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Theorem List for Metamath Proof Explorer - 29201-29300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj1083 29201 Technical lemma for bnj69 29233. 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 29202* Technical lemma for bnj69 29233. 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 29203* Technical lemma for bnj69 29233. 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 29204 Technical lemma for bnj69 29233. 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 29205* Technical lemma for bnj69 29233. 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 29206* Technical lemma for bnj69 29233. 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 29207* Technical lemma for bnj69 29233. 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 29208 Technical lemma for bnj69 29233. 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 29209* Technical lemma for bnj69 29233. 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 29210* Technical lemma for bnj69 29233. 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 29211 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 29212* Technical lemma for bnj69 29233. 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 29213* Technical lemma for bnj69 29233. 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 29214 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 29215 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 29216* Technical lemma for bnj69 29233. 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 29217 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  trCl ( X ,  A ,  R )  C_  A
 
Theorembnj1137 29218* 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 29219 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 29220* Technical lemma for bnj69 29233. 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 29221 Technical lemma for bnj69 29233. 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 29222* 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 29223 Technical lemma for bnj69 29233. 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 29224 Technical lemma for bnj69 29233. 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 ) ) ) )
 
Theorembnj1173 29225 Technical lemma for bnj69 29233. 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 )   &    |-  ( th 
 <->  ( ( R  FrSe  A 
 /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A ) )   &    |-  ( ( ph  /\ 
 ps )  ->  R  FrSe  A )   &    |-  ( ( ph  /\ 
 ps )  ->  X  e.  A )   =>    |-  ( ( ph  /\  ps  /\  z  e.  C ) 
 ->  ( th  <->  w  e.  A ) )
 
Theorembnj1174 29226 Technical lemma for bnj69 29233. 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 )  ->  (
 z  e.  C  /\  ( th  ->  ( w R z  ->  -.  w  e.  C ) ) ) )   &    |-  ( th  ->  ( w R z  ->  w  e.  trCl ( X ,  A ,  R ) ) )   =>    |-  E. z A. w ( ( ph  /\ 
 ps )  ->  (
 ( ph  /\  ps  /\  z  e.  C )  /\  ( th  ->  ( w R z  ->  -.  w  e.  B ) ) ) )
 
Theorembnj1175 29227 Technical lemma for bnj69 29233. 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 )   &    |-  ( ch 
 <->  ( ( R  FrSe  A 
 /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  ( w  e.  A  /\  w R z ) ) )   &    |-  ( th  <->  ( ( R 
 FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A ) )   =>    |-  ( th  ->  ( w R z  ->  w  e.  trCl ( X ,  A ,  R )
 ) )
 
Theorembnj1176 29228* Technical lemma for bnj69 29233. 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 )  ->  ( R  Fr  A  /\  C  C_  A  /\  C  =/=  (/)  /\  C  e.  _V ) )   &    |-  ( ( R  Fr  A  /\  C  C_  A  /\  C  =/=  (/)  /\  C  e.  _V )  ->  E. z  e.  C  A. w  e.  C  -.  w R z )   =>    |-  E. z A. w ( ( ph  /\ 
 ps )  ->  (
 z  e.  C  /\  ( th  ->  ( w R z  ->  -.  w  e.  C ) ) ) )
 
Theorembnj1177 29229 Technical lemma for bnj69 29233. 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 
 <->  ( X  e.  B  /\  y  e.  B  /\  y R X ) )   &    |-  C  =  ( 
 trCl ( X ,  A ,  R )  i^i  B )   &    |-  ( ( ph  /\ 
 ps )  ->  R  FrSe  A )   &    |-  ( ( ph  /\ 
 ps )  ->  B  C_  A )   &    |-  ( ( ph  /\ 
 ps )  ->  X  e.  A )   =>    |-  ( ( ph  /\  ps )  ->  ( R  Fr  A  /\  C  C_  A  /\  C  =/=  (/)  /\  C  e.  _V ) )
 
Theorembnj1186 29230* Technical lemma for bnj69 29233. 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. z A. w ( ( ph  /\ 
 ps )  ->  (
 z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  E. z  e.  B  A. w  e.  B  -.  w R z )
 
Theorembnj1190 29231* Technical lemma for bnj69 29233. 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 
 <->  ( R  FrSe  A  /\  B  C_  A  /\  B  =/= 
 (/) ) )   &    |-  ( ps 
 <->  ( x  e.  B  /\  y  e.  B  /\  y R x ) )   =>    |-  ( ( ph  /\  ps )  ->  E. w  e.  B  A. z  e.  B  -.  z R w )
 
Theorembnj1189 29232* Technical lemma for bnj69 29233. 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 
 <->  ( R  FrSe  A  /\  B  C_  A  /\  B  =/= 
 (/) ) )   &    |-  ( ps 
 <->  ( x  e.  B  /\  y  e.  B  /\  y R x ) )   &    |-  ( ch  <->  A. y  e.  B  -.  y R x )   =>    |-  ( ph  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
 
19.25.3  The existence of a minimal element in certain classes
 
Theorembnj69 29233* Existence of a minimal element in certain classes: if  R is well-founded and set-like on 
A, then every non-empty subclass of  A has a minimal element. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  B  C_  A  /\  B  =/= 
 (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
 
Theorembnj1228 29234* Existence of a minimal element in certain classes: if  R is well-founded and set-like on 
A, then every non-empty subclass of  A has a minimal element. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( w  e.  B  ->  A. x  w  e.  B )   =>    |-  ( ( R  FrSe  A 
 /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
 
19.25.4  Well-founded induction
 
Theorembnj1204 29235* Well-founded induction. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. y  e.  A  ( y R x 
 ->  [. y  /  x ].
 ph ) )   =>    |-  ( ( R 
 FrSe  A  /\  A. x  e.  A  ( ps  ->  ph ) )  ->  A. x  e.  A  ph )
 
Theorembnj1234 29236* Technical lemma for bnj60 29285. 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  =  <. x ,  (
 f  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  C  =  { f  |  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  Z  =  <. x ,  (
 g  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  D  =  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) }   =>    |-  C  =  D
 
Theorembnj1245 29237* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   &    |-  Z  =  <. x ,  ( h  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  K  =  { h  |  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  Z ) ) }   =>    |-  ( ph  ->  dom  h  C_  A )
 
Theorembnj1256 29238* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   =>    |-  ( ph  ->  E. d  e.  B  g  Fn  d )
 
Theorembnj1259 29239* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   =>    |-  ( ph  ->  E. d  e.  B  h  Fn  d )
 
Theorembnj1253 29240* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   =>    |-  ( ph  ->  E  =/=  (/) )
 
Theorembnj1279 29241* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   =>    |-  ( ( x  e.  E  /\  A. y  e.  E  -.  y R x )  ->  (  pred ( x ,  A ,  R )  i^i  E )  =  (/) )
 
Theorembnj1286 29242* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   =>    |-  ( ps  ->  pred
 ( x ,  A ,  R )  C_  D )
 
Theorembnj1280 29243* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   &    |-  ( ps  ->  (  pred ( x ,  A ,  R )  i^i  E )  =  (/) )   =>    |-  ( ps  ->  (
 g  |`  pred ( x ,  A ,  R )
 )  =  ( h  |`  pred ( x ,  A ,  R )
 ) )
 
Theorembnj1296 29244* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   &    |-  ( ps  ->  ( g  |`  pred
 ( x ,  A ,  R ) )  =  ( h  |`  pred ( x ,  A ,  R ) ) )   &    |-  Z  =  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.   &    |-  K  =  { g  |  E. d  e.  B  (
 g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) }   &    |-  W  =  <. x ,  ( h  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  L  =  { h  |  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  W ) ) }   =>    |-  ( ps  ->  ( g `  x )  =  ( h `  x ) )
 
Theorembnj1309 29245* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   =>    |-  ( w  e.  B  ->  A. x  w  e.  B )
 
Theorembnj1307 29246* Technical lemma for bnj60 29285. 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  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( w  e.  B  ->  A. x  w  e.  B )   =>    |-  ( w  e.  C  ->  A. x  w  e.  C )
 
Theorembnj1311 29247* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   =>    |-  ( ( R 
 FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  (
 g  |`  D )  =  ( h  |`  D ) )
 
Theorembnj1318 29248 Technical lemma for bnj60 29285. 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.)
 |-  ( X  =  Y  ->  trCl
 ( X ,  A ,  R )  =  trCl ( Y ,  A ,  R ) )
 
Theorembnj1326 29249* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   =>    |-  ( ( R 
 FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  (
 g  |`  D )  =  ( h  |`  D ) )
 
Theorembnj1321 29250* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  E. f ta )  ->  E! f ta )
 
Theorembnj1364 29251 Property of  FrSe. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( R  FrSe  A  ->  R  Se  A )
 
Theorembnj1371 29252* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  ( ta'  <->  ( f  e.  C  /\  dom  f  =  ( {
 y }  u.  trCl ( y ,  A ,  R ) ) ) )   =>    |-  ( f  e.  H  ->  Fun  f )
 
Theorembnj1373 29253* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  ( ta'  <->  [. y  /  x ].
 ta )   =>    |-  ( ta'  <->  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
 
Theorembnj1374 29254* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   =>    |-  ( f  e.  H  ->  f  e.  C )
 
Theorembnj1384 29255* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   =>    |-  ( R  FrSe  A  ->  Fun  P )
 
Theorembnj1388 29256* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   =>    |-  ( ch  ->  A. y  e.  pred  ( x ,  A ,  R ) E. f ta' )
 
Theorembnj1398 29257* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  ( th 
 <->  ( ch  /\  z  e.  U_ y  e.  pred  ( x ,  A ,  R ) ( {
 y }  u.  trCl ( y ,  A ,  R ) ) ) )   &    |-  ( et  <->  ( th  /\  y  e.  pred ( x ,  A ,  R )  /\  z  e.  ( { y }  u.  trCl
 ( y ,  A ,  R ) ) ) )   =>    |-  ( ch  ->  U_ y  e.  pred  ( x ,  A ,  R )
 ( { y }  u.  trCl ( y ,  A ,  R ) )  =  dom  P )
 
Theorembnj1413 29258* Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  B  e.  _V )
 
Theorembnj1408 29259* Technical lemma for bnj1414 29260. 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.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   &    |-  C  =  ( 
 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 )
 
Theorembnj1414 29260* Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
 
Theorembnj1415 29261* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   =>    |-  ( ch  ->  dom  P  =  trCl ( x ,  A ,  R ) )
 
Theorembnj1416 29262 Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  dom 
 P  =  trCl ( x ,  A ,  R ) )   =>    |-  ( ch  ->  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
 
Theorembnj1418 29263 Property of  pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 y  e.  pred ( x ,  A ,  R )  ->  y R x )
 
Theorembnj1417 29264* Technical lemma for bnj60 29285. 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 
 <->  R  FrSe  A )   &    |-  ( ps 
 <->  -.  x  e.  trCl ( x ,  A ,  R ) )   &    |-  ( ch 
 <-> 
 A. y  e.  A  ( y R x 
 ->  [. y  /  x ].
 ps ) )   &    |-  ( th 
 <->  ( ph  /\  x  e.  A  /\  ch )
 )   &    |-  B  =  (  pred ( x ,  A ,  R )  u.  U_ y  e.  pred  ( x ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ph  ->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
 
Theorembnj1421 29265* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  Fun 
 P )   &    |-  ( ch  ->  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )   &    |-  ( ch  ->  dom  P  =  trCl ( x ,  A ,  R ) )   =>    |-  ( ch  ->  Fun 
 Q )
 
Theorembnj1444 29266* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e.  dom  f ) )   =>    |-  ( rh  ->  A. y rh )
 
Theorembnj1445 29267* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e.  dom  f ) )   &    |-  ( si  <->  ( rh  /\  y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl
 ( y ,  A ,  R ) ) ) )   &    |-  ( ph  <->  ( si  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  (
 f `  x )  =  ( G `  Y ) ) )   &    |-  X  =  <. z ,  (
 f  |`  pred ( z ,  A ,  R ) ) >.   =>    |-  ( si  ->  A. d si )
 
Theorembnj1446 29268* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   =>    |-  ( ( Q `
  z )  =  ( G `  W )  ->  A. d ( Q `
  z )  =  ( G `  W ) )
 
Theorembnj1447 29269* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   =>    |-  ( ( Q `
  z )  =  ( G `  W )  ->  A. y ( Q `
  z )  =  ( G `  W ) )
 
Theorembnj1448 29270* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   =>    |-  ( ( Q `
  z )  =  ( G `  W )  ->  A. f ( Q `
  z )  =  ( G `  W ) )
 
Theorembnj1449 29271* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   =>    |-  ( ze  ->  A. f ze )
 
Theorembnj1442 29272* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   =>    |-  ( et  ->  ( Q `  z )  =  ( G `  W ) )
 
Theorembnj1450 29273* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e.  dom  f ) )   &    |-  ( si  <->  ( rh  /\  y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl
 ( y ,  A ,  R ) ) ) )   &    |-  ( ph  <->  ( si  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  (
 f `  x )  =  ( G `  Y ) ) )   &    |-  X  =  <. z ,  (
 f  |`  pred ( z ,  A ,  R ) ) >.   =>    |-  ( ze  ->  ( Q `  z )  =  ( G `  W ) )
 
Theorembnj1423 29274* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   =>    |-  ( ch  ->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )
 
Theorembnj1452 29275* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   =>    |-  ( ch  ->  E  e.  B )
 
Theorembnj1466 29276* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( w  e.  Q  ->  A. f  w  e.  Q )
 
Theorembnj1467 29277* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( w  e.  Q  ->  A. d  w  e.  Q )
 
Theorembnj1463 29278* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  e.  _V )   &    |-  ( ch  ->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )   &    |-  ( ch  ->  Q  Fn  E )   &    |-  ( ch  ->  E  e.  B )   =>    |-  ( ch  ->  Q  e.  C )
 
Theorembnj1489 29279* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( ch  ->  Q  e.  _V )
 
Theorembnj1491 29280* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  ( Q  e.  C  /\  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )   =>    |-  ( ( ch  /\  Q  e.  _V )  ->  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
 
Theorembnj1312 29281* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
 
Theorembnj1493 29282* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )
 
Theorembnj1497 29283* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  A. g  e.  C  Fun  g
 
Theorembnj1498 29284* Technical lemma for bnj60 29285. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  dom 
 F  =  A )
 
19.25.5  Well-founded recursion, part 1 of 3
 
Theorembnj60 29285* Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  F  Fn  A )
 
Theorembnj1514 29286* Technical lemma for bnj1500 29291. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  (
 f  e.  C  ->  A. x  e.  dom  f
 ( f `  x )  =  ( G `  Y ) )
 
Theorembnj1518 29287* Technical lemma for bnj1500 29291. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  x  e.  A ) )   &    |-  ( ps  <->  ( ph  /\  f  e.  C  /\  x  e. 
 dom  f ) )   =>    |-  ( ps  ->  A. d ps )
 
Theorembnj1519 29288* Technical lemma for bnj1500 29291. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. )  ->  A. d
 ( F `  x )  =  ( G ` 
 <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1520 29289* Technical lemma for bnj1500 29291. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. )  ->  A. f
 ( F `  x )  =  ( G ` 
 <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1501 29290* Technical lemma for bnj1500 29291. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  x  e.  A ) )   &    |-  ( ps  <->  ( ph  /\  f  e.  C  /\  x  e. 
 dom  f ) )   &    |-  ( ch  <->  ( ps  /\  d  e.  B  /\  dom  f  =  d ) )   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
19.25.6  Well-founded recursion, part 2 of 3
 
Theorembnj1500 29291* Well-founded recursion, part 2 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1525 29292* Technical lemma for bnj1522 29295. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) ) )   &    |-  ( ps  <->  ( ph  /\  F  =/=  H ) )   =>    |-  ( ps  ->  A. x ps )
 
Theorembnj1529 29293* Technical lemma for bnj1522 29295. 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  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )   &    |-  ( w  e.  F  ->  A. x  w  e.  F )   =>    |-  ( ch  ->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
 
Theorembnj1523 29294* Technical lemma for bnj1522 29295. 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  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) ) )   &    |-  ( ps  <->  ( ph  /\  F  =/=  H ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  A  /\  ( F `
  x )  =/=  ( H `  x ) ) )   &    |-  D  =  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }   &    |-  ( th 
 <->  ( ch  /\  y  e.  D  /\  A. z  e.  D  -.  z R y ) )   =>    |-  ( ( R 
 FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) )  ->  F  =  H )
 
19.25.7  Well-founded recursion, part 3 of 3
 
Theorembnj1522 29295* Well-founded recursion, part 3 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) )  ->  F  =  H )
 
19.26  Mathbox for Norm Megill

Note: A label suffixed with "N" (after the "Atoms..." section below), such as lshpnel2N 29622, means that the definition or theorem is not used for the derivation of hlathil 32601. This is a temporary renaming to assist cleaning up the theorems needed by hlathil 32601.

Please inform me of any changes that might affect my mathbox, since I may be working on it independently of the github commits. - Norm 30-Nov-2015

 
19.26.1  Experiments to study ax-7 unbundling

This section reproduces all predicate calculus theorems through sbal2 2210 that depend on ax-7 1749. It is an experiment to see how much of predicate calculus can be derived using the weaker (unbundled) ax-7v 29296.

The theorems in this section with suffix "NEW7" are direct replacements for the existing ones without the suffix but have proofs that avoid ax-7 1749 in favor of ax-7v 29296.

Theorems with suffix "AUX7" are new theorems that do not appear in the main predicate calculus section but assist the proofs of the "NEW7" suffixed theorems. They also use at most ax-7v 29296 and not ax-7 1749.

Theorems with suffix "OLD7" are the remaining predicate calculus theorems (through sbal2 2210) that haven't been proved from ax-7v 29296. In order to isolate them, they are derived from ax-7OLD7 29517 which replicates ax-7 1749. Whenever a proof of a *OLD7 theorem is found from ax-7v 29296, the suffix is changed to "NEW7" and the theorem is moved up to the "NEW7" section.

Theorems with suffix "AUXOLD7" (currently just nfsb4tw2AUXOLD7 29585) are results of an unsuccessful attempt to prove a helper theorem from ax-7v 29296, but still needs the help of ax-7 1749.

Currently there are about 138 "NEW7" theorems (starting after ax-7v 29296) and 90 "OLD7" theorems (starting after ax-7OLD7 29517).

 
19.26.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)
 
Axiomax-7v 29296* Experiment to see if ax-7 1749 can be unbundled i.e. can be derived from ax-7v 29296. This axiom is temporary. It will be replaced with a theorem derived from ax-7 1749 if we are successful, otherwise will be deleted. (Contributed by NM, 9-Oct-2017.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax7vAUX7 29297* A weaker version of ax-7 1749 with distinct variable restrictions. In order to show that this weakening is adequate, this should be the only theorem referencing ax-7 1749 directly.

(Right now we derive this from the temporary axiom ax-7v 29296 for easier 'show trace'. If this project is successful, which is seeming more and more unlikely, we will derive this from ax-7 1749 just as we derive ax9v 1667 from ax-9 1666.)

(Contributed by NM, 9-Oct-2017.)

 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
TheoremalcomwAUX7 29298* Weak version of alcom 1752 not requiring ax-7 1749. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x A. y ph  <->  A. y A. x ph )
 
Theorema7swAUX7 29299* Weak version of a7s 1750 not requiring ax-7 1749. (Contributed by NM, 28-Oct-2017.)
 |-  ( A. x A. y ph  ->  ps )   =>    |-  ( A. y A. x ph  ->  ps )
 
Theoremcbv3hvNEW7 29300* Lemma for ax10NEW7 29327. Similar to cbv3h 1972. Requires distinct variables but avoids ax-12 1950. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 25-Nov-2017.) Revised to prove from ax-7v 29296 instead of ax-7 1749.
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
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