HomeHome Metamath Proof Explorer
Theorem List (p. 283 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21492)
  Hilbert Space Explorer  Hilbert Space Explorer
(21493-23015)
  Users' Mathboxes  Users' Mathboxes
(23016-31423)
 

Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj556 28201 Technical lemma for bnj852 28222. 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.)
 |-  ( si 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  m ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   =>    |-  ( et  ->  si )
 
Theorembnj557 28202* Technical lemma for bnj852 28222. 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.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et  /\  ze )  ->  ( G `  m )  =  L )
 
Theorembnj558 28203* Technical lemma for bnj852 28222. 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.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et  /\  ze )  ->  ( G ` 
 suc  i )  =  K )
 
Theorembnj561 28204 Technical lemma for bnj852 28222. 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.)
 |-  ( si 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  m ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  si )  ->  G  Fn  n )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et )  ->  G  Fn  n )
 
Theorembnj562 28205 Technical lemma for bnj852 28222. 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.)
 |-  ( si 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  m ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  si )  ->  ph" )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  et )  ->  ph" )
 
Theorembnj570 28206* Technical lemma for bnj852 28222. 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.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  ( rh 
 <->  ( i  e.  om  /\ 
 suc  i  e.  n  /\  m  =/=  suc  i
 ) )   &    |-  K  =  U_ y  e.  ( G `  i )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  G  Fn  n )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et  /\  rh )  ->  ( G `
  suc  i )  =  K )
 
Theorembnj571 28207* Technical lemma for bnj852 28222. 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.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )   &    |-  ( rh  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  =/= 
 suc  i ) )   &    |-  ( ( R  FrSe  A 
 /\  ta  /\  et )  ->  G  Fn  n )   &    |-  (
 ps" 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et )  ->  ps" )
 
Theorembnj605 28208* Technical lemma. 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.)
 |-  ( th 
 <-> 
 A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   &    |-  ( ph"  <->  [. f  /  f ]. ph )   &    |-  ( ps"  <->  [. f  /  f ]. ps )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  f  e.  _V   &    |-  ( ch'  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  m  /\  ph'  /\  ps' ) ) )   &    |-  ( ph"  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )   &    |-  (
 ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  f  Fn  n )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  et )  ->  ph" )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  ps" )   =>    |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E. f ( f  Fn  n  /\  ph  /\  ps ) ) )
 
Theorembnj581 28209* Technical lemma for bnj580 28214. 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). (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 9-Jul-2011.) (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( ph'  <->  [. g  /  f ]. ph )   &    |-  ( ps'  <->  [. g  /  f ]. ps )   &    |-  ( ch'  <->  [. g  /  f ]. ch )   =>    |-  ( ch'  <->  ( g  Fn  n  /\  ph'  /\  ps' ) )
 
Theorembnj589 28210* Technical lemma for bnj852 28222. 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.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ps  <->  A. k  e.  om  ( suc  k  e.  n  ->  ( f `  suc  k )  =  U_ y  e.  ( f `  k
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj590 28211 Technical lemma for bnj852 28222. 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.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( B  =  suc  i  /\  ps )  ->  ( i  e.  om  ->  ( B  e.  n  ->  ( f `  B )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) ) )
 
Theorembnj591 28212* Technical lemma for bnj852 28222. 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.)
 |-  ( th 
 <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )   =>    |-  ( [. k  /  j ]. th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  k )  =  ( g `  k ) ) )
 
Theorembnj594 28213* Technical lemma for bnj852 28222. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( f  Fn  n  /\  ph  /\  ps ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ph'  <->  ( g `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( g `  suc  i )  =  U_ y  e.  ( g `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch'  <->  ( g  Fn  n  /\  ph'  /\  ps' ) )   &    |-  ( th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )   &    |-  ( [. k  /  j ]. th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  k )  =  ( g `  k ) ) )   &    |-  ( ta  <->  A. k  e.  n  ( k  _E  j  -> 
 [. k  /  j ]. th ) )   =>    |-  ( ( j  e.  n  /\  ta )  ->  th )
 
Theorembnj580 28214* Technical lemma for bnj579 28215. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ph'  <->  [. g  /  f ]. ph )   &    |-  ( ps'  <->  [. g  /  f ]. ps )   &    |-  ( ch'  <->  [. g  /  f ]. ch )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( th 
 <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )   &    |-  ( ta  <->  A. k  e.  n  ( k  _E  j  -> 
 [. k  /  j ]. th ) )   =>    |-  ( n  e.  D  ->  E* f ch )
 
Theorembnj579 28215* Technical lemma for bnj852 28222. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   =>    |-  ( n  e.  D  ->  E* f ( f  Fn  n  /\  ph  /\  ps ) )
 
Theorembnj602 28216 Equality theorem for the  pred function constant. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( X  =  Y  ->  pred
 ( X ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
 
Theorembnj607 28217* Technical lemma for bnj852 28222. 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.)
 |-  ( th 
 <-> 
 A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  G  e.  _V   &    |-  ( ch'  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  m  /\  ph'  /\  ps' ) ) )   &    |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )   &    |-  (
 ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  G  Fn  n )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  et )  ->  ph" )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  ps" )   &    |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ph0  <->  [. h  /  f ]. ph )   &    |-  ( ps0  <->  [. h  /  f ]. ps )   &    |-  ( ph1  <->  [. G  /  h ]. ph0 )   &    |-  ( ps1  <->  [. G  /  h ]. ps0 )   =>    |-  ( ( n  =/= 
 1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E. f ( f  Fn  n  /\  ph  /\  ps ) ) )
 
Theorembnj609 28218* Technical lemma for bnj852 28222. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  G  e.  _V   =>    |-  ( ph"  <->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
 
Theorembnj611 28219* Technical lemma for bnj852 28222. 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.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  G  e.  _V   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj600 28220* Technical lemma for bnj852 28222. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ch 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( th  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   &    |-  ( ph'  <->  [. m  /  n ]. ph )   &    |-  ( ps'  <->  [. m  /  n ].
 ps )   &    |-  ( ch'  <->  [. m  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  ( ch"  <->  [. G  /  f ]. ch )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  ( rh  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  =/= 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   =>    |-  ( n  =/=  1o  ->  ( ( n  e.  D  /\  th )  ->  ch ) )
 
Theorembnj601 28221* Technical lemma for bnj852 28222. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ch 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( th  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   =>    |-  ( n  =/= 
 1o  ->  ( ( n  e.  D  /\  th )  ->  ch ) )
 
Theorembnj852 28222* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   =>    |-  (
 ( R  FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) )
 
Theorembnj864 28223* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ch 
 <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
 )   &    |-  ( th  <->  ( f  Fn  n  /\  ph  /\  ps ) )   =>    |-  ( ch  ->  E! f th )
 
Theorembnj865 28224* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ch 
 <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
 )   &    |-  ( th  <->  ( f  Fn  n  /\  ph  /\  ps ) )   =>    |- 
 E. w A. n ( ch  ->  E. f  e.  w  th )
 
Theorembnj873 28225* Technical lemma for bnj69 28309. 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.)
 |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ph'  <->  [. g  /  f ]. ph )   &    |-  ( ps'  <->  [. g  /  f ]. ps )   =>    |-  B  =  { g  |  E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' ) }
 
Theorembnj849 28226* Technical lemma for bnj69 28309. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
 )   &    |-  ( th  <->  ( f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ph'  <->  [. g  /  f ]. ph )   &    |-  ( ps'  <->  [. g  /  f ]. ps )   &    |-  ( th'  <->  [. g  /  f ]. th )   &    |-  ( ta  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  B  e.  _V )
 
Theorembnj882 28227* Definition (using hypotheses for readability) of the function giving the transitive closure of  X in  A by  R. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   =>    |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f
 ( f `  i
 )
 
Theorembnj18eq1 28228 Equality theorem for transitive closure. (Contributed by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  ( X  =  Y  ->  trCl
 ( X ,  A ,  R )  =  trCl ( Y ,  A ,  R ) )
 
Theorembnj893 28229 Property of  trCl. Under certain conditions, the transitive closure of  X in  A by  R is a set. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  X  e.  A )  -> 
 trCl ( X ,  A ,  R )  e.  _V )
 
Theorembnj900 28230* Technical lemma for bnj69 28309. 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.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  B  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |-  ( f  e.  B  -> 
 (/)  e.  dom  f )
 
Theorembnj906 28231 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  X  e.  A )  -> 
 pred ( X ,  A ,  R )  C_ 
 trCl ( X ,  A ,  R )
 )
 
Theorembnj908 28232* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ch 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( th  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   &    |-  ( ph'  <->  [. m  /  n ]. ph )   &    |-  ( ps'  <->  [. m  /  n ].
 ps )   &    |-  ( ch'  <->  [. m  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  ( ch"  <->  [. G  /  f ]. ch )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  ( rh  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  =/= 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   =>    |-  ( ( R  FrSe  A 
 /\  x  e.  A  /\  ch'  /\  et )  ->  E. f ( G  Fn  n  /\  ph"  /\  ps" ) )
 
Theorembnj911 28233* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( f  Fn  n  /\  ph  /\  ps )  ->  A. i ( f  Fn  n  /\  ph  /\  ps ) )
 
Theorembnj916 28234* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( f  Fn  n  /\  ph  /\  ps )
 )   =>    |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( n  e.  D  /\  ch 
 /\  i  e.  n  /\  y  e.  (
 f `  i )
 ) )
 
Theorembnj917 28235* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   =>    |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i
 ) ) )
 
Theorembnj934 28236* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  G  e.  _V   =>    |-  (
 ( ph  /\  ( G `
  (/) )  =  ( f `  (/) ) ) 
 ->  ph" )
 
Theorembnj929 28237* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  C  e.  _V   =>    |-  (
 ( n  e.  D  /\  p  =  suc  n 
 /\  f  Fn  n  /\  ph )  ->  ph" )
 
Theorembnj938 28238* Technical lemma for bnj69 28309. 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.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( ph'  <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A  /\  ta  /\  si )  -> 
 U_ y  e.  (
 f `  p )  pred ( y ,  A ,  R )  e.  _V )
 
Theorembnj944 28239* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  ( ta 
 <->  ( f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( si  <->  ( n  e.  D  /\  p  = 
 suc  n  /\  m  e.  n ) )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  ph" )
 
Theorembnj953 28240 Technical lemma for bnj69 28309. 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.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( G `
  i )  =  ( f `  i
 )  ->  A. y ( G `  i )  =  ( f `  i ) )   =>    |-  ( ( ( G `  i )  =  ( f `  i )  /\  ( G `
  suc  i )  =  ( f `  suc  i )  /\  ( i  e.  om  /\  suc  i  e.  n )  /\  ps )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) )
 
Theorembnj958 28241* Technical lemma for bnj69 28309. 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.)
 |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  (
 ( G `  i
 )  =  ( f `
  i )  ->  A. y ( G `  i )  =  (
 f `  i )
 )
 
Theorembnj1000 28242* Technical lemma for bnj852 28222. 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.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  G  e.  _V   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj965 28243* Technical lemma for bnj852 28222. 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.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  C  =  U_ y  e.  ( f `  m )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj964 28244* Technical lemma for bnj69 28309. 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.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  C  =  U_ y  e.  ( f `  m )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  (
 ( ( R  FrSe  A 
 /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) 
 /\  ( i  e. 
 om  /\  suc  i  e.  p  /\  suc  i  e.  n ) )  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) )   &    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  ( i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  ->  ( G ` 
 suc  i )  = 
 U_ y  e.  ( G `  i )  pred ( y ,  A ,  R ) )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  ps" )
 
Theorembnj966 28245* Technical lemma for bnj69 28309. 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.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  (
 ( ( R  FrSe  A 
 /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )   &    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  G  Fn  p )   =>    |-  ( ( ( R 
 FrSe  A  /\  X  e.  A )  /\  ( ch 
 /\  n  =  suc  m 
 /\  p  =  suc  n )  /\  ( i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  ->  ( G ` 
 suc  i )  = 
 U_ y  e.  ( G `  i )  pred ( y ,  A ,  R ) )
 
Theorembnj967 28246* Technical lemma for bnj69 28309. 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.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  (
 ( ( R  FrSe  A 
 /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )   =>    |-  ( ( ( R 
 FrSe  A  /\  X  e.  A )  /\  ( ch 
 /\  n  =  suc  m 
 /\  p  =  suc  n )  /\  ( i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n ) )  ->  ( G ` 
 suc  i )  = 
 U_ y  e.  ( G `  i )  pred ( y ,  A ,  R ) )
 
Theorembnj969 28247* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  ( ta  <->  ( f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( si  <->  ( n  e.  D  /\  p  = 
 suc  n  /\  m  e.  n ) )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  C  e.  _V )
 
Theorembnj970 28248 Technical lemma for bnj69 28309. 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.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  D  =  ( om  \  { (/) } )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  p  e.  D )
 
Theorembnj910 28249* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  ( ta 
 <->  ( f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( si  <->  ( n  e.  D  /\  p  = 
 suc  n  /\  m  e.  n ) )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  ch" )
 
Theorembnj978 28250* Technical lemma for bnj69 28309. 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.)
 |-  ( th 
 <->  ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e.  pred ( y ,  A ,  R ) ) )   &    |-  ( th  ->  z  e.  trCl
 ( X ,  A ,  R ) )   =>    |-  ( ( R 
 FrSe  A  /\  X  e.  A )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
 
Theorembnj981 28251* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   =>    |-  ( Z  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
 ) ) )
 
Theorembnj983 28252* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   =>    |-  ( Z  e.  trCl ( X ,  A ,  R )  <->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) ) )
 
Theorembnj984 28253 Technical lemma for bnj69 28309. 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.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  B  =  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |-  ( G  e.  A  ->  ( G  e.  B  <->  [. G  /  f ]. E. n ch ) )
 
Theorembnj985 28254* Technical lemma for bnj69 28309. 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.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  B  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  ( G  e.  B  <->  E. p ch" )
 
Theorembnj986 28255* Technical lemma for bnj69 28309. 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.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ta 
 <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )   =>    |-  ( ch  ->  E. m E. p ta )
 
Theorembnj996 28256* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i
 ) ) )   &    |-  D  =  ( om  \  { (/)
 } )   &    |-  B  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |- 
 E. f E. n E. i E. m E. p ( th  ->  ( ch  /\  ta  /\  et ) )
 
Theorembnj998 28257* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  (
 ( th  /\  ch  /\  ta 
 /\  et )  ->  ch" )
 
Theorembnj999 28258* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  C  =  U_ y  e.  ( f `  m )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  (
 ( ch"  /\  i  e.  om 
 /\  suc  i  e.  p  /\  y  e.  ( G `  i ) ) 
 ->  pred ( y ,  A ,  R ) 
 C_  ( G `  suc  i ) )
 
Theorembnj1001 28259 Technical lemma for bnj69 28309. 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.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i
 ) ) )   &    |-  D  =  ( om  \  { (/)
 } )   &    |-  ( ( th  /\ 
 ch  /\  ta  /\  et )  ->  ch" )   =>    |-  ( ( th  /\  ch 
 /\  ta  /\  et )  ->  ( ch"  /\  i  e.  om 
 /\  suc  i  e.  p ) )
 
Theorembnj1006 28260* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i
 ) ) )   &    |-  ( ph'  <->  [. p  /  n ]. ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  (
 ( th  /\  ch  /\  ta 
 /\  et )  ->  ( ch"  /\  i  e.  om  /\  suc  i  e.  p ) )   =>    |-  ( ( th  /\  ch 
 /\  ta  /\  et )  -> 
 pred ( y ,  A ,  R ) 
 C_  ( G `  suc  i ) )
 
Theorembnj1014 28261* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   =>    |-  ( ( g  e.  B  /\  j  e.  dom  g )  ->  ( g `  j
 )  C_  trCl ( X ,  A ,  R ) )
 
Theorembnj1015 28262* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  G  e.  V   &    |-  J  e.  V   =>    |-  (
 ( G  e.  B  /\  J  e.  dom  G )  ->  ( G `  J )  C_  trCl ( X ,  A ,  R ) )
 
Theorembnj1018 28263* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  ( ch"  <->  ( p  e.  D  /\  G  Fn  p  /\  ph"  /\  ps" ) )   &    |-  ( ( th  /\  ch 
 /\  ta  /\  et )  ->  ch" )   &    |-  ( ( th  /\ 
 ch  /\  ta  /\  et )  ->  ( ch"  /\  i  e.  om  /\  suc  i  e.  p ) )   =>    |-  ( ( th  /\ 
 ch  /\  et  /\  E. p ta )  ->  ( G `  suc  i ) 
 C_  trCl ( X ,  A ,  R )
 )
 
Theorembnj1020 28264* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i
 ) ) )   &    |-  ( ph'  <->  [. p  /  n ]. ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  ( ch"  <->  ( p  e.  D  /\  G  Fn  p  /\  ph"  /\  ps" ) )   =>    |-  ( ( th  /\  ch 
 /\  et  /\  E. p ta )  ->  pred (
 y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
 
Theorembnj1021 28265* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i
 ) ) )   &    |-  D  =  ( om  \  { (/)
 } )   &    |-  B  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |- 
 E. f E. n E. i E. m ( th  ->  ( th  /\ 
 ch  /\  et  /\  E. p ta ) )
 
Theorembnj907 28266* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i
 ) ) )   &    |-  ( ph'  <->  [. p  /  n ]. ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  (
 ( R  FrSe  A  /\  X  e.  A )  -> 
 TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
 
Theorembnj1029 28267 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  X  e.  A )  -> 
 TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
 
Theorembnj1033 28268* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( et 
 <->  z  e.  trCl ( X ,  A ,  R ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( E. f E. n E. i ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B )   =>    |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1034 28269* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( E. f E. n E. i ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B )   =>    |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1039 28270 Technical lemma for bnj69 28309. 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.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ps'  <->  [. j  /  i ]. ps )   =>    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj1040 28271* Technical lemma for bnj69 28309. 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.)
 |-  ( ph'  <->  [. j  /  i ]. ph )   &    |-  ( ps'  <->  [. j  /  i ]. ps )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ch'  <->  [. j  /  i ]. ch )   =>    |-  ( ch'  <->  ( n  e.  D  /\  f  Fn  n  /\  ph'  /\  ps' ) )
 
Theorembnj1047 28272 Technical lemma for bnj69 28309. 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.)
 |-  ( rh 
 <-> 
 A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  ( et'  <->  [. j  /  i ]. et )   =>    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  et' ) )
 
Theorembnj1049 28273 Technical lemma for bnj69 28309. 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.)
 |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  ( et  <->  ( ( th  /\ 
 ta  /\  ch  /\  ze )  ->  z  e.  B ) )   =>    |-  ( A. i  e.  n  et  <->  A. i et )
 
Theorembnj1052 28274* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( et 
 <->  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B ) )   &    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  (
 ( th  /\  ta  /\  ch 
 /\  ze )  ->  (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et ) ) )   =>    |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1053 28275* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( et 
 <->  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B ) )   &    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  (
 ( th  /\  ta  /\  ch 
 /\  ze )  ->  A. i  e.  n  ( rh  ->  et ) )   =>    |-  ( ( th  /\ 
 ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1071 28276 Technical lemma for bnj69 28309. 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.)
 |-  D  =  ( om  \  { (/)
 } )   =>    |-  ( n  e.  D  ->  _E  Fr  n )
 
Theorembnj1083 28277 Technical lemma for bnj69 28309. 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.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  K  =  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |-  ( f  e.  K  <->  E. n ch )
 
Theorembnj1090 28278* Technical lemma for bnj69 28309. 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.)
 |-  ( et 
 <->  ( ( f  e.  K  /\  i  e. 
 dom  f )  ->  ( f `  i
 )  C_  B )
 )   &    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  ( et'  <->  [. j  /  i ]. et )   &    |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i )  ->  et' ) )   &    |-  ( ph0  <->  ( i  e.  n  /\  si  /\  f  e.  K  /\  i  e.  dom  f ) )   &    |-  ( ( th  /\ 
 ta  /\  ch  /\  ze )  ->  A. i E. j
 ( ph0  ->  ( f `  i )  C_  B ) )   =>    |-  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  A. i  e.  n  ( rh  ->  et )
 )
 
Theorembnj1093 28279* Technical lemma for bnj69 28309. 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.)
 |-  E. j
 ( ( ( th  /\ 
 ta  /\  ch )  /\  ph0 )  ->  (
 f `  i )  C_  B )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   =>    |-  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  A. i E. j
 ( ph0  ->  ( f `  i )  C_  B ) )
 
Theorembnj1097 28280 Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   =>    |-  ( ( i  =  (/)  /\  ( ( th  /\  ta  /\  ch )  /\  ph0 ) )  ->  ( f `  i
 )  C_  B )
 
Theorembnj1110 28281* Technical lemma for bnj69 28309. 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.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( si 
 <->  ( ( j  e.  n  /\  j  _E  i )  ->  et' ) )   &    |-  ( ph0  <->  ( i  e.  n  /\  si  /\  f  e.  K  /\  i  e.  dom  f ) )   &    |-  ( et'  <->  ( ( f  e.  K  /\  j  e.  dom  f )  ->  ( f `  j
 )  C_  B )
 )   =>    |- 
 E. j ( ( i  =/=  (/)  /\  (
 ( th  /\  ta  /\  ch )  /\  ph0 ) ) 
 ->  ( j  e.  n  /\  i  =  suc  j  /\  ( f `  j )  C_  B ) )
 
Theorembnj1112 28282* Technical lemma for bnj69 28309. 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.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ps  <->  A. j ( ( j  e.  om  /\  suc  j  e.  n ) 
 ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj1118 28283* Technical lemma for bnj69 28309. 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.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  D  =  ( om  \  { (/)
 } )   &    |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i )  ->  et' ) )   &    |-  ( ph0  <->  ( i  e.  n  /\  si  /\  f  e.  K  /\  i  e.  dom  f ) )   &    |-  ( et'  <->  ( ( f  e.  K  /\  j  e.  dom  f )  ->  ( f `  j
 )  C_  B )
 )   =>    |- 
 E. j ( ( i  =/=  (/)  /\  (
 ( th  /\  ta  /\  ch )  /\  ph0 ) ) 
 ->  ( f `  i
 )  C_  B )
 
Theorembnj1121 28284 Technical lemma for bnj69 28309. 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.)
 |-  ( th 
 <->  ( R  FrSe  A  /\  X  e.  A )
 )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i
 ) ) )   &    |-  ( et 
 <->  ( ( f  e.  K  /\  i  e. 
 dom  f )  ->  ( f `  i
 )  C_  B )
 )   &    |-  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  A. i  e.  n  et )   &    |-  K  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |-  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B )
 
Theorembnj1123 28285* Technical lemma for bnj69 28309. 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.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  K  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   &    |-  ( et  <->  ( ( f  e.  K  /\  i  e.  dom  f )  ->  ( f `  i
 )  C_  B )
 )   &    |-  ( et'  <->  [. j  /  i ]. et )   =>    |-  ( et'  <->  ( ( f  e.  K  /\  j  e.  dom  f )  ->  ( f `  j
 )  C_  B )
 )
 
Theorembnj1030 28286* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( et 
 <->  ( ( f  e.  K  /\  i  e. 
 dom  f )  ->  ( f `  i
 )  C_  B )
 )   &    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  ( ph'  <->  [. j  /  i ]. ph )   &    |-  ( ps'  <->  [. j  /  i ]. ps )   &    |-  ( ch'  <->  [. j  /  i ]. ch )   &    |-  ( th'  <->  [. j  /  i ]. th )   &    |-  ( ta'  <->  [. j  /  i ]. ta )   &    |-  ( ze'  <->  [. j  /  i ]. ze )   &    |-  ( et'  <->  [. j  /  i ]. et )   &    |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i )  ->  et' ) )   &    |-  ( ph0  <->  ( i  e.  n  /\  si  /\  f  e.  K  /\  i  e.  dom  f ) )   =>    |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1124 28287 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th 
 <->  ( R  FrSe  A  /\  X  e.  A )
 )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   =>    |-  ( ( th  /\ 
 ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1133 28288* Technical lemma for bnj69 28309. 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.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ta  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. th ) )   &    |-  (
 ( i  e.  n  /\  ta )  ->  th )   =>    |-  ( ch  ->  A. i  e.  n  th )
 
Theorembnj1128 28289* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( th  <->  ( ch  ->  ( f `  i ) 
 C_  A ) )   &    |-  ( ta  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. th ) )   &    |-  ( ph'  <->  [. j  /  i ]. ph )   &    |-  ( ps'  <->  [. j  /  i ]. ps )   &    |-  ( ch'  <->  [. j  /  i ]. ch )   &    |-  ( th'  <->  [. j  /  i ]. th )   =>    |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
 
Theorembnj1127 28290 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
 
Theorembnj1125 28291 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R ) )  ->  trCl ( Y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
 
Theorembnj1145 28292* Technical lemma for bnj69 28309. 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.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( th  <->  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )   =>    |-  trCl
 ( X ,  A ,  R )  C_  A
 
Theorembnj1147 28293 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  trCl ( X ,  A ,  R )  C_  A
 
Theorembnj1137 28294* Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  TrFo ( B ,  A ,  R )
 )
 
Theorembnj1148 28295 Property of  pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  X  e.  A )  -> 
 pred ( X ,  A ,  R )  e.  _V )
 
Theorembnj1136 28296* Technical lemma for bnj69 28309. 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.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   =>    |-  ( ( R 
 FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
 
Theorembnj1152 28297 Technical lemma for bnj69 28309. 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.)
 |-  ( Y  e.  pred ( X ,  A ,  R ) 
 <->  ( Y  e.  A  /\  Y R X ) )
 
Theorembnj1154 28298* Property of  Fr. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/)  /\  B  e.  _V )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
 
Theorembnj1171 28299 Technical lemma for bnj69 28309. 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.)
 |-  (
 ( ph  /\  ps )  ->  B  C_  A )   &    |-  E. z A. w ( ( ph  /\ 
 ps )  ->  (
 z  e.  B  /\  ( w  e.  A  ->  ( w R z 
 ->  -.  w  e.  B ) ) ) )   =>    |-  E. z A. w ( ( ph  /\  ps )  ->  ( z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )
 
Theorembnj1172 28300 Technical lemma for bnj69 28309. 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.)
 |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )   &    |-  E. z A. w ( ( ph  /\ 
 ps )  ->  (
 ( ph  /\  ps  /\  z  e.  C )  /\  ( th  ->  ( w R z  ->  -.  w  e.  B ) ) ) )   &    |-  ( ( ph  /\ 
 ps  /\  z  e.  C )  ->  ( th  <->  w  e.  A ) )   =>    |-  E. z A. w ( ( ph  /\  ps )  ->  ( z  e.  B  /\  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B ) ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31423
  Copyright terms: Public domain < Previous  Next >