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Theorem bnj518 34087
Description: Technical lemma for bnj852 34122. 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 33908 . . . 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 33882 . . . 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 34086 . . . . 5  |-  ( ( n  e.  om  /\  ph 
/\  ps )  ->  A. p  e.  n  ( f `  p )  C_  A
)
87r19.21bi 2826 . . . 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 3493 . . . 4  |-  ( ( f `  p ) 
C_  A  ->  (
y  e.  ( f `
 p )  -> 
y  e.  A ) )
12 bnj93 34064 . . . . 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 2869 . 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 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    C_ wss 3471   (/)c0 3793   U_ciun 4332   suc csuc 4889   ` cfv 5594   omcom 6699    /\ w-bnj17 33881    predc-bnj14 33883    FrSe w-bnj15 33887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-iota 5557  df-fv 5602  df-om 6700  df-bnj17 33882  df-bnj14 33884  df-bnj13 33886  df-bnj15 33888
This theorem is referenced by:  bnj535  34091  bnj546  34097
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