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Theorem List for Metamath Proof Explorer - 25001-25100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnotev 25001 It's false that  ph eventually holds iff  -.  ph always holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -. 
 <> ph  <->  [.]  -.  ph )
 
Theoremnotal 25002 It's false that  ph always holds iff  -.  ph eventually holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -.  [.] ph  <->  <>  -.  ph )
 
Theoremltl4ev 25003 The contrapositive of ax-ltl4 24988. If the truth of  ph in each step implies it is true in the previous step, and  ph is eventually true, then  ph is true in the first step. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  (
 ( [.] ( () ph  -> 
 ph )  /\  <> ph )  -> 
 ph )
 
Axiomax-ltl5 25004  ph holds until  ps iff  ps holds in the current step or  ph holds in the current step and in the next step  ph holds until  ps. (Contributed by FL, 27-Feb-2011.)
 |-  (
 ( ph  until  ps )  <->  ( ps  \/  ( ph  /\ 
 () ( ph  until  ps )
 ) ) )
 
Axiomax-ltl6 25005 If  ph holds until  ps then eventually  ps holds. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( ph  until  ps )  -> 
 <> ps )
 
Theoremnopsthph 25006 If  ps doesn't hold in the first step and  ph holds until  ps then  ph holds. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  (
 ( -.  ps  /\  ( ph  until  ps ) )  ->  ph )
 
Theoremphthps 25007 If  ph doesn't hold in the current step and  ph holds until  ps then  ps holds in the current step. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( -.  ph  /\  ( ph  until  ps ) )  ->  ps )
 
Theoremimunt 25008 If  ps is true, then  ph is true until  ps. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ps  ->  ( ph  until  ps )
 )
 
Theoremevpexun 25009 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 25010  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 25011 If  ph always holds, it holds in the first step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  ph )
 
Theoremalneal2 25012 If  ph always holds, it always holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  () [.] ph )
 
Theoremalne 25013 If  ph always holds, it holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  () ph )
 
Theoremalalifal 25014 It is always true that  ph always holds iff 
ph always holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] [.] ph  <->  [.] ph )
 
Theoremalneal1a 25015 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 25016 Removing boxes in the antecedents and consequent. (Contributed by FL, 16-Sep-2016.)
 |-  ( ch  ->  ( ph  ->  ps ) )   =>    |-  ( [.] ch  ->  ( [.] ph  ->  [.] ps ) )
 
Theoremboxand 25017 Distributivity of  [.] over  /\. (Contributed by FL, 1-Sep-2016.)
 |-  ( [.] ( ph  /\  ps ) 
 <->  ( [.] ph  /\  [.] ps ) )
 
Theoremboxrim 25018 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 25019 Distribute 'always' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( [.] ps  ->  [.] ch ) )
 
Theoremnxtimd 25020 Distribute 'next' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( () ps  ->  () ch ) )
 
Theoremdiaimd 25021 Distribute 'eventually' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  (
 <> ps  ->  <> ch ) )
 
Theoremboxbid 25022 Distribute 'always' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  ( [.] ps  <->  [.] ch ) )
 
Theoremnxtbid 25023 Distribute 'next' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  ( () ps  <->  () ch ) )
 
Theoremdiabid 25024 Distribute 'eventually' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  (
 <> ps  <->  <> ch ) )
 
Theoremevevifev 25025 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 25026 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 25027  T. is true in every step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  [.]  T.
 
Theoremunttr 25028 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 25029 An "induction principle" for until, roughly stating that it is the least fixed point satisfying a property like ax-ltl5 25004. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( ( ph  until  ps )  ->  th ) )
 
Theoremuntindd 25030 An "induction principle" for until, roughly stating that it is the least fixed point satisfying a property like ax-ltl5 25004. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ps  ->  th )   &    |-  ( ( ph  /\ 
 () th )  ->  th )   =>    |-  (
 ( ph  until  ps )  ->  th )
 
Theoremuntim1d 25031 Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( ( ps  until  th )  ->  ( ch  until  th )
 ) )
 
Theoremuntim2d 25032 Congruence axiom for until. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( ( th  until  ps )  ->  ( th  until  ch )
 ) )
 
Theoremuntbi12d 25033 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 25034 Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph 
 <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  until  ch )  <->  ( ps  until  th ) )
 
Theoremaxlmp1 25035 If  ph always holds, then it is a theorem. (Contributed by FL, 16-Sep-2016.)
 |-  [.] ph   =>    |-  ph
 
Theoremaxlmp2 25036 Moving a universal quantifier inside and outside a box. (Contributed by FL, 16-Sep-2016.)
 |-  A. x [.] ph   =>    |- 
 [.] A. x ph
 
Theoremaxlmp3 25037 Moving a universal quantifier inside and outside a box. (Contributed by FL, 16-Sep-2016.)
 |-  [.] A. x ph   =>    |- 
 A. x [.] ph
 
Axiomax-lll 25038 Set equality is true in all worlds. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( x  =  y  ->  [.] x  =  y )
 
Theoremaxlll2 25039 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 25040 Distribute conditional equality over 'always'. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( [.] ph  <->  [.]
 ps ) )
 
Theoremcdeqnxt 25041 Distribute conditional equality over 'next'. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( () ph  <->  ()
 ps ) )
 
Theoremcdequnt 25042 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 25043* 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 25044* The domain of an operation class abstraction. Compare dmoprabss 5931. (Contributed by FL, 24-Jan-2010.)
 |-  dom  {
 <. <. x ,  y >. ,  z >.  |  (
 <. x ,  y >.  e.  ( A  X.  B )  /\  ph ) }  C_  ( A  X.  B )
 
Theoremoprssopvg 25045 Value returned by the operation  G in terms of the value returned by the "super"-operation  F. (A version of oprssov 5991 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 25046* The domain of an operation class abstraction. (A version of dmoprabss 5931 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 25047* Existence of an operation class abstraction. (A version of mptex 5748 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 25048* Class abstraction for operations in terms of class abstraction of ordered pairs. (A version of dfoprab4 6179 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 25049* Representation of an operation class abstraction in terms of its values. (A version of fnov 5954 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 25050 Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 3849. (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 25051* 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 25052* The law of concretion for operation class abstraction. Compare with eloprabg 5937. 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 25053 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 25054 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 25055* 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 25056 A theorem of the 1st2nd 6168 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 25057* 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 25058 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 25059* 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 25060* 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 25061 A function's value belongs to its range. A more general version of fnovrn 5997. 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 25062 If an intersection is not empty, its operands are not empty. (Contributed by FL, 27-Apr-2008.)
 |-  (
 ( A  i^i  B )  =/=  (/)  ->  ( A  =/= 
 (/)  /\  B  =/=  (/) ) )
 
Theoremf2imacnv 25063 Image of a preimage. (Contributed by FL, 1-Jun-2008.)
 |-  (
 ( F : A -1-1-onto-> B  /\  C  C_  B )  ->  ( F " ( `' F " C ) )  =  C )
 
Theoremoooeqim2 25064 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 25065 A theorem about things which don't exist  _V and  ( _V  X.  _V ). (Contributed by FL, 22-Sep-2008.)
 |-  ( _V  X.  _V )  =/= 
 _V
 
Theoremducidu 25066 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 25067 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 25068* 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 25069* 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 25070* 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 25071* 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 25072* 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 25073* 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 25074* The domain of the intersection of two reflexive classes is the intersection of their domains. Compare with dmin 4888. (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 25075* 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 25076* 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 25077* 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 25078* 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 25079 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
 |-  (
 ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( F : A --> B 
 <->  G : C --> D ) )
 
Theoremunfinsef 25080 A class whose union is finite is finite. (Contributed by FL, 22-Dec-2008.)
 |-  ( U. A  e.  Fin  ->  A  e.  Fin )
 
Theoremf1ofi 25081 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 25082 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 25083 A square cross product  ( A  X.  A
) is reflexive. (Contributed by FL, 31-Jul-2009.)
 |-  (  _I  |`  A )  C_  ( A  X.  A )
 
Theoremsqpsym 25084 A square cross product is symmetric. (Contributed by FL, 31-Jul-2009.)
 |-  `' ( A  X.  A ) 
 C_  ( A  X.  A )
 
Theoremisunscov 25085* 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 25086* ac5 8106 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 25087 Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.)
 |-  (  _I  |`  { A }
 )  =  ( { A }  X.  { A } )
 
Theoremresidcp 25088 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 25089* 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 25090* 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 25091* 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 25092 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 25093 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 25094 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 25095 Projection of a cross product. (Contributed by FL, 5-Oct-2009.)
 |-  ( B  =/=  (/)  ->  ( 1st " ( A  X.  B ) )  =  A )
 
Theoremprjcp2 25096 Projection of a cross product. (Contributed by FL, 15-Oct-2012.)
 |-  ( A  =/=  (/)  ->  ( 2nd " ( A  X.  B ) )  =  B )
 
Theoremeloi 25097* 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 25098 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 25099 A class builder defined by an implication. (Contributed by FL, 18-Sep-2010.)
 |-  { x  |  ( ph  ->  ps ) }  =  ( { x  |  -.  ph }  u.  { x  |  ps }
 )
 
Theoremmappow 25100 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 ) ) )
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