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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprelpwi 23201 A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)
 |-  (
 ( A  e.  C  /\  B  e.  C ) 
 ->  { A ,  B }  e.  ~P C )
 
Theorempwundif2 23202 A shorter proof for pwundif 4316 (Contributed by Thierry Arnoux, 27-Jul-2016.)
 |-  ~P ( A  u.  B )  =  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )
 
Theoremeqri 23203 Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  ( x  e.  A  <->  x  e.  B )   =>    |-  A  =  B
 
Theoremelpreq 23204 Equality wihin a pair (Contributed by Thierry Arnoux, 23-Aug-2017.)
 |-  ( ph  ->  X  e.  { A ,  B }
 )   &    |-  ( ph  ->  Y  e.  { A ,  B } )   &    |-  ( ph  ->  ( X  =  A  <->  Y  =  A ) )   =>    |-  ( ph  ->  X  =  Y )
 
Theoremrnpropg 23205 The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ran  { <. A ,  C >. ,  <. B ,  D >. }  =  { C ,  D }
 )
 
Theoremopfv 23206 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 ) >. )
 
Theoremabrexexd 23207* Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)
 |-  F/_ x A   &    |-  ( ph  ->  A  e.  _V )   =>    |-  ( ph  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
 
Theoremelabreximdv 23208* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  ( A  =  B  ->  ( ch  <->  ps ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  C )  ->  ps )   =>    |-  (
 ( ph  /\  A  e.  { y  |  E. x  e.  C  y  =  B } )  ->  ch )
 
Theorempreqsnd 23209 Equivalence for a pair equal to a singleton, deduction form. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   =>    |-  ( ph  ->  ( { A ,  B }  =  { C } 
 <->  ( A  =  C  /\  B  =  C ) ) )
 
Theoremunidmvol 23210 The union of the Lebesgue measurable sets is  RR. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  U. dom  vol 
 =  RR
 
Theoremitgeq12dv 23211* Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  =  D )   =>    |-  ( ph  ->  S. A C  _d x  =  S. B D  _d x )
 
18.3.11  Functions and relations - misc additions
 
Theoremxpdisjres 23212 Restriction of a constant function (or other cross product) outside of its domain (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  (
 ( A  i^i  C )  =  (/)  ->  (
 ( A  X.  B )  |`  C )  =  (/) )
 
Theoremcsbcnvg 23213 Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
 |-  ( A  e.  V  ->  `'
 [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F )
 
Theoremfimacnvinrn 23214 Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( Fun  F  ->  ( `' F " A )  =  ( `' F "
 ( A  i^i  ran  F ) ) )
 
Theoremfimacnvinrn2 23215 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 23216* 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 )
 
Theoremxpima 23217 The image of a constant function (or other cross product). (Contributed by Thierry Arnoux, 4-Feb-2017.)
 |-  (
 ( A  X.  B ) " C )  =  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )
 
Theoremfovcld 23218 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 )
 
Theoremdfrel4 23219* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5584 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.)
 |-  F/_ x R   &    |-  F/_ y R   =>    |-  ( Rel  R  <->  R  =  { <. x ,  y >.  |  x R y }
 )
 
Theoremelovimad 23220 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 23221 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 ) --> B )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ran  ( F  o F  .+  G )  C_  B )
 
Theoremofrn2 23222 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 ) --> B )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ran  ( F  o F  .+  G )  C_  (  .+  " ( ran  F  X.  ran  G ) ) )
 
Theoremoff2 23223* 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 )
 
Theoremunipreima 23224* Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)
 |-  ( Fun  F  ->  ( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
 
Theoremsspreima 23225 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 ) )
 
Theoremxppreima 23226 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 23227* 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 23228* 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 23229* 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 23230 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 23231* 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 23232* 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 23233* 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 23234* 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 23235* 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 23236 Domain and co-domain of the mapping operation; deduction form. This version of fmptd 5700 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 23237* 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 23238 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 23239 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 )
 
Theoremfvmpt2d 23240* Deduction version of fvmpt2 5624. (Contributed by Thierry Arnoux, 8-Dec-2016.)
 |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   =>    |-  ( ( ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
 
Theoremmptfnf 23241 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 23242 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 23243 Deduction form of dffn5f 5593. (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 23244* 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 23245* Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 5710. (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 23246* 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 23247* 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 23248* 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 ) ) )
 
TheoremfuncnvmptOLD 23249* 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 ( x  e.  A  /\  y  =  B ) ) )
 
Theoremfuncnvmpt 23250* 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 23251* 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 23252* 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 ) ) )
 
Theoremrnmptss 23253* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  e.  C  ->  ran  F  C_  C )
 
Theoremrnmpt2ss 23254* 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 23255 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 ) )
 
Theoremgtiso 23256 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 23257* 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 ) ) )
 
18.3.12  First and second members of an ordered pair - misc additions
 
Theoremdf1stres 23258* 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 23259* 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 23260 Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  ( -.  A  e.  ( _V 
 X.  _V )  ->  ( 1st `  A )  =  (/) )
 
Theorem2ndnpr 23261 Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  ( -.  A  e.  ( _V 
 X.  _V )  ->  ( 2nd `  A )  =  (/) )
 
Theoremcurry2ima 23262* 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 ) } )
 
18.3.13  Supremum - misc additions
 
Theoremsupssd 23263* 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 ) )
 
18.3.14  Ordering on reals - misc additions
 
Theoremlt2addrd 23264* 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 ) )
 
18.3.15  Extended reals - misc additions
 
Theoremxrlelttric 23265 Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremxrre3FL 23266 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.) (TODO remove and use xrre3 10516, which was imported )
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  <  +oo )
 )  ->  A  e.  RR )
 
Theoremxraddge02 23267 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 23268* 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 23269 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 23270 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 23271 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 23272 The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  (ordTop ` 
 <_  )  e.  Haus
 
18.3.16  Real number intervals - misc additions
 
Theoremicossicc 23273 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 23274 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 23275 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
 |-  ( A (,) B )  C_  ( A [,) B )
 
Theoremiocssioo 23276 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 23277 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 ) )
 
Theoremicossico 23278 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B )
 )  ->  ( C [,) D )  C_  ( A [,) B ) )
 
Theoremioossioo 23279 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 23280 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 23281 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 23282 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 23283 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 23284 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 23285 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 23286 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 23287 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 23288 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 23289 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 23290 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 ) )
 
18.3.17  Finite intervals of integers - misc additions
 
Theoremfzssnn 23291 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 )
 
Theoremssnnssfz 23292* For any finite subset of  NN, find a superset in the form of a set of sequential integers. (Contributed by Thierry Arnoux, 13-Sep-2017.)
 |-  ( A  e.  ( ~P NN  i^i  Fin )  ->  E. n  e.  NN  A  C_  (
 1 ... n ) )
 
18.3.18  Half-open integer ranges - misc additions
 
Theoremfzossnn 23293 Half-opened integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  (
 1..^ N )  C_  NN
 
Theoremelfzo1 23294 Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  ( N  e.  ( 1..^ M )  <->  ( N  e.  NN  /\  M  e.  NN  /\  N  <  M ) )
 
18.3.19  Closed unit
 
Theoremunitsscn 23295 The closed unit is a subset of the set of the complex numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  (
 0 [,] 1 )  C_  CC
 
Theoremelunitrn 23296 The closed unit is a subset of the set of the real numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 21-Dec-2016.)
 |-  ( A  e.  ( 0 [,] 1 )  ->  A  e.  RR )
 
Theoremelunitcn 23297 The closed unit is a subset of the set of the complext numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 21-Dec-2016.)
 |-  ( A  e.  ( 0 [,] 1 )  ->  A  e.  CC )
 
Theoremelunitge0 23298 An element of the closed unit is positive Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 20-Dec-2016.)
 |-  ( A  e.  ( 0 [,] 1 )  ->  0  <_  A )
 
Theoremunitssxrge0 23299 The closed unit is a subset of the set of the extended non-negative reals. Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  (
 0 [,] 1 )  C_  ( 0 [,]  +oo )
 
Theoremunitdivcld 23300 Necessary conditions for a quotient to be in the closed unit. (somewhat too strong, it would be sufficient that A and B are in RR+) (Contributed by Thierry Arnoux, 20-Dec-2016.)
 |-  (
 ( A  e.  (
 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  ->  ( A  <_  B  <->  ( A  /  B )  e.  (
 0 [,] 1 ) ) )
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