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Theorem List for Metamath Proof Explorer - 25101-25200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremevpexun 25101 Eventually  ph expressed with the  until operator. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  ( <> ph  <->  (  T.  until  ph ) )
 
Theoremalbineal 25102  ph always holds iff  ph holds in the first step and always holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  <->  ( ph  /\  () [.] ph ) )
 
Theoremalneal1 25103 If  ph always holds, it holds in the first step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  ph )
 
Theoremalneal2 25104 If  ph always holds, it always holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  () [.] ph )
 
Theoremalne 25105 If  ph always holds, it holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  () ph )
 
Theoremalalifal 25106 It is always true that  ph always holds iff 
ph always holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] [.] ph  <->  [.] ph )
 
Theoremalneal1a 25107 Removing a box in the consequent. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  ( [.] ph  ->  ps )   =>    |-  ( [.] ph  ->  [.] ps )
 
Theoremimpbox2 25108 Removing boxes in the antecedents and consequent. (Contributed by FL, 16-Sep-2016.)
 |-  ( ch  ->  ( ph  ->  ps ) )   =>    |-  ( [.] ch  ->  ( [.] ph  ->  [.] ps ) )
 
Theoremboxand 25109 Distributivity of  [.] over  /\. (Contributed by FL, 1-Sep-2016.)
 |-  ( [.] ( ph  /\  ps ) 
 <->  ( [.] ph  /\  [.] ps ) )
 
Theoremboxrim 25110 If  [.] ph implies  ps in the current world, then it implies  ps in every world. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ps )   =>    |-  ( [.] ph  ->  [.] ps )
 
Theoremboximd 25111 Distribute 'always' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( [.] ps  ->  [.] ch ) )
 
Theoremnxtimd 25112 Distribute 'next' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( () ps  ->  () ch ) )
 
Theoremdiaimd 25113 Distribute 'eventually' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  (
 <> ps  ->  <> ch ) )
 
Theoremboxbid 25114 Distribute 'always' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  ( [.] ps  <->  [.] ch ) )
 
Theoremnxtbid 25115 Distribute 'next' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  ( () ps  <->  () ch ) )
 
Theoremdiabid 25116 Distribute 'eventually' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  (
 <> ps  <->  <> ch ) )
 
Theoremevevifev 25117 It is eventually true that  ph eventually holds iff  ph eventually holds. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( <> <> ph  <->  <> ph )
 
Theoremalthalne 25118 If  ph is always true, then it is always true that  ph holds in the next step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  [.] () ph )
 
Theoremtrtrst 25119  T. is true in every step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  [.]  T.
 
Theoremunttr 25120 It's true that  ph is true until true is true. (Contributed by FL, 27-Feb-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph  until  T.  )
 
Theoremuntind 25121 An "induction principle" for until, roughly stating that it is the least fixed point satisfying a property like ax-ltl5 25096. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( ( ph  until  ps )  ->  th ) )
 
Theoremuntindd 25122 An "induction principle" for until, roughly stating that it is the least fixed point satisfying a property like ax-ltl5 25096. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ps  ->  th )   &    |-  ( ( ph  /\ 
 () th )  ->  th )   =>    |-  (
 ( ph  until  ps )  ->  th )
 
Theoremuntim1d 25123 Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( ( ps  until  th )  ->  ( ch  until  th )
 ) )
 
Theoremuntim2d 25124 Congruence axiom for until. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( ( th  until  ps )  ->  ( th  until  ch )
 ) )
 
Theoremuntbi12d 25125 Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   &    |-  ( [.] ph  ->  ( th  <->  ta ) )   =>    |-  ( [.] ph  ->  ( ( ps  until  th )  <->  ( ch  until  ta ) ) )
 
Theoremuntbi12i 25126 Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph 
 <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  until  ch )  <->  ( ps  until  th ) )
 
Theoremaxlmp1 25127 If  ph always holds, then it is a theorem. (Contributed by FL, 16-Sep-2016.)
 |-  [.] ph   =>    |-  ph
 
Theoremaxlmp2 25128 Moving a universal quantifier inside and outside a box. (Contributed by FL, 16-Sep-2016.)
 |-  A. x [.] ph   =>    |- 
 [.] A. x ph
 
Theoremaxlmp3 25129 Moving a universal quantifier inside and outside a box. (Contributed by FL, 16-Sep-2016.)
 |-  [.] A. x ph   =>    |- 
 A. x [.] ph
 
Axiomax-lll 25130 Set equality is true in all worlds. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( x  =  y  ->  [.] x  =  y )
 
Theoremaxlll2 25131 One can add or remove a box in front of  x  =  y. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  ( [.] x  =  y  <->  x  =  y
 )
 
Theoremcdeqbox 25132 Distribute conditional equality over 'always'. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( [.] ph  <->  [.]
 ps ) )
 
Theoremcdeqnxt 25133 Distribute conditional equality over 'next'. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( () ph  <->  ()
 ps ) )
 
Theoremcdequnt 25134 Distribute conditional equality over 'until'. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   &    |- CondEq ( x  =  y  ->  ( ch 
 <-> 
 th ) )   =>    |- CondEq ( x  =  y  ->  ( ( ph  until  ch )  <->  ( ps  until  th ) ) )
 
18.13.4  Operations
 
Theoremssoprab2g 25135* Inference of operation class abstraction subclass from implication. (Contributed by FL, 24-Jan-2010.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  { <. <. x ,  y >. ,  z >.  |  ps } 
 C_  { <. <. x ,  y >. ,  z >.  |  ch } )
 
Theoremdmoprabsss 25136* The domain of an operation class abstraction. Compare dmoprabss 5945. (Contributed by FL, 24-Jan-2010.)
 |-  dom  {
 <. <. x ,  y >. ,  z >.  |  (
 <. x ,  y >.  e.  ( A  X.  B )  /\  ph ) }  C_  ( A  X.  B )
 
Theoremoprssopvg 25137 Value returned by the operation  G in terms of the value returned by the "super"-operation  F. (A version of oprssov 6005 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 25138* The domain of an operation class abstraction. (A version of dmoprabss 5945 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 25139* Existence of an operation class abstraction. (A version of mptex 5762 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 25140* Class abstraction for operations in terms of class abstraction of ordered pairs. (A version of dfoprab4 6193 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 25141* Representation of an operation class abstraction in terms of its values. (A version of fnov 5968 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 ) ) } )
 )
 
18.13.5  General Set Theory
 
Theoremuninqs 25142 Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 3863. (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 25143* 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 25144* The law of concretion for operation class abstraction. Compare with eloprabg 5951. 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 25145 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 25146 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 25147* 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 25148 A theorem of the 1st2nd 6182 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 25149* 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 25150 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 25151* 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 25152* 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 25153 A function's value belongs to its range. A more general version of fnovrn 6011. 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 25154 If an intersection is not empty, its operands are not empty. (Contributed by FL, 27-Apr-2008.)
 |-  (
 ( A  i^i  B )  =/=  (/)  ->  ( A  =/= 
 (/)  /\  B  =/=  (/) ) )
 
Theoremf2imacnv 25155 Image of a preimage. (Contributed by FL, 1-Jun-2008.)
 |-  (
 ( F : A -1-1-onto-> B  /\  C  C_  B )  ->  ( F " ( `' F " C ) )  =  C )
 
Theoremoooeqim2 25156 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 25157 A theorem about things which don't exist  _V and  ( _V  X.  _V ). (Contributed by FL, 22-Sep-2008.)
 |-  ( _V  X.  _V )  =/= 
 _V
 
Theoremducidu 25158 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 25159 The field of a class equals the field of its converse. (Contributed by FL, 16-Apr-2012.)
 |-  ( dom  A  u.  ran  A )  =  ( dom  `' A  u.  ran  `' A )
 
Theoremdomfldrefc 25160* 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 25161* 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 25162* 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 25163* 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 25164* 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 25165* 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 25166* The domain of the intersection of two reflexive classes is the intersection of their domains. Compare with dmin 4902. (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 25167* 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 25168* 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 25169* 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 25170* 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 25171 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
 |-  (
 ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( F : A --> B 
 <->  G : C --> D ) )
 
Theoremunfinsef 25172 A class whose union is finite is finite. (Contributed by FL, 22-Dec-2008.)
 |-  ( U. A  e.  Fin  ->  A  e.  Fin )
 
Theoremf1ofi 25173 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 25174 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 25175 A square cross product  ( A  X.  A
) is reflexive. (Contributed by FL, 31-Jul-2009.)
 |-  (  _I  |`  A )  C_  ( A  X.  A )
 
Theoremsqpsym 25176 A square cross product is symmetric. (Contributed by FL, 31-Jul-2009.)
 |-  `' ( A  X.  A ) 
 C_  ( A  X.  A )
 
Theoremisunscov 25177* 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 25178* ac5 8120 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 25179 Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.)
 |-  (  _I  |`  { A }
 )  =  ( { A }  X.  { A } )
 
Theoremresidcp 25180 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 25181* 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 25182* 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 25183* 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 25184 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 25185 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 25186 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 25187 Projection of a cross product. (Contributed by FL, 5-Oct-2009.)
 |-  ( B  =/=  (/)  ->  ( 1st " ( A  X.  B ) )  =  A )
 
Theoremprjcp2 25188 Projection of a cross product. (Contributed by FL, 15-Oct-2012.)
 |-  ( A  =/=  (/)  ->  ( 2nd " ( A  X.  B ) )  =  B )
 
Theoremeloi 25189* 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 25190 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 25191 A class builder defined by an implication. (Contributed by FL, 18-Sep-2010.)
 |-  { x  |  ( ph  ->  ps ) }  =  ( { x  |  -.  ph }  u.  { x  |  ps }
 )
 
Theoremmappow 25192 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 25193* 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 25194 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 25195* 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 25196 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 25197 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 25198 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 25199 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 25200 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 )
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