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Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfimacnvinrn2 24001 Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 17-Feb-2017.)
 |-  (
 ( Fun  F  /\  ran 
 F  C_  B )  ->  ( `' F " A )  =  ( `' F " ( A  i^i  B ) ) )
 
Theoremsuppss2f 24002* Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.)
 |-  F/ k ph   &    |-  F/_ k A   &    |-  F/_ k W   &    |-  ( ( ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )   =>    |-  ( ph  ->  ( `' ( k  e.  A  |->  B ) " ( _V  \  { Z }
 ) )  C_  W )
 
Theoremfovcld 24003 Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)
 |-  ( ph  ->  F : ( R  X.  S ) --> C )   =>    |-  ( ( ph  /\  A  e.  R  /\  B  e.  S )  ->  ( A F B )  e.  C )
 
Theoremelovimad 24004 Elementhood of the image set of an operation value (Contributed by Thierry Arnoux, 13-Mar-2017.)
 |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  D )   &    |-  Fun  F   &    |-  ( ph  ->  ( C  X.  D ) 
 C_  dom  F )   =>    |-  ( ph  ->  ( A F B )  e.  ( F " ( C  X.  D ) ) )
 
Theoremofrn 24005 The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : A --> B )   &    |-  ( ph  ->  .+  : ( B  X.  B ) --> C )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ran  ( F  o F  .+  G )  C_  C )
 
Theoremofrn2 24006 The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : A --> B )   &    |-  ( ph  ->  .+  : ( B  X.  B ) --> C )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ran  ( F  o F  .+  G )  C_  (  .+  " ( ran  F  X.  ran  G ) ) )
 
Theoremoff2 24007* The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)
 |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  T ) )  ->  ( x R y )  e.  U )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : B --> T )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  ( A  i^i  B )  =  C )   =>    |-  ( ph  ->  ( F  o F R G ) : C --> U )
 
Theoremofresid 24008 Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : A --> B )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ( F  o F R G )  =  ( F  o F ( R  |`  ( B  X.  B ) ) G ) )
 
Theoremunipreima 24009* Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)
 |-  ( Fun  F  ->  ( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
 
Theoremsspreima 24010 The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)
 |-  (
 ( Fun  F  /\  A  C_  B )  ->  ( `' F " A ) 
 C_  ( `' F " B ) )
 
Theoremopfv 24011 Value of a function producing ordered pairs. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  (
 ( ( Fun  F  /\  ran  F  C_  ( _V  X.  _V ) ) 
 /\  x  e.  dom  F )  ->  ( F `  x )  =  <. ( ( 1st  o.  F ) `  x ) ,  ( ( 2nd  o.  F ) `  x ) >. )
 
Theoremxppreima 24012 The preimage of a cross product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.)
 |-  (
 ( Fun  F  /\  ran 
 F  C_  ( _V  X. 
 _V ) )  ->  ( `' F " ( Y  X.  Z ) )  =  ( ( `' ( 1st  o.  F ) " Y )  i^i  ( `' ( 2nd 
 o.  F ) " Z ) ) )
 
Theoremxppreima2 24013* The preimage of a cross product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : A --> C )   &    |-  H  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )   =>    |-  ( ph  ->  ( `' H " ( Y  X.  Z ) )  =  ( ( `' F " Y )  i^i  ( `' G " Z ) ) )
 
Theoremfmptapd 24014* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  ( R  u.  { A } )  =  S )   &    |-  ( ( ph  /\  x  =  A )  ->  C  =  B )   =>    |-  ( ph  ->  (
 ( x  e.  R  |->  C )  u.  { <. A ,  B >. } )  =  ( x  e.  S  |->  C ) )
 
Theoremfmptpr 24015* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   &    |-  ( ( ph  /\  x  =  A )  ->  E  =  C )   &    |-  ( ( ph  /\  x  =  B ) 
 ->  E  =  D )   =>    |-  ( ph  ->  { <. A ,  C >. ,  <. B ,  D >. }  =  ( x  e.  { A ,  B }  |->  E ) )
 
Theoremelunirn2 24016 Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.)
 |-  (
 ( Fun  F  /\  B  e.  ( F `  A ) )  ->  B  e.  U. ran  F )
 
Theoremabfmpunirn 24017* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)
 |-  F  =  ( x  e.  V  |->  { y  |  ph } )   &    |-  { y  |  ph }  e.  _V   &    |-  (
 y  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( B  e.  U.
 ran  F  <->  ( B  e.  _V 
 /\  E. x  e.  V  ps ) )
 
Theoremrabfmpunirn 24018* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)
 |-  F  =  ( x  e.  V  |->  { y  e.  W  |  ph
 } )   &    |-  W  e.  _V   &    |-  (
 y  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( B  e.  U.
 ran  F  <->  E. x  e.  V  ( B  e.  W  /\  ps ) )
 
Theoremabfmpeld 24019* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
 |-  F  =  ( x  e.  V  |->  { y  |  ps }
 )   &    |-  ( ph  ->  { y  |  ps }  e.  _V )   &    |-  ( ph  ->  (
 ( x  =  A  /\  y  =  B )  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <->  ch ) ) )
 
Theoremabfmpel 24020* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
 |-  F  =  ( x  e.  V  |->  { y  |  ph } )   &    |-  { y  |  ph }  e.  _V   &    |-  (
 ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <->  ps ) )
 
Theoremcbvmptf 24021* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Thierry Arnoux, 9-Mar-2017.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/_ y B   &    |-  F/_ x C   &    |-  ( x  =  y 
 ->  B  =  C )   =>    |-  ( x  e.  A  |->  B )  =  (
 y  e.  A  |->  C )
 
TheoremfmptdF 24022 Domain and co-domain of the mapping operation; deduction form. This version of fmptd 5852 usex bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x C   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  C )   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( ph  ->  F : A --> C )
 
Theoremfmpt3d 24023* Domain and co-domain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  C )   =>    |-  ( ph  ->  F : A
 --> C )
 
Theoremresmptf 24024 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( B  C_  A  ->  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
 
Theoremfvmpt2f 24025 Value of a function given by the "maps to" notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)
 |-  F/_ x A   =>    |-  ( ( x  e.  A  /\  B  e.  C )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
 
Theoremmptfnf 24026 The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Thierry Arnoux, 10-May-2017.)
 |-  F/_ x A   =>    |-  ( A. x  e.  A  B  e.  _V  <->  ( x  e.  A  |->  B )  Fn  A )
 
Theoremfnmptf 24027 The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
 |-  F/_ x A   =>    |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  Fn  A )
 
Theoremfeqmptdf 24028 Deduction form of dffn5f 5740. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.)
 |-  F/_ x A   &    |-  F/_ x F   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
 
Theoremfmptcof2 24029* Composition of two functions expressed as ordered-pair class abstractions. (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.) (Revised by Thierry Arnoux, 10-May-2017.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/ x ph   &    |-  ( ph  ->  A. x  e.  A  R  e.  B )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )   &    |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )   &    |-  ( y  =  R  ->  S  =  T )   =>    |-  ( ph  ->  ( G  o.  F )  =  ( x  e.  A  |->  T ) )
 
Theoremfcomptf 24030* Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 5863. (Contributed by Thierry Arnoux, 30-Jun-2017.)
 |-  F/_ x B   =>    |-  ( ( A : D
 --> E  /\  B : C
 --> D )  ->  ( A  o.  B )  =  ( x  e.  C  |->  ( A `  ( B `
  x ) ) ) )
 
Theoremcofmpt 24031* Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)
 |-  ( ph  ->  F : C --> D )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  C )   =>    |-  ( ph  ->  ( F  o.  ( x  e.  A  |->  B ) )  =  ( x  e.  A  |->  ( F `  B ) ) )
 
Theoremofoprabco 24032* Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  F/_ a M   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : A --> C )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  M  =  ( a  e.  A  |->  <.
 ( F `  a
 ) ,  ( G `
  a ) >. ) )   &    |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) )   =>    |-  ( ph  ->  ( F  o F R G )  =  ( N  o.  M ) )
 
Theoremoffval2f 24033* The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  X )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )   =>    |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  A  |->  ( B R C ) ) )
 
Theoremofpreima 24034* Express the preimage of a function operation as a union of preimages. (Contributed by Thierry Arnoux, 8-Mar-2018.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : A --> C )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  R  Fn  ( B  X.  C ) )   =>    |-  ( ph  ->  ( `' ( F  o F R G ) " D )  =  U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p ) }
 )  i^i  ( `' G " { ( 2nd `  p ) } )
 ) )
 
TheoremfuncnvmptOLD 24035* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x F   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   =>    |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x ( x  e.  A  /\  y  =  B ) ) )
 
Theoremfuncnvmpt 24036* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x F   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   =>    |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x  e.  A y  =  B ) )
 
Theoremfuncnv5mpt 24037* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x F   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( x  =  z  ->  B  =  C )   =>    |-  ( ph  ->  ( Fun  `' F  <->  A. x  e.  A  A. z  e.  A  ( x  =  z  \/  B  =/=  C ) ) )
 
Theoremfuncnv4mpt 24038* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x F   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   =>    |-  ( ph  ->  ( Fun  `' F  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B ) ) )
 
Theoremrnmpt2ss 24039* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 23-May-2017.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( A. x  e.  A  A. y  e.  B  C  e.  D  ->  ran  F  C_  D )
 
Theorempartfun 24040 Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. (Contributed by Thierry Arnoux, 30-Mar-2017.)
 |-  ( x  e.  A  |->  if ( x  e.  B ,  C ,  D )
 )  =  ( ( x  e.  ( A  i^i  B )  |->  C )  u.  ( x  e.  ( A  \  B )  |->  D ) )
 
19.3.4.3  Isomorphisms - misc. add.
 
Theoremgtiso 24041 Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  (
 ( A  C_  RR*  /\  B  C_  RR* )  ->  ( F 
 Isom  <  ,  `'  <  ( A ,  B )  <->  F  Isom  <_  ,  `'  <_  ( A ,  B ) ) )
 
Theoremisoun 24042* Infer an isomorphism from for a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.)
 |-  ( ph  ->  H  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  G 
 Isom  R ,  S  ( C ,  D ) )   &    |-  ( ( ph  /\  x  e.  A  /\  y  e.  C )  ->  x R y )   &    |-  ( ( ph  /\  z  e.  B  /\  w  e.  D )  ->  z S w )   &    |-  ( ( ph  /\  x  e.  C  /\  y  e.  A )  ->  -.  x R y )   &    |-  ( ( ph  /\  z  e.  D  /\  w  e.  B )  ->  -.  z S w )   &    |-  ( ph  ->  ( A  i^i  C )  =  (/) )   &    |-  ( ph  ->  ( B  i^i  D )  =  (/) )   =>    |-  ( ph  ->  ( H  u.  G )  Isom  R ,  S  ( ( A  u.  C ) ,  ( B  u.  D ) ) )
 
19.3.4.4  Disjointness (additional proof requiring functions)
 
Theoremdisjdsct 24043* A disjoint collection is distinct, i.e. each set in this collection is different of all others, provided that it does not contain the empty set This can be expressed as "the converse of the mapping function is a function", or "the mapping function is single-rooted". (Cf. funcnv 5470) (Contributed by Thierry Arnoux, 28-Feb-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( V  \  { (/)
 } ) )   &    |-  ( ph  -> Disj  x  e.  A B )   =>    |-  ( ph  ->  Fun  `' ( x  e.  A  |->  B ) )
 
19.3.4.5  First and second members of an ordered pair - misc additions
 
Theoremdf1stres 24044* Definition for a restriction of the  1st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  ( 1st  |`  ( A  X.  B ) )  =  ( x  e.  A ,  y  e.  B  |->  x )
 
Theoremdf2ndres 24045* Definition for a restriction of the  2nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  ( 2nd  |`  ( A  X.  B ) )  =  ( x  e.  A ,  y  e.  B  |->  y )
 
Theorem1stnpr 24046 Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  ( -.  A  e.  ( _V 
 X.  _V )  ->  ( 1st `  A )  =  (/) )
 
Theorem2ndnpr 24047 Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  ( -.  A  e.  ( _V 
 X.  _V )  ->  ( 2nd `  A )  =  (/) )
 
Theorem1stpreima 24048 The preimage by  1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
 |-  ( A  C_  B  ->  ( `' ( 1st  |`  ( B  X.  C ) )
 " A )  =  ( A  X.  C ) )
 
Theorem2ndpreima 24049 The preimage by  2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
 |-  ( A  C_  C  ->  ( `' ( 2nd  |`  ( B  X.  C ) )
 " A )  =  ( B  X.  A ) )
 
Theoremcurry2ima 24050* The image of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 25-Sep-2017.)
 |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  B  /\  D  C_  A )  ->  ( G " D )  =  { y  |  E. x  e.  D  y  =  ( x F C ) } )
 
19.3.4.6  Supremum - misc additions
 
Theoremsupssd 24051* Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  B  C_  C )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )   =>    |-  ( ph  ->  -.  sup ( C ,  A ,  R ) R sup ( B ,  A ,  R ) )
 
19.3.4.7  Countable Sets
 
Theoremnnct 24052  NN is countable (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  NN  ~<_  om
 
Theoremctex 24053 A countable set is a set (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  A  e.  _V )
 
Theoremssct 24054 The subset of a countable set is countable (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  (
 ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
 
Theoremxpct 24055 The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  (
 ( A  ~<_  om  /\  B 
 ~<_  om )  ->  ( A  X.  B )  ~<_  om )
 
Theoremsnct 24056 A singleton is countable (Contributed by Thierry Arnoux, 16-Sep-2016.)
 |-  ( A  e.  V  ->  { A }  ~<_  om )
 
Theoremprct 24057 An unordered pair is countable (Contributed by Thierry Arnoux, 16-Sep-2016.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  { A ,  B } 
 ~<_  om )
 
Theoremfnct 24058 If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  (
 ( F  Fn  A  /\  A  ~<_  om )  ->  F  ~<_  om )
 
Theoremdmct 24059 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  dom  A  ~<_  om )
 
Theoremcnvct 24060 If a set is countable, its converse is as well. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  `' A  ~<_  om )
 
Theoremrnct 24061 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  ran  A  ~<_  om )
 
Theoremmptct 24062* A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  ( x  e.  A  |->  B )  ~<_  om )
 
Theoremmpt2cti 24063* An operation is countable if both its domains are countable. (Contributed by Thierry Arnoux, 17-Sep-2017.)
 |-  A. x  e.  A  A. y  e.  B  C  e.  V   =>    |-  (
 ( A  ~<_  om  /\  B 
 ~<_  om )  ->  ( x  e.  A ,  y  e.  B  |->  C )  ~<_ 
 om )
 
Theoremabrexct 24064* An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  { y  |  E. x  e.  A  y  =  B }  ~<_  om )
 
Theoremmptctf 24065 A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  ( A  ~<_  om  ->  ( x  e.  A  |->  B )  ~<_  om )
 
Theoremabrexctf 24066* An image set of a countable set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  ( A  ~<_  om  ->  { y  |  E. x  e.  A  y  =  B } 
 ~<_  om )
 
19.3.5  Real and Complex Numbers
 
19.3.5.1  Complex addition - misc. additions
 
Theoremaddeq0 24067 Two complex which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 2-May-2017.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  =  0  <->  A  =  -u B ) )
 
19.3.5.2  Ordering on reals - misc additions
 
Theoremlt2addrd 24068* If the right-side of a 'less-than' relationship is an addition, then we can express the left-side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  ( B  +  C )
 )   =>    |-  ( ph  ->  E. b  e.  RR  E. c  e. 
 RR  ( A  =  ( b  +  c
 )  /\  b  <  B 
 /\  c  <  C ) )
 
19.3.5.3  Extended reals - misc additions
 
Theoremxgepnf 24069 An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( A  e.  RR*  ->  (  +oo  <_  A  <->  A  =  +oo ) )
 
Theoremxlemnf 24070 An extended real which is less than minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
 |-  ( A  e.  RR*  ->  ( A  <_  -oo  <->  A  =  -oo ) )
 
Theoremxrlelttric 24071 Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremxaddeq0 24072 Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A + e B )  =  0  <->  A  =  - e B ) )
 
Theoreminfxrmnf 24073 The infinimum of a set of extended reals containing minus infnity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
 |-  (
 ( A  C_  RR*  /\  -oo  e.  A )  ->  sup ( A ,  RR* ,  `'  <  )  =  -oo )
 
Theoremxrinfm 24074 The extended real numbers are unbounded below. (Contributed by Thierry Arnoux, 18-Feb-2018.)
 |-  sup ( RR* ,  RR* ,  `'  <  )  =  -oo
 
Theoremle2halvesd 24075 A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  ( C  /  2 ) )   &    |-  ( ph  ->  B  <_  ( C  /  2 ) )   =>    |-  ( ph  ->  ( A  +  B )  <_  C )
 
Theoremxraddge02 24076 A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( 0  <_  B  ->  A  <_  ( A + e B ) ) )
 
Theoremxlt2addrd 24077* If the right-side of a 'less-than' relationship is an addition, then we can express the left-side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  B  =/=  -oo )   &    |-  ( ph  ->  C  =/=  -oo )   &    |-  ( ph  ->  A  <  ( B + e C ) )   =>    |-  ( ph  ->  E. b  e.  RR*  E. c  e.  RR*  ( A  =  ( b + e
 c )  /\  b  <  B  /\  c  <  C ) )
 
Theoremxrsupssd 24078 Inequality deduction for supremum of an extended real subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |-  ( ph  ->  B  C_  C )   &    |-  ( ph  ->  C  C_  RR* )   =>    |-  ( ph  ->  sup ( B ,  RR* ,  <  ) 
 <_  sup ( C ,  RR*
 ,  <  ) )
 
Theoremxrofsup 24079 The supremum is preserved by extended addition set operation. (provided minus infinity is not involved as it does not behave well with addition) (Contributed by Thierry Arnoux, 20-Mar-2017.)
 |-  ( ph  ->  X  C_  RR* )   &    |-  ( ph  ->  Y  C_  RR* )   &    |-  ( ph  ->  sup ( X ,  RR*
 ,  <  )  =/=  -oo )   &    |-  ( ph  ->  sup ( Y ,  RR* ,  <  )  =/=  -oo )   &    |-  ( ph  ->  Z  =  ( + e "
 ( X  X.  Y ) ) )   =>    |-  ( ph  ->  sup ( Z ,  RR* ,  <  )  =  ( sup ( X ,  RR*
 ,  <  ) + e sup ( Y ,  RR*
 ,  <  ) )
 )
 
Theoremsupxrnemnf 24080 The supremum of a non-empty set of extended reals which does not contain minus infinity is not minus infinity. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |-  (
 ( A  C_  RR*  /\  A  =/= 
 (/)  /\  -.  -oo  e.  A )  ->  sup ( A ,  RR* ,  <  )  =/=  -oo )
 
Theoremxrhaus 24081 The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  (ordTop ` 
 <_  )  e.  Haus
 
19.3.5.4  Real number intervals - misc additions
 
Theoremicossicc 24082 A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
 |-  ( A [,) B )  C_  ( A [,] B )
 
Theoremiocssicc 24083 A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  ( A (,] B )  C_  ( A [,] B )
 
Theoremioossico 24084 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
 |-  ( A (,) B )  C_  ( A [,) B )
 
Theoremiocssioo 24085 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <  B ) )  ->  ( C (,] D )  C_  ( A (,) B ) )
 
Theoremicossioo 24086 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  C  /\  D  <_  B )
 )  ->  ( C [,) D )  C_  ( A (,) B ) )
 
Theoremioossioo 24087 Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B )
 )  ->  ( C (,) D )  C_  ( A (,) B ) )
 
Theoremjoiniooico 24088 Disjoint joining an open interval with a closed below, open above interval to form a closed below, open above interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  ( ( ( A (,) B )  i^i  ( B [,) C ) )  =  (/)  /\  (
 ( A (,) B )  u.  ( B [,) C ) )  =  ( A (,) C ) ) )
 
Theoremiccgelb 24089 An element of a closed interval is more than or equal to its lower bound (Contributed by Thierry Arnoux, 23-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  A  <_  C )
 
Theoremsnunioc 24090 The closure of the open end of a left-open real interval. (Contributed by Thierry Arnoux, 28-Mar-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( { A }  u.  ( A (,] B ) )  =  ( A [,] B ) )
 
Theoremubico 24091 A right-open interval does not contain its right endpoint. (Contributed by Thierry Arnoux, 5-Apr-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR* )  ->  -.  B  e.  ( A [,) B ) )
 
Theoremxeqlelt 24092 Equality in terms of 'less than or equal to', 'less than'. (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( A  <_  B  /\  -.  A  <  B ) ) )
 
Theoremeliccelico 24093 Relate elementhood to a closed interval with elementhood to the same closed-below, opened-above interval or to its upper bound. (Contributed by Thierry Arnoux, 3-Jul-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  ( A [,) B )  \/  C  =  B ) ) )
 
Theoremelicoelioo 24094 Relate elementhood to a closed-below, opened-above interval with elementhood to the same opened interval or to its lower bound. (Contributed by Thierry Arnoux, 6-Jul-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  ( C  e.  ( A [,) B )  <->  ( C  =  A  \/  C  e.  ( A (,) B ) ) ) )
 
Theoremiocinioc2 24095 Intersection between two opened below, closed above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 7-Aug-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  (
 ( A (,] C )  i^i  ( B (,] C ) )  =  ( B (,] C ) )
 
Theoremxrdifh 24096 Set difference of a half-opened interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017.)
 |-  A  e.  RR*   =>    |-  ( RR*  \  ( A [,]  +oo ) )  =  (  -oo [,) A )
 
Theoremiocinif 24097 Relate intersection of two opened below, closed above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A (,] C )  i^i  ( B (,] C ) )  =  if ( A  <  B ,  ( B (,] C ) ,  ( A (,] C ) ) )
 
Theoremdifioo 24098 The difference between two opened intervals sharing the same lower bound (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  (
 ( A (,) C )  \  ( A (,) B ) )  =  ( B [,) C ) )
 
Theoremdifico 24099 The difference between two closed below, opened above intervals sharing the same upper bound (Contributed by Thierry Arnoux, 13-Oct-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <_  C ) )  ->  ( ( A [,) C )  \  ( B [,) C ) )  =  ( A [,) B ) )
 
19.3.5.5  Finite intervals of integers - misc additions
 
Theoremfzssnn 24100 Finite sets of sequential integers starting from a natural are a subset of the natural numbers. (Contributed by Thierry Arnoux, 4-Aug-2017.)
 |-  ( M  e.  NN  ->  ( M ... N ) 
 C_  NN )
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