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Theorem List for Metamath Proof Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcaofdir 6201* 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 6202* Transfer nncan 9166-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 6203* 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 6205, allowing it to be used as a function or structure argument. By ofmresval 6204, 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 6204 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 6205 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 6206* 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 6207 Extend the definition of a class to include the first member an ordered pair function.
 class  1st
 
Syntaxc2nd 6208 Extend the definition of a class to include the second member an ordered pair function.
 class  2nd
 
Definitiondf-1st 6209 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6215 proves that it does this. For example,  ( 1st ` 
<. 3 ,  4
>. )  =  3. Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5236 and op1stb 4651). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 1st  =  ( x  e.  _V  |->  U. dom  { x } )
 
Definitiondf-2nd 6210 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6216 proves that it does this. For example,  ( 2nd ` 
<. 3 ,  4
>. )  =  4. Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5239 and op2ndb 5238). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 2nd  =  ( x  e.  _V  |->  U. ran  { x } )
 
Theorem1stval 6211 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 6212 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 6213 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 1st `  (/) )  =  (/)
 
Theorem2nd0 6214 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 2nd `  (/) )  =  (/)
 
Theoremop1st 6215 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 6216 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 6217 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 6218 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 6219 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 6220 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 6221 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 6221, ot2ndg 6222, ot3rdg 6223.) (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 6222 Extract the second member of an ordered triple. (See ot1stg 6221 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 6223 Extract the third member of an ordered triple. (See ot1stg 6221 comment.) (Contributed by NM, 3-Apr-2015.)
 |-  ( C  e.  V  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
 
Theorem1stval2 6224 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 6225 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 6226 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 6227 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 6228 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 6229 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 6230 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 6231 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 6232* 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 6233* 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 6234 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 6235 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 6236 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 6237 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 6238 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5242. (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 6239 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5242. (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 6240 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 6241 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 6242 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 6243 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 6244* 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 6245 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 6246 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 6247 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 6248 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 6249 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 6250 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 6251* 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 6252 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 6253 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 6254 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 6255 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 6256 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 6257* 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 6258* 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 6259 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3077 that avoids the existential quantifiers of copsexg 4334). (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 6260 Equality theorem for substitution of a class  A for an ordered pair  <. x ,  y >. in  B (analog of csbeq1a 3165). (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 6261* 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 6262* 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 6263* 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 6264* 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 6265* 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 6266* Define the cross product of three classes. Compare df-xp 4777. (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 6267* Implicit substitution inference for ordered pairs. Compare copsex2ga 6268. (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 6268* Implicit substitution inference for ordered pairs. Compare copsex2g 4336. (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 6269* 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 6270* 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 6271* 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 6272* 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 6273* 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 6274* 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 6275* 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 6276* 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 6277* 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 6278* 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 6279* 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 6280* 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 6281* 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 6282* 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 6283* 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 6284* 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 6285* 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 6286 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 6287* 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 )
 
Theoremrelmpt2opab 6288* Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  { <. z ,  w >.  |  ph } )   =>    |-  Rel  ( C F D )
 
Theoremfmpt2co 6289* Composition of two functions. Variation of fmptco 5774 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  R  e.  C )   &    |-  ( ph  ->  F  =  ( x  e.  A ,  y  e.  B  |->  R ) )   &    |-  ( ph  ->  G  =  ( z  e.  C  |->  S ) )   &    |-  (
 z  =  R  ->  S  =  T )   =>    |-  ( ph  ->  ( G  o.  F )  =  ( x  e.  A ,  y  e.  B  |->  T ) )
 
Theoremoprabco 6290* Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
 |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  e.  D )   &    |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( H `  C ) )   =>    |-  ( H  Fn  D  ->  G  =  ( H  o.  F ) )
 
Theoremoprab2co 6291* Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
 |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  e.  R )   &    |-  ( ( x  e.  A  /\  y  e.  B )  ->  D  e.  S )   &    |-  F  =  ( x  e.  A ,  y  e.  B  |->  <. C ,  D >. )   &    |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( C M D ) )   =>    |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
 
Theoremdf1st2 6292* An alternate possible definition of the  1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
 
Theoremdf2nd2 6293* An alternate possible definition of the  2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  z  =  y }  =  ( 2nd  |`  ( _V  X.  _V ) )
 
Theorem1stconst 6294 The mapping of a restriction of the  1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
 |-  ( B  e.  V  ->  ( 1st  |`  ( A  X.  { B }
 ) ) : ( A  X.  { B } ) -1-1-onto-> A )
 
Theorem2ndconst 6295 The mapping of a restriction of the  2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
 |-  ( A  e.  V  ->  ( 2nd  |`  ( { A }  X.  B ) ) : ( { A }  X.  B ) -1-1-onto-> B )
 
Theoremdfmpt2 6296* Alternate definition for the "maps to" notation df-mpt2 5950 (although it requires that  C be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  C  e.  _V   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  = 
 U_ x  e.  A  U_ y  e.  B  { <.
 <. x ,  y >. ,  C >. }
 
Theoremcurry1 6297* Composition with  `' ( 2nd  |`  ( { C }  X.  _V ) ) turns any binary operation  F with a constant first operand into a function  G of the second operand only. This transformation is called "currying." (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)
 |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
 
Theoremcurry1val 6298 The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  A )  ->  ( G `  D )  =  ( C F D ) )
 
Theoremcurry1f 6299 Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)
 |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )   =>    |-  ( ( F :
 ( A  X.  B )
 --> D  /\  C  e.  A )  ->  G : B
 --> D )
 
Theoremcurry2 6300* Composition with  `' ( 1st  |`  ( _V  X.  { C }
) ) turns any binary operation  F with a constant second operand into a function  G of the first operand only. This transformation is called "currying." (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.)
 |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
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