Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj518 Structured version   Unicode version

Theorem bnj518 29194
Description: Technical lemma for bnj852 29229. 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 29014 . . . 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 28988 . . . 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 29193 . . . . 5  |-  ( ( n  e.  om  /\  ph 
/\  ps )  ->  A. p  e.  n  ( f `  p )  C_  A
)
87r19.21bi 2796 . . . 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 3334 . . . 4  |-  ( ( f `  p ) 
C_  A  ->  (
y  e.  ( f `
 p )  -> 
y  e.  A ) )
12 bnj93 29171 . . . . 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 2780 . 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 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    C_ wss 3312   (/)c0 3620   U_ciun 4085   suc csuc 4575   omcom 4837   ` cfv 5446    /\ w-bnj17 28987    predc-bnj14 28989    FrSe w-bnj15 28993
This theorem is referenced by:  bnj535  29198  bnj546  29204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-iota 5410  df-fv 5454  df-bnj17 28988  df-bnj14 28990  df-bnj13 28992  df-bnj15 28994
  Copyright terms: Public domain W3C validator