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Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
16.12.3  Linear temporal logic

Propositional Linear temporal logic (LTL) is a kind of modal logic. It is composed of the axioms of classical logic plus the axioms ax-ltl1 24305, ax-ltl2 24306, ax-ltl3 24307, ax-ltl4 , ax-lmp 24309, and ax-nmp 24310. In classical logic, propositions don't depend on the time. In LTL the "world" evolves. We will imagine the world as a sequence of states with a first state and future states. Instead of state I will also use the term "step" to emphasize that LTL is used to formalize the evolution of process in a computer. A proposition that is true in one state of the "world" may be false in the next one. The proposition  [.] ph means  ph is true in every state of the world, in the first state as well as in the future states. It is read "
ph is always true " or " ph always holds ". The proposition  () ph means  ph is true in the next state of the world. The proposition 
<> ph means that  ph is true in one state of the world at least but we don't know exactly which one. It can be the first state, it can be a future state. It is read " ph is eventually true " or " ph eventually holds". When no operator is used in front of a proposition, it means that  ph is unconditionnaly true or that it is true in the current state ( depending on the context).  ph  until  ps means  ph is true in every state of the world until  ps is true.

 
Syntaxwbox 24301 An always true proposition is well formed.
 wff  [.] ph
 
Syntaxwdia 24302 An eventually true proposition is well formed.
 wff  <> ph
 
Syntaxwcirc 24303 A proposition true in the next step is well formed.
 wff  ()
 ph
 
Syntaxwunt 24304 The proposition " ph is true until  ps is true " is well formed.
 wff  ( ph  until  ps )
 
Axiomax-ltl1 24305 If  ( ph  ->  ps ) and  ph always hold then  ps always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ( [.] ( ph  ->  ps )  ->  ( [.] ph  ->  [.]
 ps ) )
 
Axiomax-ltl2 24306  ph doesn't hold in the next step iff in the next step 
-.  ph holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -.  () ph  <->  ()  -.  ph )
 
Axiomax-ltl3 24307 If, in the next step,  ph  ->  ps and  ph hold then, in the next step,  ps holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( () ( ph  ->  ps )  ->  ( () ph  ->  ()
 ps ) )
 
Axiomax-ltl4 24308 Suppose that it is always true that if  ph is true in the current step then  ph is true in the next step. Suppose that  ph is true in the first step. Then  ph is always true. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( [.] ( ph  ->  ()
 ph )  /\  ph )  ->  [.] ph )
 
Axiomax-lmp 24309 If  ph is a theorem then it always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ph   =>    |- 
 [.] ph
 
Axiomax-nmp 24310 If  ph is a theorem then it holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ph   =>    |- 
 () ph
 
Definitiondf-dia 24311  ph eventually holds iff it is not true that  -.  ph always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ( <> ph  <->  -. 
 [.]  -.  ph )
 
Theoremimpbox 24312 If  ph  ->  ps is unconditionally true and if  ph is always true then  ps is always true. (Contributed by FL, 22-Feb-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( [.] ph  ->  [.]
 ps )
 
Theorembibox 24313 If  ph  <->  ps is unconditionally true then  ph is always true is equivalent to  ps is always true. (Contributed by FL, 22-Feb-2011.)
 |-  ( ph 
 <->  ps )   =>    |-  ( [.] ph  <->  [.] ps )
 
Theoremimpxt 24314 If  ph  ->  ps holds unconditionally and if  ph holds in the next state then  ps holds in the next state. (Contributed by FL, 20-Mar-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( () ph  ->  ()
 ps )
 
Theorembinxt 24315 If  ph  <->  ps holds unconditionally then  ph holds in the next state of the world iff  ps holds in the next state. (Contributed by FL, 20-Mar-2011.)
 |-  ( ph 
 <->  ps )   =>    |-  ( () ph  <->  () ps )
 
Theoremnxtor 24316  ( ph  \/  ps ) holds in the next step iff  ph holds in the next step or  ps holds in the next step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( () ( ph  \/  ps ) 
 <->  ( () ph  \/  () ps ) )
 
Theoremnxtand 24317  ( ph  /\ 
ps ) holds in the next step iff  ph holds in the next step and  ps holds in the next step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( () ( ph  /\  ps ) 
 <->  ( () ph  /\  () ps ) )
 
Theoremboxeq 24318  ph holds now and will always hold in the future iff it is not true that  -.  ph holds now or sometimes in the future. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( [.] ph  <->  -.  <>  -.  ph )
 
Theoremdiaimi 24319 If  ph implies  ps unconditionally, then if  ph eventually holds so does  ps. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph  ->  ps )   =>    |-  ( <> ph  ->  <> ps )
 
Theorembidia 24320 If  ph  <->  ps holds then  ph eventually holds iff  ps eventually holds. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph 
 <->  ps )   =>    |-  ( <> ph  <->  <> ps )
 
Theoremnotev 24321 It's false that  ph eventually holds iff  -.  ph always holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -. 
 <> ph  <->  [.]  -.  ph )
 
Theoremnotal 24322 It's false that  ph always holds iff  -.  ph eventually holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -.  [.] ph  <->  <>  -.  ph )
 
Theoremltl4ev 24323 The contrapositive of ax-ltl4 24308. 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 24324  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 24325 If  ph holds until  ps then eventually  ps holds. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( ph  until  ps )  -> 
 <> ps )
 
Theoremnopsthph 24326 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 24327 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 24328 If  ps is true, then  ph is true until  ps. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ps  ->  ( ph  until  ps )
 )
 
Theoremevpexun 24329 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 24330  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 24331 If  ph always holds, it holds in the first step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  ph )
 
Theoremalneal2 24332 If  ph always holds, it always holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  () [.] ph )
 
Theoremalne 24333 If  ph always holds, it holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  () ph )
 
Theoremalalifal 24334 It is always true that  ph always holds iff 
ph always holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] [.] ph  <->  [.] ph )
 
Theoremalneal1a 24335 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 24336 Removing boxes in the antecedents and consequent. (Contributed by FL, 16-Sep-2016.)
 |-  ( ch  ->  ( ph  ->  ps ) )   =>    |-  ( [.] ch  ->  ( [.] ph  ->  [.] ps ) )
 
Theoremboxand 24337 Distributivity of  [.] over  /\. (Contributed by FL, 1-Sep-2016.)
 |-  ( [.] ( ph  /\  ps ) 
 <->  ( [.] ph  /\  [.] ps ) )
 
Theoremboxrim 24338 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 24339 Distribute 'always' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( [.] ps  ->  [.] ch ) )
 
Theoremnxtimd 24340 Distribute 'next' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( () ps  ->  () ch ) )
 
Theoremdiaimd 24341 Distribute 'eventually' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  (
 <> ps  ->  <> ch ) )
 
Theoremboxbid 24342 Distribute 'always' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  ( [.] ps  <->  [.] ch ) )
 
Theoremnxtbid 24343 Distribute 'next' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  ( () ps  <->  () ch ) )
 
Theoremdiabid 24344 Distribute 'eventually' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  (
 <> ps  <->  <> ch ) )
 
Theoremevevifev 24345 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 24346 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 24347  T. is true in every step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  [.]  T.
 
Theoremunttr 24348 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 24349 An "induction principle" for until, roughly stating that it is the least fixed point satisfying a property like ax-ltl5 24324. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( ( ph  until  ps )  ->  th ) )
 
Theoremuntindd 24350 An "induction principle" for until, roughly stating that it is the least fixed point satisfying a property like ax-ltl5 24324. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ps  ->  th )   &    |-  ( ( ph  /\ 
 () th )  ->  th )   =>    |-  (
 ( ph  until  ps )  ->  th )
 
Theoremuntim1d 24351 Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( ( ps  until  th )  ->  ( ch  until  th )
 ) )
 
Theoremuntim2d 24352 Congruence axiom for until. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( ( th  until  ps )  ->  ( th  until  ch )
 ) )
 
Theoremuntbi12d 24353 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 24354 Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph 
 <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  until  ch )  <->  ( ps  until  th ) )
 
Theoremaxlmp1 24355 If  ph always holds then it is a theorem. (Contributed by FL, 16-Sep-2016.)
 |-  [.] ph   =>    |-  ph
 
Theoremaxlmp2 24356 Moving a universal quantifier inside and outside a box. (Contributed by FL, 16-Sep-2016.)
 |-  A. x [.] ph   =>    |- 
 [.] A. x ph
 
Theoremaxlmp3 24357 Moving a universal quantifier inside and outside a box. (Contributed by FL, 16-Sep-2016.)
 |-  [.] A. x ph   =>    |- 
 A. x [.] ph
 
Axiomax-lll 24358 Set equality is true in all worlds. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( x  =  y  ->  [.] x  =  y )
 
Theoremaxlll2 24359 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 24360 Distribute conditional equality over 'always'. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( [.] ph  <->  [.]
 ps ) )
 
Theoremcdeqnxt 24361 Distribute conditional equality over 'next'. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( () ph  <->  ()
 ps ) )
 
Theoremcdequnt 24362 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 ) ) )
 
16.12.4  Operations
 
Theoremssoprab2g 24363* 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 24364* The domain of an operation class abstraction. Compare dmoprabss 5828. (Contributed by FL, 24-Jan-2010.)
 |-  dom  {
 <. <. x ,  y >. ,  z >.  |  (
 <. x ,  y >.  e.  ( A  X.  B )  /\  ph ) }  C_  ( A  X.  B )
 
Theoremoprssopvg 24365 Value returned by the operation  G in terms of the value returned by the "super"-operation  F. (A version of oprssov 5888 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 24366* The domain of an operation class abstraction. (A version of dmoprabss 5828 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 24367* Existence of an operation class abstraction. (A version of mptex 5645 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 24368* Class abstraction for operations in terms of class abstraction of ordered pairs. (A version of dfoprab4 6076 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 24369* Representation of an operation class abstraction in terms of its values. (A version of fnov 5851 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 ) ) } )
 )
 
16.12.5  General Set Theory
 
Theoremuninqs 24370 Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 3788. (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 24371* 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 24372* The law of concretion for operation class abstraction. Compare with eloprabg 5834. 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 24373 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 24374 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 24375* 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 24376 A theorem of the 1st2nd 6065 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 24377* 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 24378 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 24379* 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 24380* 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 24381 A function's value belongs to its range. A more general version of fnovrn 5894. 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 24382 If an intersection is not empty its operands are not empty. (Contributed by FL, 27-Apr-2008.)
 |-  (
 ( A  i^i  B )  =/=  (/)  ->  ( A  =/= 
 (/)  /\  B  =/=  (/) ) )
 
Theoremf2imacnv 24383 Image of a preimage. (Contributed by FL, 1-Jun-2008.)
 |-  (
 ( F : A -1-1-onto-> B  /\  C  C_  B )  ->  ( F " ( `' F " C ) )  =  C )
 
Theoremoooeqim2 24384 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 24385 A theorem about things which don't exist  _V and  ( _V  X.  _V ). (Contributed by FL, 22-Sep-2008.)
 |-  ( _V  X.  _V )  =/= 
 _V
 
Theoremducidu 24386 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 24387 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 24388* 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 24389* 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 24390* 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 24391* 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 24392* 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 24393* 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 24394* The domain of the intersection of two reflexive classes is the intersection of their domains. Compare with dmin 4839. (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 24395* 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 24396* 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 24397* 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 24398* 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 24399 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
 |-  (
 ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( F : A --> B 
 <->  G : C --> D ) )
 
Theoremunfinsef 24400 A class whose union is finite is finite. (Contributed by FL, 22-Dec-2008.)
 |-  ( U. A  e.  Fin  ->  A  e.  Fin )
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