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Theorem List for Metamath Proof Explorer - 24401-24500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnotal 24401 It's false that  ph always holds iff  -.  ph eventually holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -.  [.] ph  <->  <>  -.  ph )
 
Theoremltl4ev 24402 The contrapositive of ax-ltl4 24387. 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 24403  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 24404 If  ph holds until  ps then eventually  ps holds. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( ph  until  ps )  -> 
 <> ps )
 
Theoremnopsthph 24405 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 24406 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 24407 If  ps is true, then  ph is true until  ps. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ps  ->  ( ph  until  ps )
 )
 
Theoremevpexun 24408 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 24409  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 24410 If  ph always holds, it holds in the first step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  ph )
 
Theoremalneal2 24411 If  ph always holds, it always holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  () [.] ph )
 
Theoremalne 24412 If  ph always holds, it holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  () ph )
 
Theoremalalifal 24413 It is always true that  ph always holds iff 
ph always holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] [.] ph  <->  [.] ph )
 
Theoremalneal1a 24414 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 24415 Removing boxes in the antecedents and consequent. (Contributed by FL, 16-Sep-2016.)
 |-  ( ch  ->  ( ph  ->  ps ) )   =>    |-  ( [.] ch  ->  ( [.] ph  ->  [.] ps ) )
 
Theoremboxand 24416 Distributivity of  [.] over  /\. (Contributed by FL, 1-Sep-2016.)
 |-  ( [.] ( ph  /\  ps ) 
 <->  ( [.] ph  /\  [.] ps ) )
 
Theoremboxrim 24417 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 24418 Distribute 'always' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( [.] ps  ->  [.] ch ) )
 
Theoremnxtimd 24419 Distribute 'next' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( () ps  ->  () ch ) )
 
Theoremdiaimd 24420 Distribute 'eventually' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  (
 <> ps  ->  <> ch ) )
 
Theoremboxbid 24421 Distribute 'always' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  ( [.] ps  <->  [.] ch ) )
 
Theoremnxtbid 24422 Distribute 'next' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  ( () ps  <->  () ch ) )
 
Theoremdiabid 24423 Distribute 'eventually' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  (
 <> ps  <->  <> ch ) )
 
Theoremevevifev 24424 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 24425 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 24426  T. is true in every step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  [.]  T.
 
Theoremunttr 24427 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 24428 An "induction principle" for until, roughly stating that it is the least fixed point satisfying a property like ax-ltl5 24403. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( ( ph  until  ps )  ->  th ) )
 
Theoremuntindd 24429 An "induction principle" for until, roughly stating that it is the least fixed point satisfying a property like ax-ltl5 24403. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ps  ->  th )   &    |-  ( ( ph  /\ 
 () th )  ->  th )   =>    |-  (
 ( ph  until  ps )  ->  th )
 
Theoremuntim1d 24430 Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( ( ps  until  th )  ->  ( ch  until  th )
 ) )
 
Theoremuntim2d 24431 Congruence axiom for until. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( ( th  until  ps )  ->  ( th  until  ch )
 ) )
 
Theoremuntbi12d 24432 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 24433 Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph 
 <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  until  ch )  <->  ( ps  until  th ) )
 
Theoremaxlmp1 24434 If  ph always holds then it is a theorem. (Contributed by FL, 16-Sep-2016.)
 |-  [.] ph   =>    |-  ph
 
Theoremaxlmp2 24435 Moving a universal quantifier inside and outside a box. (Contributed by FL, 16-Sep-2016.)
 |-  A. x [.] ph   =>    |- 
 [.] A. x ph
 
Theoremaxlmp3 24436 Moving a universal quantifier inside and outside a box. (Contributed by FL, 16-Sep-2016.)
 |-  [.] A. x ph   =>    |- 
 A. x [.] ph
 
Axiomax-lll 24437 Set equality is true in all worlds. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( x  =  y  ->  [.] x  =  y )
 
Theoremaxlll2 24438 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 24439 Distribute conditional equality over 'always'. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( [.] ph  <->  [.]
 ps ) )
 
Theoremcdeqnxt 24440 Distribute conditional equality over 'next'. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( () ph  <->  ()
 ps ) )
 
Theoremcdequnt 24441 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 24442* 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 24443* The domain of an operation class abstraction. Compare dmoprabss 5891. (Contributed by FL, 24-Jan-2010.)
 |-  dom  {
 <. <. x ,  y >. ,  z >.  |  (
 <. x ,  y >.  e.  ( A  X.  B )  /\  ph ) }  C_  ( A  X.  B )
 
Theoremoprssopvg 24444 Value returned by the operation  G in terms of the value returned by the "super"-operation  F. (A version of oprssov 5951 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 24445* The domain of an operation class abstraction. (A version of dmoprabss 5891 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 24446* Existence of an operation class abstraction. (A version of mptex 5708 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 24447* Class abstraction for operations in terms of class abstraction of ordered pairs. (A version of dfoprab4 6139 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 24448* Representation of an operation class abstraction in terms of its values. (A version of fnov 5914 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 24449 Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 3848. (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 24450* 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 24451* The law of concretion for operation class abstraction. Compare with eloprabg 5897. 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 24452 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 24453 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 24454* 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 24455 A theorem of the 1st2nd 6128 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 24456* 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 24457 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 24458* 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 24459* 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 24460 A function's value belongs to its range. A more general version of fnovrn 5957. 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 24461 If an intersection is not empty its operands are not empty. (Contributed by FL, 27-Apr-2008.)
 |-  (
 ( A  i^i  B )  =/=  (/)  ->  ( A  =/= 
 (/)  /\  B  =/=  (/) ) )
 
Theoremf2imacnv 24462 Image of a preimage. (Contributed by FL, 1-Jun-2008.)
 |-  (
 ( F : A -1-1-onto-> B  /\  C  C_  B )  ->  ( F " ( `' F " C ) )  =  C )
 
Theoremoooeqim2 24463 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 24464 A theorem about things which don't exist  _V and  ( _V  X.  _V ). (Contributed by FL, 22-Sep-2008.)
 |-  ( _V  X.  _V )  =/= 
 _V
 
Theoremducidu 24465 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 24466 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 24467* 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 24468* 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 24469* 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 24470* 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 24471* 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 24472* 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 24473* The domain of the intersection of two reflexive classes is the intersection of their domains. Compare with dmin 4885. (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 24474* 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 24475* 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 24476* 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 24477* 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 24478 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
 |-  (
 ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( F : A --> B 
 <->  G : C --> D ) )
 
Theoremunfinsef 24479 A class whose union is finite is finite. (Contributed by FL, 22-Dec-2008.)
 |-  ( U. A  e.  Fin  ->  A  e.  Fin )
 
Theoremf1ofi 24480 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 24481 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 24482 A square cross product  ( A  X.  A
) is reflexive. (Contributed by FL, 31-Jul-2009.)
 |-  (  _I  |`  A )  C_  ( A  X.  A )
 
Theoremsqpsym 24483 A square cross product is symmetric. (Contributed by FL, 31-Jul-2009.)
 |-  `' ( A  X.  A ) 
 C_  ( A  X.  A )
 
Theoremisunscov 24484* 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 24485* ac5 8100 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 24486 Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.)
 |-  (  _I  |`  { A }
 )  =  ( { A }  X.  { A } )
 
Theoremresidcp 24487 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 24488* 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 24489* 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 24490* 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 24491 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 24492 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 24493 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 24494 Projection of a cross product. (Contributed by FL, 5-Oct-2009.)
 |-  ( B  =/=  (/)  ->  ( 1st " ( A  X.  B ) )  =  A )
 
Theoremprjcp2 24495 Projection of a cross product. (Contributed by FL, 15-Oct-2012.)
 |-  ( A  =/=  (/)  ->  ( 2nd " ( A  X.  B ) )  =  B )
 
Theoremeloi 24496* 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 24497 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 24498 A class builder defined by an implication. (Contributed by FL, 18-Sep-2010.)
 |-  { x  |  ( ph  ->  ps ) }  =  ( { x  |  -.  ph }  u.  { x  |  ps }
 )
 
Theoremmappow 24499 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 24500* 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 ) ) )
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