ( ( n  e.  om  /\ 
ph  /\  ps )  /\  p  e.  n
)  ->  ( f `  p )  C_  A
)
94, 8sylbi 195 . . 3  |-  ( ( n  e.  om  /\  ph 
/\  ps  /\  p  e.  n )  ->  (
f `  p )  C_  A )
103, 9sylbi 195 . 2  |-  ( ta 
->  ( f `  p
)  C_  A )
11 ssel 3459 . . . 4  |-  ( ( f `  p ) 
C_  A  ->  (
y  e.  ( f `
 p )  -> 
y  e.  A ) )
12 bnj93 32189 . . . . 5  |-  ( ( R  FrSe  A  /\  y  e.  A )  ->  pred ( y ,  A ,  R )  e.  _V )
1312ex 434 . . . 4  |-  ( R 
FrSe  A  ->  ( y  e.  A  ->  pred (
y ,  A ,  R )  e.  _V ) )
1411, 13sylan9r 658 . . 3  |-  ( ( R  FrSe  A  /\  ( f `  p
)  C_  A )  ->  ( y  e.  ( f `  p )  ->  pred ( y ,  A ,  R )  e.  _V ) )
1514ralrimiv 2828 . 2  |-  ( ( R  FrSe  A  /\  ( f `  p
)  C_  A )  ->  A. y  e.  ( f `  p ) 
pred ( y ,  A ,  R )  e.  _V )
1610, 15sylan2 474 1  |-  ( ( R  FrSe  A  /\  ta )  ->  A. y  e.  ( f `  p
)  pred ( y ,  A ,  R )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078    C_ wss 3437   (/)c0 3746   U_ciun 4280   suc csuc 4830   ` cfv 5527   omcom 6587    /\ w-bnj17 32007    predc-bnj14 32009    FrSe w-bnj15 32013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-tr 4495  df-eprel 4741  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-iota 5490  df-fv 5535  df-om 6588  df-bnj17 32008  df-bnj14 32010  df-bnj13 32012  df-bnj15 32014
This theorem is referenced by:  bnj535  32216  bnj546  32222
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