HomeHome Metamath Proof Explorer
Theorem List (p. 62 of 322)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21498)
  Hilbert Space Explorer  Hilbert Space Explorer
(21499-23021)
  Users' Mathboxes  Users' Mathboxes
(23022-32154)
 

Theorem List for Metamath Proof Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremofc2 6101 Right operation by a constant. (Contributed by NM, 7-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ( ph  /\  X  e.  A )  ->  ( F `  X )  =  C )   =>    |-  ( ( ph  /\  X  e.  A )  ->  (
 ( F  o F R ( A  X.  { B } ) ) `
  X )  =  ( C R B ) )
 
Theoremofc12 6102 Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   =>    |-  ( ph  ->  (
 ( A  X.  { B } )  o F R ( A  X.  { C } ) )  =  ( A  X.  { ( B R C ) } ) )
 
Theoremcaofref 6103* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ( ph  /\  x  e.  S )  ->  x R x )   =>    |-  ( ph  ->  F  o R R F )
 
Theoremcaofinvl 6104* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  N : S --> S )   &    |-  ( ph  ->  G  =  ( v  e.  A  |->  ( N `  ( F `
  v ) ) ) )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( ( N `  x ) R x )  =  B )   =>    |-  ( ph  ->  ( G  o F R F )  =  ( A  X.  { B } ) )
 
Theoremcaofid0l 6105* Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( B R x )  =  x )   =>    |-  ( ph  ->  ( ( A  X.  { B }
 )  o F R F )  =  F )
 
Theoremcaofid0r 6106* Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( x R B )  =  x )   =>    |-  ( ph  ->  ( F  o F R ( A  X.  { B } ) )  =  F )
 
Theoremcaofid1 6107* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  (
 ( ph  /\  x  e.  S )  ->  ( x R B )  =  C )   =>    |-  ( ph  ->  ( F  o F R ( A  X.  { B } ) )  =  ( A  X.  { C } ) )
 
Theoremcaofid2 6108* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  (
 ( ph  /\  x  e.  S )  ->  ( B R x )  =  C )   =>    |-  ( ph  ->  (
 ( A  X.  { B } )  o F R F )  =  ( A  X.  { C } ) )
 
Theoremcaofcom 6109* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x R y )  =  ( y R x ) )   =>    |-  ( ph  ->  ( F  o F R G )  =  ( G  o F R F ) )
 
Theoremcaofrss 6110* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x R y  ->  x T y ) )   =>    |-  ( ph  ->  ( F  o R R G  ->  F  o R T G ) )
 
Theoremcaofass 6111* Transfer an associative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x R y ) T z )  =  ( x O ( y P z ) ) )   =>    |-  ( ph  ->  (
 ( F  o F R G )  o F T H )  =  ( F  o F O ( G  o F P H ) ) )
 
Theoremcaoftrn 6112* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x R y  /\  y T z )  ->  x U z ) )   =>    |-  ( ph  ->  ( ( F  o R R G  /\  G  o R T H )  ->  F  o R U H ) )
 
Theoremcaofdi 6113* Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> K )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x T ( y R z ) )  =  ( ( x T y ) O ( x T z ) ) )   =>    |-  ( ph  ->  ( F  o F T ( G  o F R H ) )  =  ( ( F  o F T G )  o F O ( F  o F T H ) ) )
 
Theoremcaofdir 6114* Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> K )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  K )
 )  ->  ( ( x R y ) T z )  =  ( ( x T z ) O ( y T z ) ) )   =>    |-  ( ph  ->  (
 ( G  o F R H )  o F T F )  =  ( ( G  o F T F )  o F O ( H  o F T F ) ) )
 
Theoremcaonncan 6115* Transfer nncan 9076-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  A : I --> S )   &    |-  ( ph  ->  B : I
 --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x M ( x M y ) )  =  y )   =>    |-  ( ph  ->  ( A  o F M ( A  o F M B ) )  =  B )
 
Theoremofmres 6116* Equivalent expressions for a restriction of the function operation map. Unlike  o F R which is a proper class,  (  o F R  |  `  ( A  X.  B
) ) can be a set by ofmresex 6118, allowing it to be used as a function or structure argument. By ofmresval 6117, the restricted operation map values are the same as the original values, allowing theorems for  o F R to be reused. (Contributed by NM, 20-Oct-2014.)
 |-  (  o F R  |`  ( A  X.  B ) )  =  (
 f  e.  A ,  g  e.  B  |->  ( f  o F R g ) )
 
Theoremofmresval 6117 Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
 |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F (  o F R  |`  ( A  X.  B ) ) G )  =  ( F  o F R G ) )
 
Theoremofmresex 6118 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   =>    |-  ( ph  ->  (  o F R  |`  ( A  X.  B ) )  e.  _V )
 
Theoremsuppssof1 6119* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  ( ph  ->  ( `' A " ( _V  \  { Y } )
 )  C_  L )   &    |-  (
 ( ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )   &    |-  ( ph  ->  A : D --> V )   &    |-  ( ph  ->  B : D
 --> R )   &    |-  ( ph  ->  D  e.  W )   =>    |-  ( ph  ->  ( `' ( A  o F O B ) " ( _V  \  { Z }
 ) )  C_  L )
 
2.4.13  First and second members of an ordered pair
 
Syntaxc1st 6120 Extend the definition of a class to include the first member an ordered pair function.
 class  1st
 
Syntaxc2nd 6121 Extend the definition of a class to include the second member an ordered pair function.
 class  2nd
 
Definitiondf-1st 6122 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6128 proves that it does this. For example,  ( 1st ` 
<. 3 ,  4
>. )  =  3. Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5154 and op1stb 4569). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 1st  =  ( x  e.  _V  |->  U. dom  { x } )
 
Definitiondf-2nd 6123 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6129 proves that it does this. For example,  ( 2nd ` 
<. 3 ,  4
>. )  =  4. Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5157 and op2ndb 5156). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 2nd  =  ( x  e.  _V  |->  U. ran  { x } )
 
Theorem1stval 6124 The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( 1st `  A )  =  U. dom  { A }
 
Theorem2ndval 6125 The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( 2nd `  A )  =  U. ran  { A }
 
Theorem1st0 6126 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 1st `  (/) )  =  (/)
 
Theorem2nd0 6127 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 2nd `  (/) )  =  (/)
 
Theoremop1st 6128 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( 1st `  <. A ,  B >. )  =  A
 
Theoremop2nd 6129 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( 2nd `  <. A ,  B >. )  =  B
 
Theoremop1std 6130 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( C  =  <. A ,  B >.  ->  ( 1st `  C )  =  A )
 
Theoremop2ndd 6131 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( C  =  <. A ,  B >.  ->  ( 2nd `  C )  =  B )
 
Theoremop1stg 6132 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
 
Theoremop2ndg 6133 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
 
Theoremot1stg 6134 Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 6134, ot2ndg 6135, ot3rdg 6136.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A )
 
Theoremot2ndg 6135 Extract the second member of an ordered triple. (See ot1stg 6134 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B )
 
Theoremot3rdg 6136 Extract the third member of an ordered triple. (See ot1stg 6134 comment.) (Contributed by NM, 3-Apr-2015.)
 |-  ( C  e.  V  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
 
Theorem1stval2 6137 Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )
 
Theorem2ndval2 6138 Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  ( 2nd `  A )  =  |^| |^| |^| `' { A } )
 
Theoremfo1st 6139 The  1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |- 
 1st : _V -onto-> _V
 
Theoremfo2nd 6140 The  2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |- 
 2nd : _V -onto-> _V
 
Theoremf1stres 6141 Mapping of a restriction of the 
1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) --> A
 
Theoremf2ndres 6142 Mapping of a restriction of the 
2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B ) --> B
 
Theoremfo1stres 6143 Onto mapping of a restriction of the  1st (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)
 |-  ( B  =/=  (/)  ->  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> A )
 
Theoremfo2ndres 6144 Onto mapping of a restriction of the  2nd (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)
 |-  ( A  =/=  (/)  ->  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> B )
 
Theorem1st2val 6145* Value of an alternate definition of the  1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( { <. <. x ,  y >. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A )
 
Theorem2nd2val 6146* Value of an alternate definition of the  2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( { <. <. x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( 2nd `  A )
 
Theorem1stcof 6147 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
 |-  ( F : A --> ( B  X.  C ) 
 ->  ( 1st  o.  F ) : A --> B )
 
Theorem2ndcof 6148 Composition of the first member function with another function. (Contributed by FL, 15-Oct-2012.)
 |-  ( F : A --> ( B  X.  C ) 
 ->  ( 2nd  o.  F ) : A --> C )
 
Theoremxp1st 6149 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  ( B  X.  C )  ->  ( 1st `  A )  e.  B )
 
Theoremxp2nd 6150 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  ( B  X.  C )  ->  ( 2nd `  A )  e.  C )
 
Theoremelxp6 6151 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5160. (Contributed by NM, 9-Oct-2004.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >.  /\  (
 ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
 
Theoremelxp7 6152 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5160. (Contributed by NM, 19-Aug-2006.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
 
Theoremdifxp 6153 Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.)
 |-  ( ( C  X.  D )  \  ( A  X.  B ) )  =  ( ( ( C  \  A )  X.  D )  u.  ( C  X.  ( D  \  B ) ) )
 
Theoremdifxp1 6154 Difference law for cross product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)
 |-  ( ( A  \  B )  X.  C )  =  ( ( A  X.  C )  \  ( B  X.  C ) )
 
Theoremdifxp2 6155 Difference law for cross product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)
 |-  ( A  X.  ( B  \  C ) )  =  ( ( A  X.  B )  \  ( A  X.  C ) )
 
Theoremeqopi 6156 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( A  e.  ( V  X.  W ) 
 /\  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C )
 )  ->  A  =  <. B ,  C >. )
 
Theoremxp2 6157* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  X.  B )  =  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x )  e.  B ) }
 
Theoremunielxp 6158 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  e.  ( B  X.  C )  ->  U. A  e.  U. ( B  X.  C ) )
 
Theorem1st2nd2 6159 Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
 |-  ( A  e.  ( B  X.  C )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >. )
 
Theorem1st2ndb 6160 Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.)
 |-  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >. )
 
Theoremxpopth 6161 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
 |-  ( ( A  e.  ( C  X.  D ) 
 /\  B  e.  ( R  X.  S ) ) 
 ->  ( ( ( 1st `  A )  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B ) )  <->  A  =  B ) )
 
Theoremeqop 6162 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >.  <->  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C )
 ) )
 
Theoremeqop2 6163 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  =  <. B ,  C >.  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C ) ) )
 
Theoremop1steq 6164* Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)
 |-  ( A  e.  ( V  X.  W )  ->  ( ( 1st `  A )  =  B  <->  E. x  A  =  <. B ,  x >. ) )
 
Theorem2nd1st 6165 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
 |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  <. ( 2nd `  A ) ,  ( 1st `  A ) >. )
 
Theorem1st2nd 6166 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
 |-  ( ( Rel  B  /\  A  e.  B ) 
 ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >. )
 
Theorem1stdm 6167 The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)
 |-  ( ( Rel  R  /\  A  e.  R ) 
 ->  ( 1st `  A )  e.  dom  R )
 
Theorem2ndrn 6168 The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)
 |-  ( ( Rel  R  /\  A  e.  R ) 
 ->  ( 2nd `  A )  e.  ran  R )
 
Theorem1st2ndbr 6169 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
 |-  ( ( Rel  B  /\  A  e.  B ) 
 ->  ( 1st `  A ) B ( 2nd `  A ) )
 
Theoremreleldm2 6170* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
 |-  ( Rel  A  ->  ( B  e.  dom  A  <->  E. x  e.  A  ( 1st `  x )  =  B ) )
 
Theoremreldm 6171* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
 |-  ( Rel  A  ->  dom 
 A  =  ran  ( x  e.  A  |->  ( 1st `  x ) ) )
 
Theoremsbcopeq1a 6172 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3001 that avoids the existential quantifiers of copsexg 4252). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  =  <. x ,  y >.  ->  ( [. ( 1st `  A )  /  x ]. [. ( 2nd `  A )  /  y ]. ph  <->  ph ) )
 
Theoremcsbopeq1a 6173 Equality theorem for substitution of a class  A for an ordered pair  <. x ,  y >. in  B (analog of csbeq1a 3089). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  =  <. x ,  y >.  ->  [_ ( 1st `  A )  /  x ]_ [_ ( 2nd `  A )  /  y ]_ B  =  B )
 
Theoremdfopab2 6174* A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 { <. x ,  y >.  |  ph }  =  { z  e.  ( _V  X.  _V )  | 
 [. ( 1st `  z
 )  /  x ]. [. ( 2nd `  z )  /  y ]. ph }
 
Theoremdfoprab3s 6175* A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) }
 
Theoremdfoprab3 6176* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
 |-  ( w  =  <. x ,  y >.  ->  ( ph 
 <->  ps ) )   =>    |-  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  ph ) }  =  { <. <. x ,  y >. ,  z >.  |  ps }
 
Theoremdfoprab4 6177* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( w  =  <. x ,  y >.  ->  ( ph 
 <->  ps ) )   =>    |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) }
 
Theoremdfoprab4f 6178* Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 F/ x ph   &    |-  F/ y ph   &    |-  ( w  =  <. x ,  y >.  ->  ( ph  <->  ps ) )   =>    |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) }
 
Theoremdfxp3 6179* Define the cross product of three classes. Compare df-xp 4695. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
 |-  ( ( A  X.  B )  X.  C )  =  { <. <. x ,  y >. ,  z >.  |  ( x  e.  A  /\  y  e.  B  /\  z  e.  C ) }
 
Theoremcopsex2gb 6180* Implicit substitution inference for ordered pairs. Compare copsex2ga 6181. (Contributed by NM, 12-Mar-2014.)
 |-  ( A  =  <. x ,  y >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x E. y ( A  =  <. x ,  y >.  /\ 
 ps )  <->  ( A  e.  ( _V  X.  _V )  /\  ph ) )
 
Theoremcopsex2ga 6181* Implicit substitution inference for ordered pairs. Compare copsex2g 4254. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  =  <. x ,  y >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( A  e.  ( V  X.  W ) 
 ->  ( ph  <->  E. x E. y
 ( A  =  <. x ,  y >.  /\  ps ) ) )
 
Theoremelopaba 6182* Membership in an ordered pair class builder. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  =  <. x ,  y >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( A  e.  {
 <. x ,  y >.  |  ps }  <->  ( A  e.  ( _V  X.  _V )  /\  ph ) )
 
Theoremexopxfr 6183* Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( x  =  <. y ,  z >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x  e.  ( _V  X.  _V ) ph  <->  E. y E. z ps )
 
Theoremexopxfr2 6184* Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.)
 |-  ( x  =  <. y ,  z >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( Rel  A  ->  ( E. x  e.  A  ph  <->  E. y E. z
 ( <. y ,  z >.  e.  A  /\  ps ) ) )
 
Theoremelopabi 6185* A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
 |-  ( x  =  ( 1st `  A )  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  ( 2nd `  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( A  e.  {
 <. x ,  y >.  | 
 ph }  ->  ch )
 
Theoremeloprabi 6186* A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( x  =  ( 1st `  ( 1st `  A ) )  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  ( 2nd `  ( 1st `  A ) )  ->  ( ps  <->  ch ) )   &    |-  ( z  =  ( 2nd `  A )  ->  ( ch  <->  th ) )   =>    |-  ( A  e.  {
 <. <. x ,  y >. ,  z >.  |  ph } 
 ->  th )
 
Theoremmpt2mptsx 6187* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( x  e.  A ,  y  e.  B  |->  C )  =  (
 z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
 
Theoremmpt2mpts 6188* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
 |-  ( x  e.  A ,  y  e.  B  |->  C )  =  (
 z  e.  ( A  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
 
Theoremdmmpt2ssx 6189* The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  dom  F  C_  U_ x  e.  A  ( { x }  X.  B )
 
Theoremfmpt2x 6190* Functionality, domain and codomain of a class given by the "maps to" notation, where  B ( x ) is not constant but depends on  x. (Contributed by NM, 29-Dec-2014.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( A. x  e.  A  A. y  e.  B  C  e.  D  <->  F : U_ x  e.  A  ( { x }  X.  B ) --> D )
 
Theoremfmpt2 6191* Functionality, domain and range of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( A. x  e.  A  A. y  e.  B  C  e.  D  <->  F : ( A  X.  B ) --> D )
 
Theoremfnmpt2 6192* Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  F  Fn  ( A  X.  B ) )
 
Theoremfnmpt2i 6193* Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  C  e.  _V   =>    |-  F  Fn  ( A  X.  B )
 
Theoremdmmpt2 6194* Domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  C  e.  _V   =>    |- 
 dom  F  =  ( A  X.  B )
 
Theoremmpt2exxg 6195* Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( ( A  e.  R  /\  A. x  e.  A  B  e.  S )  ->  F  e.  _V )
 
Theoremmpt2exg 6196* Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( ( A  e.  R  /\  B  e.  S )  ->  F  e.  _V )
 
Theoremmpt2exga 6197* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 12-Sep-2011.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  A ,  y  e.  B  |->  C )  e. 
 _V )
 
Theoremmpt2ex 6198* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by Mario Carneiro, 20-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  e.  _V
 
Theoremmpt20 6199 A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  (/)
 
Theoremovmptss 6200* If all the values of the mapping are subsets of a class  X, then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( A. x  e.  A  A. y  e.  B  C  C_  X  ->  ( E F G )  C_  X )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32154
  Copyright terms: Public domain < Previous  Next >