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Theorem List for Metamath Proof Explorer - 27901-28000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj1304 27901 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  E. x ps )   &    |-  ( ps  ->  ch )   &    |-  ( ps  ->  -. 
 ch )   =>    |- 
 -.  ph
 
Theorembnj1316 27902* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 y  e.  A  ->  A. x  y  e.  A )   &    |-  ( y  e.  B  ->  A. x  y  e.  B )   =>    |-  ( A  =  B  -> 
 U_ x  e.  A  C  =  U_ x  e.  B  C )
 
Theorembnj1317 27903* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  =  { x  |  ph }   =>    |-  ( y  e.  A  ->  A. x  y  e.  A )
 
Theorembnj1322 27904 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( A  =  B  ->  ( A  i^i  B )  =  A )
 
Theorembnj1340 27905 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps  ->  E. x th )   &    |-  ( ch 
 <->  ( ps  /\  th ) )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ps  ->  E. x ch )
 
Theorembnj1345 27906 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  E. x ( ps 
 /\  ch ) )   &    |-  ( th 
 <->  ( ph  /\  ps  /\ 
 ch ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( ph  ->  E. x th )
 
Theorembnj1350 27907* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch  ->  A. x ch )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  A. x (
 ph  /\  ps  /\  ch ) )
 
Theorembnj1351 27908* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   =>    |-  (
 ( ph  /\  ps )  ->  A. x ( ph  /\ 
 ps ) )
 
Theorembnj1352 27909* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   =>    |-  (
 ( ph  /\  ps )  ->  A. x ( ph  /\ 
 ps ) )
 
Theorembnj1361 27910* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ( x  e.  A  ->  x  e.  B ) )   =>    |-  ( ph  ->  A 
 C_  B )
 
Theorembnj1366 27911* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  ( ps 
 <->  ( A  e.  _V  /\ 
 A. x  e.  A  E! y ph  /\  B  =  { y  |  E. x  e.  A  ph } )
 )   =>    |-  ( ps  ->  B  e.  _V )
 
Theorembnj1379 27912* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 A. f  e.  A  Fun  f )   &    |-  D  =  ( dom  f  i^i  dom  g )   &    |-  ( ps  <->  ( ph  /\  A. f  e.  A  A. g  e.  A  ( f  |`  D )  =  (
 g  |`  D ) ) )   &    |-  ( ch  <->  ( ps  /\  <. x ,  y >.  e. 
 U. A  /\  <. x ,  z >.  e.  U. A ) )   &    |-  ( th 
 <->  ( ch  /\  f  e.  A  /\  <. x ,  y >.  e.  f ) )   &    |-  ( ta  <->  ( th  /\  g  e.  A  /\  <. x ,  z >.  e.  g ) )   =>    |-  ( ps  ->  Fun  U. A )
 
Theorembnj1383 27913* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 A. f  e.  A  Fun  f )   &    |-  D  =  ( dom  f  i^i  dom  g )   &    |-  ( ps  <->  ( ph  /\  A. f  e.  A  A. g  e.  A  ( f  |`  D )  =  (
 g  |`  D ) ) )   =>    |-  ( ps  ->  Fun  U. A )
 
Theorembnj1385 27914* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 A. f  e.  A  Fun  f )   &    |-  D  =  ( dom  f  i^i  dom  g )   &    |-  ( ps  <->  ( ph  /\  A. f  e.  A  A. g  e.  A  ( f  |`  D )  =  (
 g  |`  D ) ) )   &    |-  ( x  e.  A  ->  A. f  x  e.  A )   &    |-  ( ph'  <->  A. h  e.  A  Fun  h )   &    |-  E  =  ( dom  h  i^i  dom  g )   &    |-  ( ps'  <->  ( ph'  /\  A. h  e.  A  A. g  e.  A  ( h  |`  E )  =  (
 g  |`  E ) ) )   =>    |-  ( ps  ->  Fun  U. A )
 
Theorembnj1386 27915* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 A. f  e.  A  Fun  f )   &    |-  D  =  ( dom  f  i^i  dom  g )   &    |-  ( ps  <->  ( ph  /\  A. f  e.  A  A. g  e.  A  ( f  |`  D )  =  (
 g  |`  D ) ) )   &    |-  ( x  e.  A  ->  A. f  x  e.  A )   =>    |-  ( ps  ->  Fun  U. A )
 
Theorembnj1397 27916 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  E. x ps )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ph  ->  ps )
 
Theorembnj1400 27917* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 y  e.  A  ->  A. x  y  e.  A )   =>    |- 
 dom  U.  A  =  U_ x  e.  A  dom  x
 
Theorembnj1405 27918* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  X  e.  U_ y  e.  A  B )   =>    |-  ( ph  ->  E. y  e.  A  X  e.  B )
 
Theorembnj1422 27919 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  Fun  A )   &    |-  ( ph  ->  dom  A  =  B )   =>    |-  ( ph  ->  A  Fn  B )
 
Theorembnj1424 27920 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  =  ( B  u.  C )   =>    |-  ( D  e.  A  ->  ( D  e.  B  \/  D  e.  C ) )
 
Theorembnj1436 27921 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  =  { x  |  ph }   =>    |-  ( x  e.  A  -> 
 ph )
 
Theorembnj1441 27922* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( x  e.  A  ->  A. y  x  e.  A )   &    |-  ( ph  ->  A. y ph )   =>    |-  ( z  e.  { x  e.  A  |  ph
 }  ->  A. y  z  e.  { x  e.  A  |  ph } )
 
Theorembnj1454 27923 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  =  { x  |  ph }   =>    |-  ( B  e.  _V  ->  ( B  e.  A  <->  [. B  /  x ]. ph ) )
 
Theorembnj1459 27924* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <->  ( ph  /\  x  e.  A ) )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  A. x  e.  A  ch )
 
Theorembnj1464 27925* Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
 
Theorembnj1465 27926* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ( ps  ->  A. x ps )   &    |-  ( ch  ->  ps )   =>    |-  ( ( ch  /\  A  e.  V )  ->  E. x ph )
 
Theorembnj1468 27927* Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  e.  A  ->  A. x  y  e.  A )   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
 
Theorembnj1476 27928 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  { x  e.  A  |  -.  ph }   &    |-  ( ps  ->  D  =  (/) )   =>    |-  ( ps  ->  A. x  e.  A  ph )
 
Theorembnj1502 27929 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  Fun  F )   &    |-  ( ph  ->  G  C_  F )   &    |-  ( ph  ->  A  e.  dom  G )   =>    |-  ( ph  ->  ( F `  A )  =  ( G `  A ) )
 
Theorembnj1503 27930 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  Fun  F )   &    |-  ( ph  ->  G  C_  F )   &    |-  ( ph  ->  A  C_ 
 dom  G )   =>    |-  ( ph  ->  ( F  |`  A )  =  ( G  |`  A ) )
 
Theorembnj1517 27931 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  =  { x  |  (
 ph  /\  ps ) }   =>    |-  ( x  e.  A  ->  ps )
 
Theorembnj1521 27932 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch  ->  E. x  e.  B  ph )   &    |-  ( th  <->  ( ch  /\  x  e.  B  /\  ph ) )   &    |-  ( ch  ->  A. x ch )   =>    |-  ( ch  ->  E. x th )
 
Theorembnj1533 27933 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th  ->  A. z  e.  B  -.  z  e.  D )   &    |-  B  C_  A   &    |-  D  =  {
 z  e.  A  |  C  =/=  E }   =>    |-  ( th  ->  A. z  e.  B  C  =  E )
 
Theorembnj1534 27934* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }   &    |-  ( w  e.  F  ->  A. x  w  e.  F )   =>    |-  D  =  { z  e.  A  |  ( F `
  z )  =/=  ( H `  z
 ) }
 
Theorembnj1536 27935* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  A )   &    |-  ( ph  ->  B 
 C_  A )   &    |-  ( ph  ->  A. x  e.  B  ( F `  x )  =  ( G `  x ) )   =>    |-  ( ph  ->  ( F  |`  B )  =  ( G  |`  B ) )
 
Theorembnj1538 27936 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  =  { x  e.  B  |  ph }   =>    |-  ( x  e.  A  -> 
 ph )
 
Theorembnj1541 27937 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( ps  /\  A  =/=  B ) )   &    |-  -.  ph   =>    |-  ( ps  ->  A  =  B )
 
Theorembnj1542 27938* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  A )   &    |-  ( ph  ->  F  =/=  G )   &    |-  ( w  e.  F  ->  A. x  w  e.  F )   =>    |-  ( ph  ->  E. x  e.  A  ( F `  x )  =/=  ( G `  x ) )
 
18.24.2  Well founded induction and recursion
 
Theorembnj110 27939* Well-founded induction restricted to a set ( A  e.  _V). 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.)
 |-  A  e.  _V   &    |-  ( ps  <->  A. y  e.  A  ( y R x 
 ->  [. y  /  x ].
 ph ) )   =>    |-  ( ( R  Fr  A  /\  A. x  e.  A  ( ps  ->  ph ) )  ->  A. x  e.  A  ph )
 
Theorembnj157 27940* Well-founded induction restricted to a set ( A  e.  _V). 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 ) )   &    |-  A  e.  _V   &    |-  R  Fr  A   =>    |-  ( A. x  e.  A  ( ps  ->  ph )  ->  A. x  e.  A  ph )
 
Theorembnj66 27941* Technical lemma for bnj60 28141. 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 ) ) }   =>    |-  (
 g  e.  C  ->  Rel  g )
 
Theorembnj91 27942* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  Z  e.  _V   =>    |-  ( [. Z  /  y ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
 
Theorembnj92 27943* First-order logic and set theory. (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 ) ) )   &    |-  Z  e.  _V   =>    |-  ( [. Z  /  n ].
 ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  Z  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj93 27944* Technical lemma for bnj97 27947. 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.)
 |-  (
 ( R  FrSe  A  /\  x  e.  A )  -> 
 pred ( x ,  A ,  R )  e.  _V )
 
Theorembnj95 27945 Technical lemma for bnj124 27952. 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.)
 |-  F  =  { <. (/) ,  pred ( x ,  A ,  R ) >. }   =>    |-  F  e.  _V
 
Theorembnj96 27946* Technical lemma for bnj150 27957. 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.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
 |-  F  =  { <. (/) ,  pred ( x ,  A ,  R ) >. }   =>    |-  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  dom  F  =  1o )
 
Theorembnj97 27947* Technical lemma for bnj150 27957. 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.)
 |-  F  =  { <. (/) ,  pred ( x ,  A ,  R ) >. }   =>    |-  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
 
Theorembnj98 27948 Technical lemma for bnj150 27957. 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.)
 |-  A. i  e.  om  ( suc  i  e.  1o  ->  ( F ` 
 suc  i )  = 
 U_ y  e.  ( F `  i )  pred ( y ,  A ,  R ) )
 
Theorembnj106 27949* First-order logic and set theory. (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 ) ) )   &    |-  F  e.  _V   =>    |-  ( [. F  /  f ]. [. 1o  /  n ].
 ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj118 27950* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ph'  <->  [. 1o  /  n ].
 ph )   =>    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
 
Theorembnj121 27951* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ze 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )   &    |-  ( ze'  <->  [. 1o  /  n ]. ze )   &    |-  ( ph'  <->  [. 1o  /  n ].
 ph )   &    |-  ( ps'  <->  [. 1o  /  n ].
 ps )   =>    |-  ( ze'  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  (
 f  Fn  1o  /\  ph'  /\  ps' ) ) )
 
Theorembnj124 27952* Technical lemma for bnj150 27957. 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.)
 |-  F  =  { <. (/) ,  pred ( x ,  A ,  R ) >. }   &    |-  ( ph"  <->  [. F  /  f ]. ph' )   &    |-  ( ps"  <->  [. F  /  f ]. ps' )   &    |-  ( ze"  <->  [. F  /  f ]. ze' )   &    |-  ( ze'  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( f  Fn  1o  /\  ph'  /\  ps' ) ) )   =>    |-  ( ze"  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  ( F  Fn  1o  /\  ph"  /\  ps" ) ) )
 
Theorembnj125 27953* Technical lemma for bnj150 27957. 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'  <->  [. 1o  /  n ].
 ph )   &    |-  ( ph"  <->  [. F  /  f ]. ph' )   &    |-  F  =  { <.
 (/) ,  pred ( x ,  A ,  R ) >. }   =>    |-  ( ph"  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
 
Theorembnj126 27954* Technical lemma for bnj150 27957. 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'  <->  [. 1o  /  n ].
 ps )   &    |-  ( ps"  <->  [. F  /  f ]. ps' )   &    |-  F  =  { <.
 (/) ,  pred ( x ,  A ,  R ) >. }   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj130 27955* Technical lemma for bnj151 27958. 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 )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( ph'  <->  [. 1o  /  n ].
 ph )   &    |-  ( ps'  <->  [. 1o  /  n ].
 ps )   &    |-  ( th'  <->  [. 1o  /  n ].
 th )   =>    |-  ( th'  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn 
 1o  /\  ph'  /\  ps' ) ) )
 
Theorembnj149 27956* Technical lemma for bnj151 27958. 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.)
 |-  ( th1  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E* f ( f  Fn  1o  /\  ph'  /\  ps' ) ) )   &    |-  ( ze0  <->  ( f  Fn 
 1o  /\  ph'  /\  ps' ) )   &    |-  ( ze1  <->  [. g  /  f ]. ze0 )   &    |-  ( ph1  <->  [. g  /  f ]. ph' )   &    |-  ( ps1  <->  [. g  /  f ]. ps' )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   =>    |-  th1
 
Theorembnj150 27957* Technical lemma for bnj151 27958. 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 ) ) )   &    |-  ( ze  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  (
 f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( ph'  <->  [. 1o  /  n ].
 ph )   &    |-  ( ps'  <->  [. 1o  /  n ].
 ps )   &    |-  ( th0  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  E. f
 ( f  Fn  1o  /\  ph'  /\  ps' ) ) )   &    |-  ( ze'  <->  [. 1o  /  n ]. ze )   &    |-  F  =  { <.
 (/) ,  pred ( x ,  A ,  R ) >. }   &    |-  ( ph"  <->  [. F  /  f ]. ph' )   &    |-  ( ps"  <->  [. F  /  f ]. ps' )   &    |-  ( ze"  <->  [. F  /  f ]. ze' )   =>    |-  th0
 
Theorembnj151 27958* Technical lemma for bnj153 27961. 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  \  { (/) } )   &    |-  ( th 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( ta  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 th ) )   &    |-  ( ze 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )   &    |-  ( ph'  <->  [. 1o  /  n ]. ph )   &    |-  ( ps'  <->  [. 1o  /  n ].
 ps )   &    |-  ( th'  <->  [. 1o  /  n ].
 th )   &    |-  ( th0  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  E. f
 ( f  Fn  1o  /\  ph'  /\  ps' ) ) )   &    |-  ( th1  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E* f ( f  Fn  1o  /\  ph'  /\  ps' ) ) )   &    |-  ( ze'  <->  [. 1o  /  n ].
 ze )   &    |-  F  =  { <.
 (/) ,  pred ( x ,  A ,  R ) >. }   &    |-  ( ph"  <->  [. F  /  f ]. ph' )   &    |-  ( ps"  <->  [. F  /  f ]. ps' )   &    |-  ( ze"  <->  [. F  /  f ]. ze' )   &    |-  ( ze0  <->  ( f  Fn 
 1o  /\  ph'  /\  ps' ) )   &    |-  ( ze1  <->  [. g  /  f ]. ze0 )   &    |-  ( ph1  <->  [. g  /  f ]. ph' )   &    |-  ( ps1  <->  [. g  /  f ]. ps' )   =>    |-  ( n  =  1o  ->  ( ( n  e.  D  /\  ta )  ->  th ) )
 
Theorembnj154 27959* Technical lemma for bnj153 27961. 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.)
 |-  ( ph1  <->  [. g  /  f ]. ph' )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   =>    |-  ( ph1  <->  ( g `  (/) )  =  pred ( x ,  A ,  R ) )
 
Theorembnj155 27960* Technical lemma for bnj153 27961. 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.)
 |-  ( ps1  <->  [. g  /  f ]. ps' )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ps1  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( g `  suc  i )  =  U_ y  e.  ( g `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj153 27961* Technical lemma for bnj852 28002. 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  \  { (/) } )   &    |-  ( th 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( ta  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 th ) )   =>    |-  ( n  =  1o  ->  ( ( n  e.  D  /\  ta )  ->  th )
 )
 
Theorembnj207 27962* Technical lemma for bnj852 28002. 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 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( ph'  <->  [. M  /  n ].
 ph )   &    |-  ( ps'  <->  [. M  /  n ].
 ps )   &    |-  ( ch'  <->  [. M  /  n ].
 ch )   &    |-  M  e.  _V   =>    |-  ( ch'  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  M  /\  ph'  /\  ps' ) ) )
 
Theorembnj213 27963 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  pred ( X ,  A ,  R )  C_  A
 
Theorembnj222 27964* Technical lemma for bnj229 27965. 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. m  e.  om  ( suc  m  e.  N  ->  ( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred ( y ,  A ,  R ) ) )
 
Theorembnj229 27965* Technical lemma for bnj517 27966. 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 ) ) )   =>    |-  ( ( n  e.  N  /\  ( suc 
 m  =  n  /\  m  e.  om  /\  ps ) )  ->  ( F `
  n )  C_  A )
 
Theorembnj517 27966* Technical lemma for bnj518 27967. 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 ) ) )   =>    |-  ( ( N  e.  om 
 /\  ph  /\  ps )  ->  A. n  e.  N  ( F `  n ) 
 C_  A )
 
Theorembnj518 27967* Technical lemma for bnj852 28002. 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 ) ) )   &    |-  ( ta  <->  ( ph  /\  ps  /\  n  e.  om  /\  p  e.  n )
 )   =>    |-  ( ( R  FrSe  A 
 /\  ta )  ->  A. y  e.  ( f `  p )  pred ( y ,  A ,  R )  e.  _V )
 
Theorembnj523 27968* Technical lemma for bnj852 28002. 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'  <->  [. M  /  n ].
 ph )   &    |-  M  e.  _V   =>    |-  ( ph'  <->  ( F `  (/) )  = 
 pred ( X ,  A ,  R )
 )
 
Theorembnj526 27969* Technical lemma for bnj852 28002. 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 ) )
 
Theorembnj528 27970 Technical lemma for bnj852 28002. 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.)
 |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   =>    |-  G  e.  _V
 
Theorembnj535 27971* Technical lemma for bnj852 28002. 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.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m ) )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  n  =  ( m  u.  { m } )  /\  f  Fn  m )  ->  G  Fn  n )
 
Theorembnj539 27972* Technical lemma for bnj852 28002. 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'  <->  [. M  /  n ].
 ps )   &    |-  M  e.  _V   =>    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  M  ->  ( F `  suc  i
 )  =  U_ y  e.  ( F `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj540 27973* Technical lemma for bnj852 28002. 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 ) ) )
 
Theorembnj543 27974* Technical lemma for bnj852 28002. 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.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  e.  m ) )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
 
Theorembnj544 27975* Technical lemma for bnj852 28002. 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.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  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 ) )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
 
Theorembnj545 27976 Technical lemma for bnj852 28002. 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 )
 )   &    |-  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 ) )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  si )  ->  G  Fn  n )   &    |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  ph" )
 
Theorembnj546 27977* Technical lemma for bnj852 28002. 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 
 /\  ta  /\  si )  -> 
 U_ y  e.  (
 f `  p )  pred ( y ,  A ,  R )  e.  _V )
 
Theorembnj548 27978* Technical lemma for bnj852 28002. 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.)
 |-  ( ta 
 <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  B  =  U_ y  e.  (
 f `  i )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  si )  ->  G  Fn  n )   =>    |-  ( ( ( R 
 FrSe  A  /\  ta  /\  si )  /\  i  e.  m )  ->  B  =  K )
 
Theorembnj553 27979* Technical lemma for bnj852 28002. 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.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  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 ) )   &    |-  C  =  U_ y  e.  (
 f `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  B  =  U_ y  e.  (
 f `  i )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i )  pred (
 y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred (
 y ,  A ,  R )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )   =>    |-  ( ( ( R 
 FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i )  ->  ( G `  m )  =  L )
 
Theorembnj554 27980* Technical lemma for bnj852 28002. 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 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  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 )   =>    |-  ( ( et  /\  ze )  ->  ( ( G `  m )  =  L  <->  ( G `  suc  i )  =  K ) )
 
Theorembnj556 27981 Technical lemma for bnj852 28002. 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 27982* Technical lemma for bnj852 28002. 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 27983* Technical lemma for bnj852 28002. 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 27984 Technical lemma for bnj852 28002. 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 27985 Technical lemma for bnj852 28002. 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 27986* Technical lemma for bnj852 28002. 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 27987* Technical lemma for bnj852 28002. 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 27988* 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 27989* Technical lemma for bnj580 27994. 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 27990* Technical lemma for bnj852 28002. 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 27991 Technical lemma for bnj852 28002. 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 27992* Technical lemma for bnj852 28002. 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 27993* Technical lemma for bnj852 28002. 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 27994* Technical lemma for bnj579 27995. 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 27995* Technical lemma for bnj852 28002. 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 27996 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 27997* Technical lemma for bnj852 28002. 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 27998* Technical lemma for bnj852 28002. 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 27999* Technical lemma for bnj852 28002. 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 28000* Technical lemma for bnj852 28002. 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 ) )
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