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Theorem bnj518 33240
Description: Technical lemma for bnj852 33275. 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 33062 . . . 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 33036 . . . 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 33239 . . . . 5  |-  ( ( n  e.  om  /\  ph 
/\  ps )  ->  A. p  e.  n  ( f `  p )  C_  A
)
87r19.21bi 2833 . . . 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 3498 . . . 4  |-  ( ( f `  p ) 
C_  A  ->  (
y  e.  ( f `
 p )  -> 
y  e.  A ) )
12 bnj93 33217 . . . . 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 2876 . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476   (/)c0 3785   U_ciun 4325   suc csuc 4880   ` cfv 5588   omcom 6685    /\ w-bnj17 33035    predc-bnj14 33037    FrSe w-bnj15 33041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-iota 5551  df-fv 5596  df-om 6686  df-bnj17 33036  df-bnj14 33038  df-bnj13 33040  df-bnj15 33042
This theorem is referenced by:  bnj535  33244  bnj546  33250
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