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Theorem List for Metamath Proof Explorer - 37001-37100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremadantlllr 37001 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ( ( ph  /\  et )  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
Theorem3adantlr3 37002 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( ( ph  /\  ( ps  /\  ch ) ) 
 /\  th )  ->  ta )   =>    |-  (
 ( ( ph  /\  ( ps  /\  ch  /\  et ) )  /\  th )  ->  ta )
 
Theoremnnxrd 37003 A natural number is an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  RR* )
 
Theorem3adantll2 37004 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( (
 ph  /\  et  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
Theorem3adantll3 37005 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( (
 ph  /\  ps  /\  et )  /\  ch )  /\  th )  ->  ta )
 
Theoremssnel 37006 If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  C_  B  /\  -.  C  e.  B )  ->  -.  C  e.  A )
 
Theoremadant423 37007 Deduction adding conjuncts to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( (
 ph  /\  th )  /\  ta )  /\  ps )  ->  ch )
 
Theoremjcn 37008 Inference joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -. 
 ch )   =>    |-  ( ph  ->  -.  ( ps  ->  ch ) )
 
Theorem3expc 37009 Exportation inference. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theoremelabrexg 37010* Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( x  e.  A  /\  B  e.  V ) 
 ->  B  e.  { y  |  E. x  e.  A  y  =  B }
 )
 
Theoremunicntop 37011 The underlying set of the standard topology on the complex numers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  CC  =  U. ( TopOpen ` fld )
 
Theoremifeq123d 37012 Equality deduction for conditional operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.) AV: This theorem already exists as ifbieq12d 3942. TODO (NM): Please replace the usage of this theorem by ifbieq12d 3942 then delete this theorem. (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ch ,  B ,  D ) )
 
Theoremsncldre 37013 A singleton is closed w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  RR  ->  { A }  e.  ( Clsd `  ( topGen `  ran  (,) ) ) )
 
Theoremneqne 37014 From non equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( -.  A  =  B  ->  A  =/=  B )
 
Theoremcnopn 37015 The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  CC  e.  ( TopOpen ` fld )
 
Theoremn0p 37016 A polynomial with a nonzero coefficient is not the zero polynomial. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  (
 ( P  e.  (Poly `  ZZ )  /\  N  e.  NN0  /\  ( (coeff `  P ) `  N )  =/=  0 )  ->  P  =/=  0p )
 
Theoremrabeqd 37017* Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ps } )
 
Theorempm2.65ni 37018 Inference rule for proof by contradiction. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( -.  ph  ->  ps )   &    |-  ( -.  ph  ->  -.  ps )   =>    |-  ph
 
TheoremraleqdOLD 37019* Equality theorem for restricted universal quantifier. (Contributed by Glauco Siliprandi, 5-Apr-2020.) Obsolete as of 18-Jul-2020. Use raleqdv 3038 instead, which is the same. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A. x  e.  A  ps 
 <-> 
 A. x  e.  B  ps ) )
 
Theoremelini 37020 Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  A  e.  B   &    |-  A  e.  C   =>    |-  A  e.  ( B  i^i  C )
 
Theorempwssfi 37021 Every element of the power set of 
A is finite if and only if  A is finite (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( A  e.  V  ->  ( A  e.  Fin  <->  ~P A  C_  Fin )
 )
 
Theoremiuneq2df 37022 Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  =  C )   =>    |-  ( ph  ->  U_ x  e.  A  B  =  U_ x  e.  A  C )
 
Theoremnnfoctb 37023* There exists a mapping from  NN onto any (nonempty) countable set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( A  ~<_  om  /\  A  =/=  (/) )  ->  E. f  f : NN -onto-> A )
 
Theoremprssd 37024 A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  C )   =>    |-  ( ph  ->  { A ,  B }  C_  C )
 
Theoremssinss1d 37025 Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  C_  C )   =>    |-  ( ph  ->  ( A  i^i  B )  C_  C )
 
Theoremprcnel 37026 A proper class doesn't belong to any class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( -.  A  e.  _V  ->  -.  A  e.  V )
 
Theorem0un 37027 The union of the empty set with a class is itself. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( (/) 
 u.  A )  =  A
 
Theoremelpwinss 37028 An element of the powerset of  B intersected with anything, is a subset of  B. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( A  e.  ( ~P B  i^i  C )  ->  A  C_  B )
 
Theoremunidmex 37029 If  F is a set, then  U. dom  F is a set (common case) (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F  e.  V )   &    |-  X  =  U. dom  F   =>    |-  ( ph  ->  X  e.  _V )
 
Theoremndisj2 37030* A non disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( x  =  y  ->  B  =  C )   =>    |-  ( -. Disj  x  e.  A  B  <->  E. x  e.  A  E. y  e.  A  ( x  =/=  y  /\  ( B  i^i  C )  =/=  (/) ) )
 
Theoremzenom 37031 The set of integer numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ZZ  ~~ 
 om
 
Theoremrexsngf 37032* Restricted existential quantification over a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( E. x  e. 
 { A } ph  <->  ps ) )
 
Theoremuzwo4 37033* Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ j ps   &    |-  ( j  =  k  ->  ( ph  <->  ps ) )   =>    |-  ( ( S  C_  ( ZZ>= `  M )  /\  E. j  e.  S  ph )  ->  E. j  e.  S  ( ph  /\  A. k  e.  S  (
 k  <  j  ->  -. 
 ps ) ) )
 
Theorem0in 37034 The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( (/) 
 i^i  A )  =  (/)
 
Theoremunisn0 37035 The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  U. { (/)
 }  =  (/)
 
Theoremssin0 37036 If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( ( A  i^i  B )  =  (/)  /\  C  C_  A  /\  D  C_  B )  ->  ( C  i^i  D )  =  (/) )
 
Theoreminabs3 37037 Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( C  C_  B  ->  (
 ( A  i^i  B )  i^i  C )  =  ( A  i^i  C ) )
 
Theorempwpwuni 37038 Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( A  e.  V  ->  ( A  e.  ~P ~P B 
 <-> 
 U. A  e.  ~P B ) )
 
Theoremdisjiun2 37039* In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  -> Disj  x  e.  A  B )   &    |-  ( ph  ->  C  C_  A )   &    |-  ( ph  ->  D  e.  ( A  \  C ) )   &    |-  ( x  =  D  ->  B  =  E )   =>    |-  ( ph  ->  (
 U_ x  e.  C  B  i^i  E )  =  (/) )
 
Theorem0pwfi 37040 The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (/)  e.  ( ~P A  i^i  Fin )
 
Theoremssinss2d 37041 Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  ( A  i^i  B )  C_  C )
 
Theoremnelsn 37042 If an element doesn't match the item in an singleton it is not in the singleton (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( A  =/=  B  ->  -.  A  e.  { B } )
 
Theoremzct 37043 The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ZZ  ~<_  om
 
Theoremiunxsngf2 37044* A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ x C   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  ( A  e.  V  -> 
 U_ x  e.  { A } B  =  C )
 
Theorempwfin0 37045 A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ~P A  i^i  Fin )  =/= 
 (/)
 
Theoremuzct 37046 An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  Z  =  ( ZZ>= `  N )   =>    |-  Z  ~<_  om
 
Theoremiunxsnf 37047* A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ x C   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  U_ x  e.  { A } B  =  C
 
Theoremfiiuncl 37048* If a set is closed under the union of two sets, then it is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  D )   &    |-  ( ( ph  /\  y  e.  D  /\  z  e.  D )  ->  (
 y  u.  z )  e.  D )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/= 
 (/) )   =>    |-  ( ph  ->  U_ x  e.  A  B  e.  D )
 
Theoremiunp1 37049* The addition of the next set to a union indexed by a finite set of sequential integers. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ k B   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( k  =  ( N  +  1 ) 
 ->  A  =  B )   =>    |-  ( ph  ->  U_ k  e.  ( M ... ( N  +  1 )
 ) A  =  (
 U_ k  e.  ( M ... N ) A  u.  B ) )
 
Theoremfiunicl 37050* If a set is closed under the union of two sets, then it is closed under finite union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( ph  /\  x  e.  A  /\  y  e.  A )  ->  ( x  u.  y )  e.  A )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/=  (/) )   =>    |-  ( ph  ->  U. A  e.  A )
 
21.30.2  Functions
 
Theoremunima 37051 Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( F  Fn  A  /\  B  C_  A  /\  C  C_  A )  ->  ( F " ( B  u.  C ) )  =  ( ( F
 " B )  u.  ( F " C ) ) )
 
Theoremfeq1dd 37052 Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  G : A --> B )
 
Theoremfnresdmss 37053 A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( F  Fn  A  /\  A  C_  B )  ->  ( F  |`  B )  =  F )
 
Theoremfmptsnxp 37054* Maps-to notation and cross product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( x  e.  { A }  |->  B )  =  ( { A }  X.  { B }
 ) )
 
Theoremmptex2 37055* If a class given as a map-to notation is a set, it's image values are set (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  ( t  e.  A  |->  B ) : A --> C )   =>    |-  ( ( ph  /\  t  e.  A ) 
 ->  B  e.  C )
 
Theoremfvmpt2bd 37056* Value of a function given by the "maps to" notation. Deduction version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   =>    |-  ( ( ph  /\  x  e.  A  /\  B  e.  C )  ->  ( F `
  x )  =  B )
 
Theoremrnmptfi 37057* The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  A  =  ( x  e.  B  |->  C )   =>    |-  ( B  e.  Fin  ->  ran  A  e.  Fin )
 
Theoremfresin2 37058 Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( F : A --> B  ->  ( F  |`  ( C  i^i  A ) )  =  ( F  |`  C ) )
 
Theoremrnmptc 37059* Range of a constant function in map to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  C )   &    |-  ( ph  ->  A  =/=  (/) )   =>    |-  ( ph  ->  ran  F  =  { B } )
 
Theoremffi 37060 A function with finite domain is finite (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( F : A --> B  /\  A  e.  Fin )  ->  F  e.  Fin )
 
Theoremsuprnmpt 37061* An explicit bound for the range of a bounded function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  =/=  (/) )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  E. y  e.  RR  A. x  e.  A  B  <_  y )   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  C  =  sup ( ran  F ,  RR ,  <  )   =>    |-  ( ph  ->  ( C  e.  RR  /\  A. x  e.  A  B  <_  C ) )
 
Theoremrnffi 37062 The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( F : A --> B  /\  A  e.  Fin )  ->  ran  F  e.  Fin )
 
Theoremmptelpm 37063* A function in maps-to notation is a partial map . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  C )   &    |-  ( ph  ->  A 
 C_  D )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  W )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  ( C  ^pm  D ) )
 
Theoremf1oeq3d 37064 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : C -1-1-onto-> A  <->  F : C -1-1-onto-> B ) )
 
Theoremfimass 37065 The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( F : A --> B  ->  ( F " X ) 
 C_  B )
 
Theoremrnmptpr 37066* Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  F  =  ( x  e.  { A ,  B }  |->  C )   &    |-  ( x  =  A  ->  C  =  D )   &    |-  ( x  =  B  ->  C  =  E )   =>    |-  ( ph  ->  ran  F  =  { D ,  E }
 )
 
Theoremffnd 37067 A mapping is a function. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  F  Fn  A )
 
Theoremresmpti 37068* Restriction of the mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  B  C_  A   =>    |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C )
 
Theoremfouniiun 37069* Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( F : A -onto-> B  ->  U. B  =  U_ x  e.  A  ( F `  x ) )
 
Theoremf1oeq2d 37070 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )
 
Theoremrnresun 37071 Distribution law for range of a restriction over a union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ran  ( F  |`  ( A  u.  B ) )  =  ( ran  ( F  |`  A )  u. 
 ran  ( F  |`  B ) )
 
Theoremf1oeq1d 37072 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F : A -1-1-onto-> B  <->  G : A -1-1-onto-> B ) )
 
Theoremdffo3f 37073* An onto mapping expressed in terms of function values. As dffo3 6052 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ x F   =>    |-  ( F : A -onto-> B 
 <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
 
Theoremrnresss 37074 The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ran  ( A  |`  B ) 
 C_  ran  A
 
Theoremmpteq1i 37075* An equality theorem for the maps to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  A  =  B   =>    |-  ( x  e.  A  |->  C )  =  ( x  e.  B  |->  C )
 
Theoremelrnmptd 37076* The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ph  ->  E. x  e.  A  C  =  B )   &    |-  ( ph  ->  C  e.  V )   =>    |-  ( ph  ->  C  e.  ran  F )
 
Theoremelrnmptf 37077 The range of a function in maps-to notation. Same as elrnmpt 5101, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ x C   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( C  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B )
 )
 
Theoremmptss 37078* Sufficient condition for inclusion in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( A  C_  B  ->  ( x  e.  A  |->  C ) 
 C_  ( x  e.  B  |->  C ) )
 
Theoremrnmptssrn 37079* Inclusion relation for two ranges expressed in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  E. y  e.  C  B  =  D )   =>    |-  ( ph  ->  ran  ( x  e.  A  |->  B )  C_  ran  ( y  e.  C  |->  D ) )
 
Theoremdisjf1 37080* A 1 to 1 mapping built from disjoint, nonempty sets . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  =/= 
 (/) )   &    |-  ( ph  -> Disj  x  e.  A  B )   =>    |-  ( ph  ->  F : A -1-1-> V )
 
Theoremrnsnf 37081 The range of a function whose domain is a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : { A } --> B )   =>    |-  ( ph  ->  ran  F  =  { ( F `  A ) } )
 
Theoremwessf1ornlem 37082* Given a function  F on a well ordered domain  A there exists a subset of  A such that  F restricted to such subset is injective and onto the range of  F (without using the axiom of choice) (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  R  We  A )   &    |-  G  =  ( y  e.  ran  F 
 |->  ( iota_ x  e.  ( `' F " { y } ) A. z  e.  ( `' F " { y } )  -.  z R x ) )   =>    |-  ( ph  ->  E. x  e.  ~P  A ( F  |`  x ) : x -1-1-onto-> ran  F )
 
Theoremwessf1orn 37083* Given a function  F on a well ordered domain  A there exists a subset of  A such that  F restricted to such subset is injective and onto the range of  F (without using the axiom of choice) (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  R  We  A )   =>    |-  ( ph  ->  E. x  e.  ~P  A ( F  |`  x ) : x -1-1-onto-> ran  F )
 
Theoremfoelrnf 37084* Property of a surjective function. As foelrn 6056 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ x F   =>    |-  ( ( F : A -onto-> B  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
 
Theoremnelrnres 37085 If  A is not in the range, it is not in the range of any restriction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( -.  A  e.  ran  B  ->  -.  A  e.  ran  ( B  |`  C ) )
 
Theoremdisjrnmpt2 37086* Disjointness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  (Disj  x  e.  A  B  -> Disj  y  e.  ran  F  y )
 
Theoremelrnmpt1sf 37087* Elementhood in an image set. Same as elrnmpt1s 5102, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ x C   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( x  =  D  ->  B  =  C )   =>    |-  ( ( D  e.  A  /\  C  e.  V )  ->  C  e.  ran  F )
 
Theoremfouniiun0 37088* Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( F : A -onto-> ( B  u.  { (/) } )  ->  U. B  =  U_ x  e.  A  ( F `  x ) )
 
Theoremdisjf1o 37089* A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   &    |-  ( ph  -> Disj  x  e.  A  B )   &    |-  C  =  { x  e.  A  |  B  =/=  (/) }   &    |-  D  =  ( ran  F  \  { (/)
 } )   =>    |-  ( ph  ->  ( F  |`  C ) : C -1-1-onto-> D )
 
Theoremfompt 37090* Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F  =  ( x  e.  A  |->  C )   =>    |-  ( F : A -onto-> B 
 <->  ( A. x  e.  A  C  e.  B  /\  A. y  e.  B  E. x  e.  A  y  =  C )
 )
 
Theoremdisjinfi 37091* Only a finite number of disjoint sets can have a non empty intersection with a finite set  C (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  -> Disj  x  e.  A  B )   &    |-  ( ph  ->  C  e.  Fin )   =>    |-  ( ph  ->  { x  e.  A  |  ( B  i^i  C )  =/=  (/) }  e.  Fin )
 
21.30.3  Ordering on real numbers - Real and complex numbers basic operations
 
Theoremsub2times 37092 Subtracting from a number, twice the number itself, gives negative the number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  CC  ->  ( A  -  ( 2  x.  A ) )  =  -u A )
 
Theoremxrltled 37093 'Less than' implies 'less than or equal to', for extended reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A 
 <_  B )
 
Theoremabssubrp 37094 The distance of two distinct complex number is a strictly positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  A  =/=  B )  ->  ( abs `  ( A  -  B ) )  e.  RR+ )
 
Theoremelfzfzo 37095 Relationship between membership in a half open finite set of sequential integers and membership in a finite set of sequential intergers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  ( M..^ N )  <->  ( A  e.  ( M ... N ) 
 /\  A  <  N ) )
 
Theoremoddfl 37096 Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( K  e.  ZZ  /\  ( K  mod  2
 )  =/=  0 )  ->  K  =  ( ( 2  x.  ( |_ `  ( K  /  2
 ) ) )  +  1 ) )
 
Theoremabscosbd 37097 Bound for the absolute value of the cosine of a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  RR  ->  ( abs `  ( cos `  A ) )  <_ 
 1 )
 
Theoremmul13d 37098 Commutative/associative law that swaps the first and the third factor in a triple product. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( B  x.  C ) )  =  ( C  x.  ( B  x.  A ) ) )
 
Theoremnegpilt0 37099 Negative  pi is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  -u pi  <  0
 
Theoremdstregt0 37100* A complex number  A that is not real, has a distance from the reals that is strictly larger than  0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  ( CC  \  RR ) )   =>    |-  ( ph  ->  E. x  e.  RR+  A. y  e.  RR  x  <  ( abs `  ( A  -  y ) ) )
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