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Theorem bnj518 28588
Description: Technical lemma for bnj852 28623. 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 28408 . . . 4  |-  ( (
ph  /\  ps  /\  n  e.  om  /\  p  e.  n )  <->  ( n  e.  om  /\  ph  /\  ps  /\  p  e.  n
) )
31, 2bitri 241 . . 3  |-  ( ta  <->  ( n  e.  om  /\  ph 
/\  ps  /\  p  e.  n ) )
4 df-bnj17 28382 . . . 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 28587 . . . . 5  |-  ( ( n  e.  om  /\  ph 
/\  ps )  ->  A. p  e.  n  ( f `  p )  C_  A
)
87r19.21bi 2740 . . . 4  |-  ( ( ( n  e.  om  /\ 
ph  /\  ps )  /\  p  e.  n
)  ->  ( f `  p )  C_  A
)
94, 8sylbi 188 . . 3  |-  ( ( n  e.  om  /\  ph 
/\  ps  /\  p  e.  n )  ->  (
f `  p )  C_  A )
103, 9sylbi 188 . 2  |-  ( ta 
->  ( f `  p
)  C_  A )
11 ssel 3278 . . . 4  |-  ( ( f `  p ) 
C_  A  ->  (
y  e.  ( f `
 p )  -> 
y  e.  A ) )
12 bnj93 28565 . . . . 5  |-  ( ( R  FrSe  A  /\  y  e.  A )  ->  pred ( y ,  A ,  R )  e.  _V )
1312ex 424 . . . 4  |-  ( R 
FrSe  A  ->  ( y  e.  A  ->  pred (
y ,  A ,  R )  e.  _V ) )
1411, 13sylan9r 640 . . 3  |-  ( ( R  FrSe  A  /\  ( f `  p
)  C_  A )  ->  ( y  e.  ( f `  p )  ->  pred ( y ,  A ,  R )  e.  _V ) )
1514ralrimiv 2724 . 2  |-  ( ( R  FrSe  A  /\  ( f `  p
)  C_  A )  ->  A. y  e.  ( f `  p ) 
pred ( y ,  A ,  R )  e.  _V )
1610, 15sylan2 461 1  |-  ( ( R  FrSe  A  /\  ta )  ->  A. y  e.  ( f `  p
)  pred ( y ,  A ,  R )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642   _Vcvv 2892    C_ wss 3256   (/)c0 3564   U_ciun 4028   suc csuc 4517   omcom 4778   ` cfv 5387    /\ w-bnj17 28381    predc-bnj14 28383    FrSe w-bnj15 28387
This theorem is referenced by:  bnj535  28592  bnj546  28598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-iota 5351  df-fv 5395  df-bnj17 28382  df-bnj14 28384  df-bnj13 28386  df-bnj15 28388
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