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Theorem List for Metamath Proof Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremunxpdomlem2 7001* Lemma for unxpdom 7003. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |-  F  =  ( x  e.  ( a  u.  b )  |->  G )   &    |-  G  =  if ( x  e.  a ,  <. x ,  if ( x  =  m ,  t ,  s ) >. ,  <. if ( x  =  t ,  n ,  m ) ,  x >. )   &    |-  ( ph  ->  w  e.  ( a  u.  b ) )   &    |-  ( ph  ->  -.  m  =  n )   &    |-  ( ph  ->  -.  s  =  t )   =>    |-  ( ( ph  /\  (
 z  e.  a  /\  -.  w  e.  a ) )  ->  -.  ( F `  z )  =  ( F `  w ) )
 
Theoremunxpdomlem3 7002* Lemma for unxpdom 7003. (Contributed by Mario Carneiro, 13-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  F  =  ( x  e.  ( a  u.  b )  |->  G )   &    |-  G  =  if ( x  e.  a ,  <. x ,  if ( x  =  m ,  t ,  s ) >. ,  <. if ( x  =  t ,  n ,  m ) ,  x >. )   =>    |-  ( ( 1o  ~<  a 
 /\  1o  ~<  b ) 
 ->  ( a  u.  b
 )  ~<_  ( a  X.  b ) )
 
Theoremunxpdom 7003 Cross product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 13-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( 1o  ~<  A 
 /\  1o  ~<  B ) 
 ->  ( A  u.  B ) 
 ~<_  ( A  X.  B ) )
 
Theoremunxpdom2 7004 Corollary of unxpdom 7003. (Contributed by NM, 16-Sep-2004.)
 |-  ( ( 1o  ~<  A 
 /\  B  ~<_  A ) 
 ->  ( A  u.  B ) 
 ~<_  ( A  X.  A ) )
 
Theoremsucxpdom 7005 Cross product dominates successor for set with cardinality greater than 1. Proposition 10.38 of [TakeutiZaring] p. 93 (but generalized to arbitrary sets, not just ordinals). (Contributed by NM, 3-Sep-2004.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
 |-  ( 1o  ~<  A  ->  suc 
 A  ~<_  ( A  X.  A ) )
 
Theorempssinf 7006 A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.)
 |-  ( ( A  C.  B  /\  A  ~~  B )  ->  -.  B  e.  Fin )
 
Theoremfisseneq 7007 A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
 |-  ( ( B  e.  Fin  /\  A  C_  B  /\  A  ~~  B )  ->  A  =  B )
 
Theoremominf 7008 The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.)
 |- 
 -.  om  e.  Fin
 
Theoremisinf 7009* Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by Mario Carneiro, 15-Jan-2013.)
 |-  ( -.  A  e.  Fin 
 ->  A. n  e.  om  E. x ( x  C_  A  /\  x  ~~  n ) )
 
Theoremfineqvlem 7010 Lemma for fineqv 7011. (Contributed by Mario Carneiro, 20-Jan-2013.) (Proof shortened by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  om  ~<_  ~P ~P A )
 
Theoremfineqv 7011 If the Axiom of Infinity is denied, then all sets are finite (which implies the Axiom of Choice). (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 3-Jan-2015.)
 |-  ( -.  om  e.  _V  <->  Fin 
 =  _V )
 
Theoremenfi 7012 Equinmerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
 |-  ( A  ~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
 
Theoremenfii 7013 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( B  e.  Fin  /\  A  ~~  B ) 
 ->  A  e.  Fin )
 
Theorempssnn 7014* A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137. (Contributed by NM, 22-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  C.  A )  ->  E. x  e.  A  B  ~~  x )
 
Theoremssnnfi 7015 A subset of a natural number is finite. (Contributed by NM, 24-Jun-1998.)
 |-  ( ( A  e.  om 
 /\  B  C_  A )  ->  B  e.  Fin )
 
Theoremssfi 7016 A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138. (Contributed by NM, 24-Jun-1998.)
 |-  ( ( A  e.  Fin  /\  B  C_  A )  ->  B  e.  Fin )
 
Theoremdomfi 7017 A set dominated by a finite set is finite. (Contributed by NM, 23-Mar-2006.) (Revised by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( A  e.  Fin  /\  B  ~<_  A )  ->  B  e.  Fin )
 
Theoremxpfir 7018 The components of a non-empty finite cross product are finite. (Contributed by Paul Chapman, 11-Apr-2009.) (Proof shortened by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( ( A  X.  B )  e. 
 Fin  /\  ( A  X.  B )  =/=  (/) )  ->  ( A  e.  Fin  /\  B  e.  Fin )
 )
 
Theoremf1finf1o 7019 Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.)
 |-  ( ( A  ~~  B  /\  B  e.  Fin )  ->  ( F : A -1-1-> B  <->  F : A -1-1-onto-> B ) )
 
Theorem0fin 7020 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
 |-  (/)  e.  Fin
 
Theoremen1eqsn 7021 A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
 |-  ( ( A  e.  B  /\  B  ~~  1o )  ->  B  =  { A } )
 
Theoremdiffi 7022 If  A is finite,  ( A 
\  B ) is finite. (Contributed by FL, 3-Aug-2009.)
 |-  ( A  e.  Fin  ->  ( A  \  B )  e.  Fin )
 
Theoremdif1enOLD 7023 If a set  A is equinumerous to the successor of a natural number  M, then  A with an element removed is equinumerous to  M. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( ( M  e.  om  /\  A  ~~ 
 suc  M  /\  X  e.  A )  ->  ( A 
 \  { X }
 )  ~~  M )
 
Theoremdif1en 7024 If a set  A is equinumerous to the successor of a natural number  M, then  A with an element removed is equinumerous to  M. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)
 |-  ( ( M  e.  om 
 /\  A  ~~  suc  M 
 /\  X  e.  A )  ->  ( A  \  { X } )  ~~  M )
 
Theoremenp1ilem 7025 Lemma for uses of enp1i 7026. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  T  =  ( { x }  u.  S )   =>    |-  ( x  e.  A  ->  ( ( A  \  { x } )  =  S  ->  A  =  T ) )
 
Theoremenp1i 7026* Proof induction for en2i 6832 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  M  e.  om   &    |-  N  =  suc  M   &    |-  ( ( A 
 \  { x }
 )  ~~  M  ->  ph )   &    |-  ( x  e.  A  ->  ( ph  ->  ps ) )   =>    |-  ( A  ~~  N  ->  E. x ps )
 
Theoremen2 7027* A set equinumerous to ordinal 2 is a pair. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( A  ~~  2o  ->  E. x E. y  A  =  { x ,  y } )
 
Theoremen3 7028* A set equinumerous to ordinal 3 is a triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( A  ~~  3o  ->  E. x E. y E. z  A  =  { x ,  y ,  z } )
 
Theoremen4 7029* A set equinumerous to ordinal 4 is a quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( A  ~~  4o  ->  E. x E. y E. z E. w  A  =  ( { x ,  y }  u.  { z ,  w } ) )
 
Theoremfindcard 7030* Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  ( y  \  { z } )  ->  ( ph  <->  ch ) )   &    |-  ( x  =  y  ->  (
 ph 
 <-> 
 th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  Fin  ->  (
 A. z  e.  y  ch  ->  th ) )   =>    |-  ( A  e.  Fin 
 ->  ta )
 
Theoremfindcard2 7031* Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  Fin  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  Fin  ->  ta )
 
Theoremfindcard2s 7032* Variation of findcard2 7031 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 ( y  e.  Fin  /\ 
 -.  z  e.  y
 )  ->  ( ch  ->  th ) )   =>    |-  ( A  e.  Fin 
 ->  ta )
 
Theoremfindcard3 7033* Schema for strong induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on any proper subset. The result is then proven to be true for all finite sets. (Contributed by Mario Carneiro, 13-Dec-2013.)
 |-  ( x  =  y 
 ->  ( ph  <->  ch ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  (
 y  e.  Fin  ->  (
 A. x ( x 
 C.  y  ->  ph )  ->  ch ) )   =>    |-  ( A  e.  Fin 
 ->  ta )
 
Theoremac6sfi 7034* A version of ac6s 8044 for finite sets. (Contributed by Jeffrey Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( y  =  ( f `  x ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  Fin  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. f ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremfrfi 7035 A partial order is well-founded on a finite set. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Po  A  /\  A  e.  Fin )  ->  R  Fr  A )
 
Theoremfimax2g 7036* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
Theoremfimaxg 7037* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  y R x ) )
 
Theoremfisupg 7038* Lemma showing existence and closure of supremum of a finite set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A. y  e.  A  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  A  y R z ) ) )
 
Theoremwofi 7039 A total order on a finite set is a well order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
 
Theoremordunifi 7040 The maximum of a finite collection of ordinals is in the set. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( A  C_  On  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  U. A  e.  A )
 
Theoremnnunifi 7041 The union (supremum) of a finite set of finite ordinals is a finite ordinal. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ( S  C_  om 
 /\  S  e.  Fin )  ->  U. S  e.  om )
 
Theoremunblem1 7042* Lemma for unbnn 7046. After removing the successor of an element from an unbounded set of natural numbers, the intersection of the result belongs to the original unbounded set. (Contributed by NM, 3-Dec-2003.)
 |-  ( ( ( B 
 C_  om  /\  A. x  e.  om  E. y  e.  B  x  e.  y
 )  /\  A  e.  B )  ->  |^| ( B  \  suc  A )  e.  B )
 
Theoremunblem2 7043* Lemma for unbnn 7046. The value of the function  F belongs to the unbounded set of natural numbers  A. (Contributed by NM, 3-Dec-2003.)
 |-  F  =  ( rec ( ( x  e. 
 _V  |->  |^| ( A  \  suc  x ) ) , 
 |^| A )  |`  om )   =>    |-  ( ( A  C_  om 
 /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
 z  e.  om  ->  ( F `  z )  e.  A ) )
 
Theoremunblem3 7044* Lemma for unbnn 7046. The value of the function  F is less than its value at a successor. (Contributed by NM, 3-Dec-2003.)
 |-  F  =  ( rec ( ( x  e. 
 _V  |->  |^| ( A  \  suc  x ) ) , 
 |^| A )  |`  om )   =>    |-  ( ( A  C_  om 
 /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
 z  e.  om  ->  ( F `  z )  e.  ( F `  suc  z ) ) )
 
Theoremunblem4 7045* Lemma for unbnn 7046. The function  F maps the set of natural numbers one-to-one to the set of unbounded natural numbers  A. (Contributed by NM, 3-Dec-2003.)
 |-  F  =  ( rec ( ( x  e. 
 _V  |->  |^| ( A  \  suc  x ) ) , 
 |^| A )  |`  om )   =>    |-  ( ( A  C_  om 
 /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  F : om -1-1-> A )
 
Theoremunbnn 7046* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. Part of the proof of Theorem 42 of [Suppes] p. 151. See unbnn3 7292 for a stronger version without the first assumption. (Contributed by NM, 3-Dec-2003.)
 |-  ( ( om  e.  _V 
 /\  A  C_  om  /\  A. x  e.  om  E. y  e.  A  x  e.  y )  ->  A  ~~ 
 om )
 
Theoremunbnn2 7047* Version of unbnn 7046 that does not require a strict upper bound. (Contributed by NM, 24-Apr-2004.)
 |-  ( ( om  e.  _V 
 /\  A  C_  om  /\  A. x  e.  om  E. y  e.  A  x  C_  y )  ->  A  ~~ 
 om )
 
Theoremisfinite2 7048 Any set strictly dominated by the class of natural numbers is finite. Sufficiency part of Theorem 42 of [Suppes] p. 151. This theorem does not require the Axiom of Infinity. (Contributed by NM, 24-Apr-2004.)
 |-  ( A  ~<  om  ->  A  e.  Fin )
 
Theoremnnsdomg 7049 Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 15-Jun-1998.)
 |-  ( ( om  e.  _V 
 /\  A  e.  om )  ->  A  ~<  om )
 
Theoremisfiniteg 7050 A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 3-Nov-2002.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( om  e.  _V  ->  ( A  e.  Fin  <->  A  ~<  om ) )
 
Theoreminfsdomnn 7051 An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( om  ~<_  A  /\  B  e.  om )  ->  B  ~<  A )
 
Theoreminfn0 7052 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
 |-  ( om  ~<_  A  ->  A  =/=  (/) )
 
Theoremfin2inf 7053 This (useless) theorem, which was proved without the Axiom of Infinity, demonstrates an artifact of our definition of binary relation, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless 
om exists. (Contributed by NM, 13-Nov-2003.)
 |-  ( A  ~<  om  ->  om  e.  _V )
 
Theoremunfilem1 7054* Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  om   &    |-  B  e.  om   &    |-  F  =  ( x  e.  B  |->  ( A  +o  x ) )   =>    |- 
 ran  F  =  (
 ( A  +o  B )  \  A )
 
Theoremunfilem2 7055* Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  om   &    |-  B  e.  om   &    |-  F  =  ( x  e.  B  |->  ( A  +o  x ) )   =>    |-  F : B -1-1-onto-> ( ( A  +o  B )  \  A )
 
Theoremunfilem3 7056 Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 16-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  B  ~~  (
 ( A  +o  B )  \  A ) )
 
Theoremunfi 7057 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  u.  B )  e.  Fin )
 
Theoremunfir 7058 If a union is finite, the operands are finite. Converse of unfi 7057. (Contributed by FL, 3-Aug-2009.)
 |-  ( ( A  u.  B )  e.  Fin  ->  ( A  e.  Fin  /\  B  e.  Fin )
 )
 
Theoremunfi2 7059 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 7057 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 7053). (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( A  ~<  om 
 /\  B  ~<  om )  ->  ( A  u.  B )  ~<  om )
 
Theoremdifinf 7060 An infinite set  A minus a finite set is infinite. (Contributed by FL, 3-Aug-2009.)
 |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  -.  ( A  \  B )  e. 
 Fin )
 
Theoremxpfi 7061 The Cartesian product of two finite sets is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  X.  B )  e.  Fin )
 
Theoremdomunfican 7062 A finite set union cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  ( ( ( A  e.  Fin  /\  B  ~~  A )  /\  ( ( A  i^i  X )  =  (/)  /\  ( B  i^i  Y )  =  (/) ) )  ->  ( ( A  u.  X )  ~<_  ( B  u.  Y ) 
 <->  X  ~<_  Y ) )
 
Theoreminfcntss 7063* Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.) (Contributed by NM, 23-Oct-2004.)
 |-  A  e.  _V   =>    |-  ( om  ~<_  A  ->  E. x ( x  C_  A  /\  x  ~~  om ) )
 
Theoremprfi 7064 An unordered pair is finite. (Contributed by NM, 22-Aug-2008.)
 |- 
 { A ,  B }  e.  Fin
 
Theoremtpfi 7065 An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.)
 |- 
 { A ,  B ,  C }  e.  Fin
 
Theoremfiint 7066* Equivalent ways of stating the finite intersection property. We show two ways of saying, "the intersection of elements in every finite non-empty subcollection of 
A is in  A." This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally. (Contributed by NM, 22-Sep-2002.)
 |-  ( A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A  <->  A. x ( ( x 
 C_  A  /\  x  =/= 
 (/)  /\  x  e.  Fin )  ->  |^| x  e.  A ) )
 
Theoremfnfi 7067 A version of fnex 5640 for finite sets that does not require Replacement. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
 
Theoremfodomfi 7068 An onto function implies dominance of domain over range, for finite sets. Unlike fodom 8082 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )
 
Theoremfodomfib 7069* Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 8084 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.)
 |-  ( A  e.  Fin  ->  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B 
 ~<_  A ) ) )
 
Theoremfofinf1o 7070 Any surjection from one finite set to another of equal size must be a bijection. (Contributed by Mario Carneiro, 19-Aug-2014.)
 |-  ( ( F : A -onto-> B  /\  A  ~~  B  /\  B  e.  Fin )  ->  F : A -1-1-onto-> B )
 
Theoremfidomdm 7071 Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( F  e.  Fin  ->  dom  F  ~<_  F )
 
Theoremdmfi 7072 The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)
 |-  ( A  e.  Fin  ->  dom  A  e.  Fin )
 
Theoremcnvfi 7073 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( A  e.  Fin  ->  `' A  e.  Fin )
 
Theoremrnfi 7074 The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( A  e.  Fin  ->  ran  A  e.  Fin )
 
Theoremfofi 7075 If a function has a finite domain, its range is finite. Theorem 37 of [Suppes] p. 104. (Contributed by NM, 25-Mar-2007.)
 |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  B  e.  Fin )
 
Theoremf1fi 7076 If a 1-to-1 function has a finite codomain its domain is finite. (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( B  e.  Fin  /\  F : A -1-1-> B )  ->  A  e.  Fin )
 
Theoremiunfi 7077* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 7078. Note that  B depends on  x, i.e. can be thought of as  B ( x ). (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( A  e.  Fin  /\  A. x  e.  A  B  e.  Fin )  ->  U_ x  e.  A  B  e.  Fin )
 
Theoremunifi 7078 The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. (Contributed by NM, 22-Aug-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( A  e.  Fin  /\  A  C_  Fin )  ->  U. A  e.  Fin )
 
Theoremunifi2 7079* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 7078 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 7053). (Contributed by NM, 11-Mar-2006.)
 |-  ( ( A  ~<  om 
 /\  A. x  e.  A  x  ~<  om )  ->  U. A  ~<  om )
 
Theoremunirnffid 7080 The union of the range of a function from a finite set into the class of finite sets is finite. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  F : T --> Fin )   &    |-  ( ph  ->  T  e.  Fin )   =>    |-  ( ph  ->  U.
 ran  F  e.  Fin )
 
Theoremimafi 7081 Images of finite sets are finite. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ( Fun  F  /\  X  e.  Fin )  ->  ( F " X )  e.  Fin )
 
Theoremsuppfif1 7082 Formula building theorem for finite supports: rearranging the index set. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  ( ph  ->  ( `' F " ( _V  \  { Z } )
 )  e.  Fin )   &    |-  ( ph  ->  G : X -1-1-> Y )   =>    |-  ( ph  ->  ( `' ( F  o.  G ) " ( _V  \  { Z } ) )  e. 
 Fin )
 
Theorempwfilem 7083* Lemma for pwfi 7084. (Contributed by NM, 26-Mar-2007.)
 |-  F  =  ( c  e.  ~P b  |->  ( c  u.  { x } ) )   =>    |-  ( ~P b  e.  Fin  ->  ~P (
 b  u.  { x } )  e.  Fin )
 
Theorempwfi 7084 The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105. (Contributed by NM, 26-Mar-2007.)
 |-  ( A  e.  Fin  <->  ~P A  e.  Fin )
 
Theoremmapfi 7085 Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  ^m  B )  e.  Fin )
 
Theoremixpfi 7086* A cross product of finitely many finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ( A  e.  Fin  /\  A. x  e.  A  B  e.  Fin )  ->  X_ x  e.  A  B  e.  Fin )
 
Theoremixpfi2 7087* A cross product of finite sets such that all but finitely many are singletons is finite. (Note that  B ( x ) and 
D ( x ) are both possibly dependent on  x. ) (Contributed by Mario Carneiro, 25-Jan-2015.)
 |-  ( ph  ->  C  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  Fin )   &    |-  (
 ( ph  /\  x  e.  ( A  \  C ) )  ->  B  C_  { D } )   =>    |-  ( ph  ->  X_ x  e.  A  B  e.  Fin )
 
Theoremmptfi 7088* A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e.  Fin  ->  ( x  e.  A  |->  B )  e.  Fin )
 
Theoremabrexfi 7089* An image set from a finite set is finite. (Contributed by Mario Carneiro, 13-Feb-2014.)
 |-  ( A  e.  Fin  ->  { y  |  E. x  e.  A  y  =  B }  e.  Fin )
 
Theoremelfpw 7090 Membership in a class of finite subsets. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( A  e.  ( ~P B  i^i  Fin )  <->  ( A  C_  B  /\  A  e.  Fin ) )
 
Theoremunifpw 7091 A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |- 
 U. ( ~P A  i^i  Fin )  =  A
 
Theoremf1opwfi 7092* A one-to-one mapping induces a one-to-one mapping on finite subsets. (Contributed by Mario Carneiro, 25-Jan-2015.)
 |-  ( F : A -1-1-onto-> B  ->  ( b  e.  ( ~P A  i^i  Fin )  |->  ( F " b
 ) ) : ( ~P A  i^i  Fin )
 -1-1-onto-> ( ~P B  i^i  Fin ) )
 
Theoremfissuni 7093* A finite subset of a union is covered by finitely many elements. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( ( A  C_  U. B  /\  A  e.  Fin )  ->  E. c  e.  ( ~P B  i^i  Fin ) A  C_  U. c
 )
 
Theoremfipreima 7094* Given a finite subset  A of the range of a function, there exists a finite subset of the domain whose image is  A. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  E. c  e.  ( ~P B  i^i  Fin )
 ( F " c
 )  =  A )
 
Theoremfinsschain 7095* A finite subset of the union of a superset chain is a subset of some element of the chain. A useful preliminary result for alexsub 17666 and others. (Contributed by Jeff Hankins, 25-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 18-May-2015.)
 |-  ( ( ( A  =/=  (/)  /\ [ C.]  Or  A )  /\  ( B  e.  Fin  /\  B  C_  U. A ) )  ->  E. z  e.  A  B  C_  z
 )
 
Theoremindexfi 7096* If for every element of a finite indexing set  A there exists a corresponding element of another set  B, then there exists a finite subset of  B consisting only of those elements which are indexed by  A. Proven without the Axiom of Choice, unlike indexdom 25745. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  ( ( A  e.  Fin  /\  B  e.  M  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. c  e.  Fin  ( c  C_  B  /\  A. x  e.  A  E. y  e.  c  ph  /\ 
 A. y  e.  c  E. x  e.  A  ph ) )
 
2.4.33  Finite intersections
 
Syntaxcfi 7097 Extend class notation with the function whose value is the class of all the finite intersections of the elements of a given set.
 class  fi
 
Definitiondf-fi 7098* Function whose value is the class of all the finite intersections of the elements of  x. (Contributed by FL, 27-Apr-2008.)
 |- 
 fi  =  ( x  e.  _V  |->  { z  |  E. y  e.  ( ~P x  i^i  Fin )
 z  =  |^| y } )
 
Theoremfival 7099* The set of all the finite intersections of the elements of  A. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( A  e.  V  ->  ( fi `  A )  =  { y  |  E. x  e.  ( ~P A  i^i  Fin )
 y  =  |^| x } )
 
Theoremelfi 7100* Specific properties of an element of 
( fi `  B
). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
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