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Theorem List for Metamath Proof Explorer - 24401-24500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoprssopvg 24401 Value returned by the operation  G in terms of the value returned by the "super"-operation  F. (A version of oprssov 5923 adapted to partial operations.) (Contributed by FL, 5-Oct-2009.)
 |-  (
 ( Fun  F  /\  G  C_  F  /\  <. A ,  B >.  e.  dom  G )  ->  ( A F B )  =  ( A G B ) )
 
Theoremdmoprabss6 24402* The domain of an operation class abstraction. (A version of dmoprabss 5863 adapted to partial operations.) (Contributed by FL, 18-Apr-2010.)
 |-  B  e.  C   =>    |-  ( Rel  A  ->  dom  { <. <. x ,  y >. ,  z >.  |  (
 <. x ,  y >.  e.  A  /\  z  =  B ) }  =  A )
 
Theoremoprabex2gpop 24403* Existence of an operation class abstraction. (A version of mptex 5680 adapted to partial operations.) (Contributed by FL, 18-Apr-2010.)
 |-  (
 ( R  e.  B  /\  Rel  R )  ->  { <. <. x ,  y >. ,  z >.  |  (
 <. x ,  y >.  e.  R  /\  z  =  A ) }  e.  _V )
 
Theoremdfoprab4pop 24404* Class abstraction for operations in terms of class abstraction of ordered pairs. (A version of dfoprab4 6111 adapted to partial operations.) (Contributed by FL, 18-Apr-2010.)
 |-  ( w  =  <. x ,  y >.  ->  ( ph  <->  ps ) )   =>    |-  ( Rel  R  ->  { <. w ,  z >.  |  ( w  e.  R  /\  ph ) }  =  { <. <. x ,  y >. ,  z >.  |  ( <. x ,  y >.  e.  R  /\  ps ) } )
 
Theoremfnovpop 24405* Representation of an operation class abstraction in terms of its values. (A version of fnov 5886 adapted to partial operations.) (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  R  ->  ( F  Fn  R  <->  F  =  { <.
 <. x ,  y >. ,  z >.  |  ( <. x ,  y >.  e.  R  /\  z  =  ( x F y ) ) } )
 )
 
18.12.5  General Set Theory
 
Theoremuninqs 24406 Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 3821. (Contributed by FL, 25-May-2007.) (Proof shortened by Mario Carneiro, 11-Jul-2014.)
 |-  R  Er  X   =>    |-  ( ( B  C_  ( A /. R ) 
 /\  C  C_  ( A /. R ) ) 
 ->  U. ( B  i^i  C )  =  ( U. B  i^i  U. C ) )
 
Theoremdifeqri2 24407* Inference from membership to difference. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( A. x ( ( x  e.  A  /\  -.  x  e.  B )  <->  x  e.  C )  ->  ( A  \  B )  =  C )
 
Theoremelo 24408* The law of concretion for operation class abstraction. Compare with eloprabg 5869. This version is to be used with categories. (Contributed by FL, 14-Jul-2007.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 y  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 z  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 v  =  C  ->  ( ch  <->  th ) )   &    |-  ( w  =  D  ->  ( th  <->  ta ) )   =>    |-  ( ( ( A  e.  Q  /\  B  e.  R  /\  C  e.  S )  /\  D  e.  T ) 
 ->  ( <. <. A ,  B >. ,  <. C ,  D >.
 >.  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >.
 >.  /\  ph ) }  <->  ta ) )
 
Theoreminpws1 24409 An intersection with a member of a powerset belongs to this powerset. (Contributed by FL, 25-Sep-2007.)
 |-  ( A  e.  ~P C  ->  ( A  i^i  B )  e.  ~P C )
 
Theoreminpws2 24410 An intersection with a member of a powerset belongs to this powerset. (Contributed by FL, 26-Oct-2007.)
 |-  ( B  e.  ~P C  ->  ( A  i^i  B )  e.  ~P C )
 
Theoremstcat 24411* Structure of the class abstraction used by  Alg, 
Cat and  Ded. (Contributed by FL, 26-Oct-2007.)
 |-  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
 
Theorem11st22nd 24412 A theorem of the 1st2nd 6100 family. (Contributed by FL, 26-Oct-2007.)
 |-  (
 ( ( Rel  B  /\  Rel  dom  B  /\  Rel 
 ran  B )  /\  A  e.  B )  ->  A  =  <. <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
 >. ,  <. ( 1st `  ( 2nd `  A ) ) ,  ( 2nd `  ( 2nd `  A ) )
 >. >. )
 
Theoremump 24413* The union of a part of a powerset belongs to it. (Contributed by FL, 16-Nov-2007.)
 |-  ( A  e.  V  ->  U.
 { x  e.  ~P A  |  ph }  e.  ~P A )
 
Theoremmoec 24414 Moving an element  B out from the intersection of a class  A. (Contributed by FL, 29-Nov-2007.)
 |-  ( B  e.  A  ->  |^|
 A  =  ( B  i^i  |^| ( A  \  { B } ) ) )
 
Theoremsplint 24415* Splitting an intersection. (Contributed by FL, 15-Oct-2012.)
 |-  ( B  C_  A  ->  |^|_ x  e.  A  C  =  (
 |^|_ x  e.  ( A  \  B ) C  i^i  |^|_ x  e.  B  C ) )
 
Theoremsplintx 24416* Moving an element out from an intersection. (Contributed by FL, 15-Oct-2012.)
 |-  ( x  =  B  ->  C  =  D )   =>    |-  ( B  e.  A  ->  |^|_ x  e.  A  C  =  ( |^|_ x  e.  ( A  \  { B } ) C  i^i  D ) )
 
Theoremfnovrn2 24417 A function's value belongs to its range. A more general version of fnovrn 5929. To be used with partial operations. (Contributed by FL, 10-Mar-2008.)
 |-  (
 ( Fun  F  /\  <. A ,  B >.  e. 
 dom  F )  ->  ( A F B )  e. 
 ran  F )
 
Theoremneiopne 24418 If an intersection is not empty its operands are not empty. (Contributed by FL, 27-Apr-2008.)
 |-  (
 ( A  i^i  B )  =/=  (/)  ->  ( A  =/= 
 (/)  /\  B  =/=  (/) ) )
 
Theoremf2imacnv 24419 Image of a preimage. (Contributed by FL, 1-Jun-2008.)
 |-  (
 ( F : A -1-1-onto-> B  /\  C  C_  B )  ->  ( F " ( `' F " C ) )  =  C )
 
Theoremoooeqim2 24420 Symmetrical equality of the images and of their antecedents when the mapping is one to one. (Contributed by FL, 1-Jun-2008.)
 |-  (
 ( F : A -1-1-> B 
 /\  X  C_  A  /\  Y  C_  A )  ->  ( ( F " X )  =  ( F " Y )  <->  X  =  Y ) )
 
Theoremvxveqv 24421 A theorem about things which don't exist  _V and  ( _V  X.  _V ). (Contributed by FL, 22-Sep-2008.)
 |-  ( _V  X.  _V )  =/= 
 _V
 
Theoremducidu 24422 The double union of the converse of a class  A is included in the double union of the class. (Contributed by FL, 31-Jul-2009.)
 |-  U. U. `' A  C_  U. U. A
 
Theoremfldcnv 24423 The field of a class equals the field of the its converse. (Contributed by FL, 16-Apr-2012.)
 |-  ( dom  A  u.  ran  A )  =  ( dom  `'  A  u.  ran  `'  A )
 
Theoremdomfldrefc 24424* The domain of a reflexive class equals its field. (Contributed by FL, 29-Dec-2011.)
 |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  ->  dom  R  =  ( dom 
 R  u.  ran  R ) )
 
Theoremranfldrefc 24425* The range of a reflexive class equals its field. (Contributed by FL, 29-Dec-2011.)
 |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  ->  ran  R  =  ( dom 
 R  u.  ran  R ) )
 
Theoremdranfldrefc 24426* The domain and range of a reflexive class are equal. (Contributed by FL, 29-Dec-2011.)
 |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  ->  dom  R  =  ran  R )
 
Theoremdomrngref 24427* Domain and range of a reflexive relation are equal. (Contributed by FL, 6-Oct-2008.)
 |-  (
 ( Rel  R  /\  A. x  e.  U. U. R x R x ) 
 ->  dom  R  =  ran  R )
 
Theoremdomfldref 24428* The domain of a reflexive relation is equal to its field . (Contributed by FL, 6-Oct-2008.)
 |-  (
 ( Rel  R  /\  A. x  e.  U. U. R x R x ) 
 ->  dom  R  =  U. U. R )
 
Theoremdomintreflemb 24429* In a reflexive class  R, an element  A belongs to the field iff the pair  <. A ,  A >. belongs to  R. (Contributed by FL, 30-Dec-2011.)
 |-  (
 ( A  e.  B  /\  A. x  e.  dom  R  x R x ) 
 ->  ( A  e.  dom  R  <->  A R A ) )
 
Theoremdomintrefb 24430* The domain of the intersection of two reflexive classes is the intersection of their domains. Compare with dmin 4874. (Contributed by FL, 30-Dec-2011.)
 |-  (
 ( A. x  e.  dom  R  x R x  /\  A. x  e.  dom  S  x S x )  ->  dom  (  R  i^i  S )  =  ( dom  R  i^i  dom  S )
 )
 
Theoremimgfldref2 24431* If  R is a reflexive relation and  A a part of its field,  A is a part of the image of  A by  R. (Contributed by FL, 3-Jul-2009.)
 |-  (
 ( A. x  e.  U. U. R x R x 
 /\  A  C_  U. U. R )  ->  A  C_  ( R " A ) )
 
Theoremcnvref 24432* The converse of a reflexive class is reflexive. (Contributed by FL, 31-Jul-2009.)
 |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  <->  A. x  e.  ( dom  `'  R  u.  ran  `'  R ) x `' R x )
 
Theoremcnvref2 24433* The converse of a reflexive relation is reflexive. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  R  ->  ( A. x  e.  U. U. R x R x  <->  A. x  e.  U. U. `' R x `' R x ) )
 
Theoremsrefwref 24434* Strong reflexivity implies weak reflexivity. (Strong and weak reflexivity is the difference between a toset and a poset). (Contributed by FL, 29-Dec-2011.)
 |-  ( A. x  e.  ( dom  R  u.  ran  R ) A. y  e.  ( dom  R  u.  ran  R ) ( x R y  \/  y R x )  ->  A. x  e.  ( dom  R  u.  ran 
 R ) x R x )
 
Theoremfeq123 24435 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
 |-  (
 ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( F : A --> B 
 <->  G : C --> D ) )
 
Theoremunfinsef 24436 A class whose union is finite is finite. (Contributed by FL, 22-Dec-2008.)
 |-  ( U. A  e.  Fin  ->  A  e.  Fin )
 
Theoremf1ofi 24437 If the domain of a bijection is finite its range is finite and reciprocally. (Contributed by FL, 31-Jul-2009.)
 |-  ( F : A -1-1-onto-> B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
 
Theoremscprefat 24438 A square cross product  ( A  X.  A
) is reflexive. (Contributed by FL, 31-Jul-2009.)
 |-  (  _I  |`  U. U. ( A  X.  A ) ) 
 C_  ( A  X.  A )
 
Theoremscprefat2 24439 A square cross product  ( A  X.  A
) is reflexive. (Contributed by FL, 31-Jul-2009.)
 |-  (  _I  |`  A )  C_  ( A  X.  A )
 
Theoremsqpsym 24440 A square cross product is symmetric. (Contributed by FL, 31-Jul-2009.)
 |-  `' ( A  X.  A ) 
 C_  ( A  X.  A )
 
Theoremisunscov 24441* If an infinite set  A is included in the underlying set of a finite cover  B then there exists a set of the cover that contains an infinite number of element of  A. (Contributed by FL, 2-Aug-2009.)
 |-  (
 ( -.  A  e.  Fin  /\  B  e.  Fin  /\  A  C_  U. B ) 
 ->  E. x  e.  B  -.  ( A  i^i  x )  e.  Fin )
 
Theoremac5g 24442* ac5 8072 with the premisse transformed into an antecedent. (Contributed by FL, 2-Aug-2009.)
 |-  ( A  e.  _V  ->  E. f ( f  Fn  A  /\  A. x  e.  A  ( x  =/=  (/)  ->  ( f `  x )  e.  x ) ) )
 
Theoremrestidsing 24443 Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.)
 |-  (  _I  |`  { A }
 )  =  ( { A }  X.  { A } )
 
Theoremresidcp 24444 The intersection of the identity function with a square cross product. (Contributed by FL, 2-Aug-2009.)
 |-  (  _I  i^i  ( A  X.  A ) )  =  (  _I  |`  A )
 
Theoremtwsymr 24445* Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.)
 |-  ( Rel  R  ->  ( R  =  `' R  <->  A. x A. y
 ( x R y 
 ->  y R x ) ) )
 
Theoremprj1b 24446* Projection of the first elements of the pairs of a relation  R. (Contributed by FL, 15-Oct-2012.)
 |-  ( Rel  R  ->  ( 1st " R )  =  { x  |  E. y <. x ,  y >.  e.  R } )
 
Theoremprj3 24447* Projection of the second elements of the pairs of a relation  R. (Contributed by FL, 15-Oct-2012.)
 |-  ( Rel  R  ->  ( 2nd " R )  =  {
 y  |  E. x <. x ,  y >.  e.  R } )
 
Theoremimfstnrelc 24448 The image under  1st of a class with no pairs inside. (Contributed by FL, 31-Aug-2009.)
 |-  (
 ( ( A  i^i  ( _V  X.  _V )
 )  =  (/)  /\  A  =/= 
 (/) )  ->  ( 1st " A )  =  { (/) } )
 
Theoremprjdmn 24449 The projection of the first elements of the pairs of a relation  R is its domain. (Contributed by FL, 5-Oct-2009.)
 |-  ( Rel  R  ->  ( 1st " R )  =  dom  R )
 
Theoremprjrn 24450 The projection of the second elements of the pairs of a relation  R is its range. (Contributed by FL, 15-Oct-2012.)
 |-  ( Rel  R  ->  ( 2nd " R )  =  ran  R )
 
Theoremprjcp1 24451 Projection of a cross product. (Contributed by FL, 5-Oct-2009.)
 |-  ( B  =/=  (/)  ->  ( 1st " ( A  X.  B ) )  =  A )
 
Theoremprjcp2 24452 Projection of a cross product. (Contributed by FL, 15-Oct-2012.)
 |-  ( A  =/=  (/)  ->  ( 2nd " ( A  X.  B ) )  =  B )
 
Theoremeloi 24453* A consequence of membership in a class abstraction whose elements belong to  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) ) using ordered pair extractors. (Used by category theory). (Contributed by FL, 24-Sep-2007.)
 |-  (
 y  =  ( 1st `  ( 1st `  A ) )  ->  ( ph  <->  ps ) )   &    |-  ( z  =  ( 2nd `  ( 1st `  A ) ) 
 ->  ( ps  <->  ch ) )   &    |-  (
 v  =  ( 1st `  ( 2nd `  A ) )  ->  ( ch  <->  th ) )   &    |-  ( w  =  ( 2nd `  ( 2nd `  A ) ) 
 ->  ( th  <->  ta ) )   =>    |-  ( A  e.  { x  |  E. y E. z E. v E. w ( x  = 
 <. <. y ,  z >. ,  <. v ,  w >.
 >.  /\  ph ) }  ->  ta )
 
Theoremuuniin 24454 The double union of an intersection is a part of the intersections of the unions. (Contributed by FL, 24-Jan-2010.)
 |-  U. U. ( A  i^i  B ) 
 C_  ( U. U. A  i^i  U. U. B )
 
Theoremclsbldimp 24455 A class builder defined by an implication. (Contributed by FL, 18-Sep-2010.)
 |-  { x  |  ( ph  ->  ps ) }  =  ( { x  |  -.  ph }  u.  { x  |  ps }
 )
 
Theoremmappow 24456 A mapping is a member of the powerset of the cross product of its domain and codomain. (Contributed by FL, 30-Dec-2010.)
 |-  (
 ( A  e.  R  /\  B  e.  S ) 
 ->  ( F : A --> B  ->  F  e.  ~P ( A  X.  B ) ) )
 
Theoremelintabg 24457* Membership in the intersection of a class abstraction. (Contributed by FL, 30-Dec-2010.)
 |-  ( A  e.  V  ->  ( A  e.  |^| { x  |  ph }  <->  A. x ( ph  ->  A  e.  x ) ) )
 
Theoremsnelpwg 24458 A singleton of a set belongs to the power class of a class containing the set. (Contributed by FL, 30-Dec-2010.)
 |-  ( A  e.  V  ->  ( A  e.  B  <->  { A }  e.  ~P B ) )
 
Theoremdff1o6f 24459* A one-to-one onto function in terms of function values. (Contributed by FL, 1-Jan-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x F   &    |-  F/_ y F   =>    |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  ran 
 F  =  B  /\  A. x  e.  A  A. y  e.  A  (
 ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
 
Theoremfixpb 24460 The non-empty components of a finite cross product are finite. (Contributed by FL, 22-Feb-2011.)
 |-  (
 ( A  =/=  (/)  /\  B  =/= 
 (/) )  ->  (
 ( A  X.  B )  e.  Fin  <->  ( A  e.  Fin  /\  B  e.  Fin )
 ) )
 
Theoremfixpc 24461 A cross product is finite iff one of its components is empty or both its components are finite. (Contributed by FL, 22-Feb-2011.)
 |-  (
 ( A  X.  B )  e.  Fin  <->  ( A  =  (/) 
 \/  B  =  (/)  \/  ( A  e.  Fin  /\  B  e.  Fin )
 ) )
 
Theoreminfxpg 24462 Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by FL, 22-Feb-2011.)
 |-  (
 ( om  ~<_  A  /\  B  =/=  (/)  /\  ( A  e.  C  /\  B  e.  D ) )  ->  ( A  X.  B ) 
 ~~  ( A  u.  B ) )
 
Theoreminfsdomnng 24463 An infinite set strictly dominates a natural number. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( A  e.  V  ->  ( ( om  ~<_  A  /\  B  e.  om )  ->  B  ~<  A )
 )
 
Theoremresid2 24464 Any operation can be restricted to  ( _V  X.  _V ). (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( Rel  A  /\  Rel 
 dom  A )  ->  ( A  |`  ( _V  X.  _V ) )  =  A )
 
Theoremimrestr 24465 Image of an element of transitive class  B under a class restricted by  B. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( Tr  B  /\  A  e.  B )  ->  ( ( C  |`  B )
 " A )  =  ( C " A ) )
 
Theoremimresord 24466 Image of an element of a ordinal  B under a class restricted by  B. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( Ord  B  /\  A  e.  B )  ->  ( ( C  |`  B )
 " A )  =  ( C " A ) )
 
Theoremsndw 24467 If  A is a part of  B and  B a part of  C and 
A is equipotent to  C then  A is equipotent to  B. The art of sandwich applied to set theory. (Contributed by FL, 16-Apr-2011.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 ( A  C_  B  /\  B  C_  C  /\  C  e.  _V )  ->  ( A  ~~  C  ->  A  ~~  B ) )
 
Theoremsndw2 24468 If  A is a part of  B and  B a part of  C and 
A is equipotent to  C then  A is equipotent to  B. The art of sandwich applied to set theory. (Contributed by FL, 16-Apr-2011.)
 |-  (
 ( A  C_  B  /\  B  C_  C  /\  C  e.  D )  ->  ( A  ~~  C  ->  A  ~~  B ) )
 
Theoremordsuccl 24469 If a successor of  A belongs to an ordinal, so does  A. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( Ord  B  /\  suc 
 A  e.  B ) 
 ->  A  e.  B )
 
Theoremordsuccl2 24470 If a successor of  A belongs to an ordinal, so does  A. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( B  e.  On  /\ 
 suc  A  e.  B )  ->  A  e.  B )
 
Theoremordsuccl3 24471 If a successor of  A belongs to an ordinal,  A is a part of the ordinal. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( B  e.  On  /\ 
 suc  A  e.  B )  ->  A  C_  B )
 
Theoremdomtri3 24472 Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by FL, 16-Apr-2011.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( -.  A  ~<_  B  <->  B  ~<  A ) )
 
Theoremisfinite1b 24473 Omega strictly dominates a finite set. (Contributed by FL, 16-Apr-2011.)
 |-  ( A  e.  Fin  ->  A  ~<  om )
 
Theoremcptwff 24474 The cross product of two finite sets is finite. (Contributed by FL, 16-Apr-2011.)
 |-  (
 ( A  ~<  om  /\  B  ~<  om )  ->  ( A  X.  B )  ~<  om )
 
Theoreminttrp 24475 The intersection of a non-empty element of a transitive class is a part of the class. (Contributed by FL, 15-Apr-2011.)
 |-  (
 ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  |^| B  C_  A )
 
Theoremtrunitr 24476 The union of two transitive classes is transitive. JFM CLASSES1. th. 55 (Contributed by FL, 16-Apr-2011.)
 |-  (
 ( Tr  A  /\  Tr  B )  ->  Tr  ( A  u.  B ) )
 
Theoremuncum2 24477* Union of a cumulative hierarchy of sets. (Contributed by FL, 23-Apr-2011.)
 |-  ( A  e.  On  ->  U_ x  e.  A  ( R1 `  x ) 
 C_  ( R1 `  A ) )
 
Theoremcelsor 24478* If all the elements of a set  A are ordinal numbers and are parts of the set then  A is an ordinal number. (Contributed by FL, 20-Apr-2011.)
 |-  (
 ( A  e.  B  /\  A. x  e.  A  ( x  e.  On  /\  x  C_  A )
 )  ->  A  e.  On )
 
Theoremreflincror 24479 If a relation  R is reflexive, it is included in  ( R  o.  R
). (Contributed by FL, 8-May-2011.)
 |-  (
 ( Rel  R  /\  (  _I  |`  U. U. R )  C_  R )  ->  R  C_  ( R  o.  R ) )
 
Theoremfldrels 24480 The field of a relation is a set. (Contributed by FL, 23-May-2011.)
 |-  X  =  U. U. R   =>    |-  ( R  e.  S  ->  X  e.  _V )
 
Theoremfvsnn 24481 Value when  C doesn't belong to the domain. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( C  =/=  A  ->  ( { <. A ,  B >. } `  C )  =  (/) )
 
Theoremfvsn2a 24482 Value of a function with a domain of two different values. (Contributed by FL, 26-Jun-2011.)
 |-  A  e.  E   &    |-  B  e.  F   &    |-  C  e.  G   &    |-  D  e.  H   =>    |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  A )  =  C )
 
Theoremfvsn2b 24483 Value of a function with a domain of two different values. (Contributed by FL, 26-Jun-2011.)
 |-  A  e.  E   &    |-  B  e.  F   &    |-  C  e.  G   &    |-  D  e.  H   =>    |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  B )  =  D )
 
Theoremcnveq3 24484 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  `' A  =  (/) ) )
 
Theoremrelrefcnv 24485 A relation is reflexive iff its converse is reflexive. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  R  ->  ( (  _I  |`  U. U. R )  C_  R  <->  (  _I  |`  U. U. `' R )  C_  `' R ) )
 
Theoremeqfnung2 24486* If a family of sets  A indexed by  I covers the common domain  B of two functions  F and  G, the restrictions of  F and  G to  ( A  i^i  B ) are equal iff  F  =  G. Compare eqfnun 25755. (Contributed by FL, 5-Nov-2011.)
 |-  (
 ( B  C_  U_ i  e.  I  A  /\  F  Fn  B  /\  G  Fn  B )  ->  ( A. i  e.  I  ( F  |`  A )  =  ( G  |`  A )  <->  F  =  G )
 )
 
Theoreminjrec2 24487* A function is an injection iff a retraction exists. Bourbaki E.II.18 prop. 8. (Contributed by FL, 11-Nov-2011.)
 |-  (
 ( F : A --> B  /\  A  e.  C )  ->  ( F : A -1-1-> B  <->  E. r ( Fun  r  /\  ( r  o.  F )  =  (  _I  |`  A ) ) ) )
 
Theoremsurjsec2 24488* A function is an surjection iff a section exists. Bourbaki E.II.18 prop. 8. (Contributed by FL, 18-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  (
 ( F : A --> B  /\  A  e.  C  /\  B  e.  D ) 
 ->  ( F : A -onto-> B 
 <-> 
 E. s ( s : B --> A  /\  ( F  o.  s
 )  =  (  _I  |`  B ) ) ) )
 
Theoremab2rexexg2 24489* Existence of a class abstraction of existentially restricted sets. (Contributed by FL, 15-Oct-2012.)
 |-  (
 ( A  e.  D  /\  A. x  e.  A  B  e.  E )  ->  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V )
 
Theoremab2rexexg 24490* Existence of a class abstraction of existentially restricted sets. (Contributed by FL, 19-Apr-2012.)
 |-  (
 ( A  e.  D  /\  B  e.  E ) 
 ->  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V )
 
Theoremov2gc 24491* Value of a composition. ovmpt2g 5916 adapted to this special case of a composite. (Contributed by FL, 14-Jul-2012.)
 |-  O  =  ( x  e.  C ,  y  e.  D  |->  ( x  o.  y
 ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A O B )  =  ( A  o.  B ) )
 
Theoremov4gc 24492* Value of a composition. ovmpt4g 5904 adapted to the special case of a composite. (Contributed by FL, 14-Jul-2012.)
 |-  O  =  ( x  e.  C ,  y  e.  D  |->  ( x  o.  y
 ) )   =>    |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( x O y )  =  ( x  o.  y
 ) )
 
Theoremdomintrefc 24493* The domain of the intersection of a family of reflexive classes is the intersection of the domains. (Contributed by FL, 15-Oct-2012.)
 |-  ( A. i  e.  A  A. x  e.  dom  R  x R x  ->  dom  |^|_  i  e.  A  R  =  |^|_ i  e.  A  dom  R )
 
Theoremrnintintrn 24494* The range of an intersection is a part of the intersection of the ranges. (The case  A  =  (/) works as well, the intersection gives  _V). (Contributed by FL, 15-Oct-2012.)
 |-  ran  |^|_ 
 x  e.  A  B  C_  |^|_ x  e.  A  ran  B
 
Theoremprjpacp1 24495 Projection of a part of a cross product. (Contributed by FL, 15-Oct-2012.)
 |-  (
 ( B  =/=  (/)  /\  C  C_  ( A  X.  B ) )  ->  ( 1st " C )  C_  A )
 
Theoremprjpacp2 24496 Projection of a part of a cross product. (Contributed by FL, 15-Oct-2012.)
 |-  (
 ( A  =/=  (/)  /\  C  C_  ( A  X.  B ) )  ->  ( 2nd " C )  C_  B )
 
Theoremrelinccppr 24497 A relation is included in the cross product of its projections. (Contributed by FL, 15-Oct-2012.)
 |-  ( Rel  A  ->  A  C_  (
 ( 1st " A )  X.  ( 2nd " A ) ) )
 
Theoremdffn5a 24498* Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  F/_ x F   =>    |-  ( F  Fn  A  <->  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
 
Theoremffvelrnb 24499 A function's value belongs to its codomain. (Contributed by FL, 14-Sep-2013.)
 |-  (
 ( A  e.  D  /\  B  e.  E ) 
 ->  ( ( F  e.  ( B  ^m  A ) 
 /\  C  e.  A )  ->  ( F `  C )  e.  B ) )
 
Theoremab2rexex2g 24500* Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x,  y, and  z. Compare abrexex2g 5702. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V  /\  A. x  e.  A  A. y  e.  B  { z  |  ph }  e.  _V )  ->  { z  | 
 E. x  e.  A  E. y  e.  B  ph
 }  e.  _V )
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