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Theorem List for Metamath Proof Explorer - 35801-35900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremifpim1 35801 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- ( -.  ph , T.  ,  ps ) )
 
Theoremifpnot 35802 Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  ( -.  ph  <-> if- ( ph , F.  , T.  ) )
 
Theoremifpid2 35803 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  ( ph 
 <-> if- ( ph , T.  , F.  ) )
 
Theoremifpim2 35804 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- ( ps , T.  ,  -.  ph ) )
 
Theoremifpbi23 35805 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  (if- ( ta ,  ph ,  ch )  <-> if- ( ta ,  ps ,  th ) ) )
 
Theoremifpdfbi 35806 Define biimplication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  <->  ps )  <-> if- ( ph ,  ps ,  -.  ps ) )
 
Theoremifpbiidcor 35807 Restatement of biid 239. (Contributed by RP, 25-Apr-2020.)
 |- if- ( ph ,  ph ,  -.  ph )
 
Theoremifpbicor 35808 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  ps ,  -.  ps )  <-> if- ( ps ,  ph ,  -.  ph ) )
 
Theoremifpxorcor 35809 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  -.  ps ,  ps )  <-> if- ( ps ,  -.  ph ,  ph ) )
 
Theoremifpbi1 35810 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ph ,  ch ,  th )  <-> if- ( ps ,  ch ,  th ) ) )
 
Theoremifpnot23 35811 Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)
 |-  ( -. if- ( ph ,  ps ,  ch )  <-> if- ( ph ,  -.  ps ,  -.  ch )
 )
 
Theoremifpnotnotb 35812 Factor conditional logic operator over negation in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  -.  ps ,  -.  ch )  <->  -. if- ( ph ,  ps ,  ch ) )
 
Theoremifpnorcor 35813 Corollary of commutation of nor. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  -.  ph ,  -.  ps )  <-> if- ( ps ,  -.  ps ,  -.  ph )
 )
 
Theoremifpnancor 35814 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  -.  ps ,  -.  ph )  <-> if- ( ps ,  -.  ph ,  -.  ps )
 )
 
Theoremifpnot23b 35815 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -. if- ( ph ,  -.  ps ,  ch )  <-> if- ( ph ,  ps ,  -.  ch ) )
 
Theoremifpbiidcor2 35816 Restatement of biid 239. (Contributed by RP, 25-Apr-2020.)
 |-  -. if- (
 ph ,  -.  ph ,  ph )
 
Theoremifpnot23c 35817 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -. if- ( ph ,  ps ,  -.  ch )  <-> if- ( ph ,  -.  ps ,  ch ) )
 
Theoremifpnot23d 35818 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -. if- ( ph ,  -.  ps ,  -.  ch )  <-> if- (
 ph ,  ps ,  ch ) )
 
Theoremifpdfnan 35819 Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  -/\  ps )  <-> if- (
 ph ,  -.  ps , T.  ) )
 
Theoremifpdfxor 35820 Define xor as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  \/_  ps )  <-> if- (
 ph ,  -.  ps ,  ps ) )
 
Theoremifpbi12 35821 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  (if- ( ph ,  ch ,  ta )  <-> if- ( ps ,  th ,  ta ) ) )
 
Theoremifpbi13 35822 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  (if- ( ph ,  ta ,  ch )  <-> if- ( ps ,  ta ,  th ) ) )
 
Theoremifpbi123 35823 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th )  /\  ( ta 
 <->  et ) )  ->  (if- ( ph ,  ch ,  ta )  <-> if- ( ps ,  th ,  et ) ) )
 
Theoremifpidg 35824 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( th  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ( ( ph  /\  ps )  ->  th )  /\  (
 ( ph  /\  th )  ->  ps ) )  /\  ( ( ch  ->  (
 ph  \/  th )
 )  /\  ( th  ->  ( ph  \/  ch ) ) ) ) )
 
Theoremifpid3g 35825 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ch  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( (
 ph  /\  ps )  ->  ch )  /\  (
 ( ph  /\  ch )  ->  ps ) ) )
 
Theoremifpid2g 35826 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ps  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ps 
 ->  ( ph  \/  ch ) )  /\  ( ch 
 ->  ( ph  \/  ps ) ) ) )
 
Theoremifpid1g 35827 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ch 
 ->  ph )  /\  ( ph  ->  ps ) ) )
 
Theoremifpim23g 35828 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  (
 ( ( ph  ->  ps )  <-> if- ( ch ,  ps ,  -.  ph ) )  <->  ( ( (
 ph  /\  ps )  ->  ch )  /\  ( ch  ->  ( ph  \/  ps ) ) ) )
 
Theoremifpim3 35829 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- (
 ph ,  ps ,  -.  ph ) )
 
Theoremifpnim1 35830 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -.  ( ph  ->  ps )  <-> if- (
 ph ,  -.  ps ,  ph ) )
 
Theoremifpim4 35831 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- ( ps ,  ps ,  -.  ph ) )
 
Theoremifpnim2 35832 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -.  ( ph  ->  ps )  <-> if- ( ps ,  -.  ps ,  ph ) )
 
Theoremifpim123g 35833 Implication of conditional logical operators. The right hand side is basically conjunctive normal form which is useful in proofs. (Contributed by RP, 16-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  ta )  -> if- ( ps ,  th ,  et ) )  <->  ( ( ( ( ph  ->  -.  ps )  \/  ( ch  ->  th ) )  /\  (
 ( ps  ->  ph )  \/  ( ta  ->  th )
 ) )  /\  (
 ( ( ph  ->  ps )  \/  ( ch 
 ->  et ) )  /\  ( ( -.  ps  -> 
 ph )  \/  ( ta  ->  et ) ) ) ) )
 
Theoremifpim1g 35834 Implication of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  th )  -> if- ( ps ,  ch ,  th ) )  <->  ( ( ( ps  ->  ph )  \/  ( th  ->  ch )
 )  /\  ( ( ph  ->  ps )  \/  ( ch  ->  th ) ) ) )
 
Theoremifp1bi 35835 Substitute the first element of conditional logical operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  th )  <-> if- ( ps ,  ch ,  th ) )  <->  ( ( ( ( ph  ->  ps )  \/  ( ch  ->  th )
 )  /\  ( ( ph  ->  ps )  \/  ( th  ->  ch ) ) ) 
 /\  ( ( ( ps  ->  ph )  \/  ( ch  ->  th )
 )  /\  ( ( ps  ->  ph )  \/  ( th  ->  ch ) ) ) ) )
 
Theoremifpbi1b 35836 When the first variable is irrelevant, it can be replaced. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  ch ,  ch )  <-> if- ( ps ,  ch ,  ch ) )
 
Theoremifpimimb 35837 Factor conditional logic operator over implication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 ->  ch ) ,  ( th  ->  ta ) )  <->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta ) ) )
 
Theoremifpororb 35838 Factor conditional logic operator over disjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 \/  ch ) ,  ( th  \/  ta ) )  <-> 
 (if- ( ph ,  ps ,  th )  \/ if-
 ( ph ,  ch ,  ta ) ) )
 
Theoremifpananb 35839 Factor conditional logic operator over conjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 /\  ch ) ,  ( th  /\  ta ) )  <-> 
 (if- ( ph ,  ps ,  th )  /\ if- (
 ph ,  ch ,  ta ) ) )
 
Theoremifpnannanb 35840 Factor conditional logic operator over nand in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps  -/\  ch ) ,  ( th  -/\  ta ) )  <-> 
 (if- ( ph ,  ps ,  th )  -/\ if- (
 ph ,  ch ,  ta ) ) )
 
Theoremifpor123g 35841 Disjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  ta )  \/ if-
 ( ps ,  th ,  et ) )  <->  ( ( ( ( ph  ->  -.  ps )  \/  ( ch  \/  th ) )  /\  (
 ( ps  ->  ph )  \/  ( ta  \/  th ) ) )  /\  ( ( ( ph  ->  ps )  \/  ( ch  \/  et ) ) 
 /\  ( ( -. 
 ps  ->  ph )  \/  ( ta  \/  et ) ) ) ) )
 
Theoremifpimim 35842 Consequnce of implication. (Contributed by RP, 17-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 ->  ch ) ,  ( th  ->  ta ) )  ->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
 ) )
 
Theoremifpbibib 35843 Factor conditional logic operator over biimplication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps  <->  ch ) ,  ( th  <->  ta ) )  <->  (if- ( ph ,  ps ,  th )  <-> if- ( ph ,  ch ,  ta ) ) )
 
Theoremifpxorxorb 35844 Factor conditional logic operator over xor in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 \/_  ch ) ,  ( th  \/_  ta ) )  <-> 
 (if- ( ph ,  ps ,  th )  \/_ if- (
 ph ,  ch ,  ta ) ) )
 
21.25.1.2  Sophisms
 
Theoremrp-fakeimass 35845 A special case where implication appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
 |-  (
 ( ph  \/  ch )  <->  ( ( ( ph  ->  ps )  ->  ch )  <->  (
 ph  ->  ( ps  ->  ch ) ) ) )
 
Theoremrp-fakeanorass 35846 A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.)
 |-  (
 ( ch  ->  ph )  <->  ( ( ( ph  /\  ps )  \/  ch )  <->  ( ph  /\  ( ps  \/  ch ) ) ) )
 
Theoremrp-fakeoranass 35847 A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
 |-  (
 ( ph  ->  ch )  <->  ( ( ( ph  \/  ps )  /\  ch )  <->  (
 ph  \/  ( ps  /\ 
 ch ) ) ) )
 
Theoremrp-fakenanass 35848 A special case where nand appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
 |-  (
 ( ph  <->  ch )  <->  ( ( (
 ph  -/\  ps )  -/\  ch )  <->  ( ph  -/\  ( ps  -/\  ch ) ) ) )
 
Theoremrp-fakeinunass 35849 A special case where a mixture of intersection and union appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.)
 |-  ( C  C_  A  <->  ( ( A  i^i  B )  u.  C )  =  ( A  i^i  ( B  u.  C ) ) )
 
Theoremrp-fakeuninass 35850 A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
 |-  ( A  C_  C  <->  ( ( A  u.  B )  i^i 
 C )  =  ( A  u.  ( B  i^i  C ) ) )
 
21.25.1.3  Finite Sets

Membership in the class of finite sets can be expressed in many ways.

 
Theoremrp-isfinite5 35851* A set is said to be finite if it can be put in one-to-one correspondence with all the natural numbers between 1 and some  n  e.  NN0. (Contributed by Richard Penner, 3-Mar-2020.)
 |-  ( A  e.  Fin  <->  E. n  e.  NN0  ( 1 ... n )  ~~  A )
 
Theoremrp-isfinite6 35852* A set is said to be finite if it is either empty or it can be put in one-to-one correspondence with all the natural numbers between 1 and some  n  e.  NN. (Contributed by Richard Penner, 10-Mar-2020.)
 |-  ( A  e.  Fin  <->  ( A  =  (/) 
 \/  E. n  e.  NN  ( 1 ... n )  ~~  A ) )
 
21.25.1.4  Infinite Sets
 
Theorempwelg 35853* The powerclass is an element of a class closed under union and powerclass operations iff the element is a member of that class. (Contributed by RP, 21-Mar-2020.)
 |-  ( A. x  e.  B  ( U. x  e.  B  /\  ~P x  e.  B )  ->  ( A  e.  B 
 <->  ~P A  e.  B ) )
 
Theorempwinfig 35854* The powerclass of an infinite set is an infinite set, and vice-versa. Here  B is a class which is closed under both the union and the powerclass operations and which may have infinite sets as members. (Contributed by RP, 21-Mar-2020.)
 |-  ( A. x  e.  B  ( U. x  e.  B  /\  ~P x  e.  B )  ->  ( A  e.  ( B  \  Fin )  <->  ~P A  e.  ( B 
 \  Fin ) ) )
 
Theorempwinfi2 35855 The powerclass of an infinite set is an infinite set, and vice-versa. Here  U is a weak universe. (Contributed by RP, 21-Mar-2020.)
 |-  ( U  e. WUni  ->  ( A  e.  ( U  \  Fin )  <->  ~P A  e.  ( U  \  Fin ) ) )
 
Theorempwinfi3 35856 The powerclass of an infinite set is an infinite set, and vice-versa. Here  T is a transitive Tarski universe. (Contributed by RP, 21-Mar-2020.)
 |-  (
 ( T  e.  Tarski  /\ 
 Tr  T )  ->  ( A  e.  ( T  \  Fin )  <->  ~P A  e.  ( T  \  Fin ) ) )
 
Theorempwinfi 35857 The powerclass of an infinite set is an infinite set, and vice-versa. (Contributed by RP, 21-Mar-2020.)
 |-  ( A  e.  ( _V  \ 
 Fin )  <->  ~P A  e.  ( _V  \  Fin ) )
 
21.25.1.5  Finite intersection property

While there is not yet a definition, the finite intersection property of a class is introduced by fiint 7854 where two textbook definitions are shown to be equivalent.

This property is seen often with ordinal numbers (onin 5473, ordelinel 5540 ), chains of sets ordered by the proper subset relation (sorpssin 6593), various sets in the field of topology (inopn 19851, incld 19980, innei 20063, ... ) and "universal" classes like weak universes (wunin 9137, tskin 9183) and the class of all sets (inex1g 4568) .

 
Theoremfipjust 35858* A definition of the finite intersection property of a class based on closure under pair-wise intersection of its elements is independent of the dummy variables. (Contributed by Richard Penner, 1-Jan-2020.)
 |-  ( A. u  e.  A  A. v  e.  A  ( u  i^i  v )  e.  A  <->  A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A )
 
Theoremcllem0 35859* The class of all sets with property  ph ( z ) is closed under the binary operation on sets defined in  R ( x ,  y ). (Contributed by Richard Penner, 3-Jan-2020.)
 |-  V  =  { z  |  ph }   &    |-  R  e.  U   &    |-  ( z  =  R  ->  ( ph  <->  ps ) )   &    |-  ( z  =  x  ->  ( ph  <->  ch ) )   &    |-  ( z  =  y  ->  ( ph  <->  th ) )   &    |-  ( ( ch 
 /\  th )  ->  ps )   =>    |-  A. x  e.  V  A. y  e.  V  R  e.  V
 
Theoremsuperficl 35860* The class of all supersets of a class has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  B  C_  z }   =>    |- 
 A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A
 
Theoremsuperuncl 35861* The class of all supersets of a class is closed under binary union. (Contributed by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  B  C_  z }   =>    |- 
 A. x  e.  A  A. y  e.  A  ( x  u.  y )  e.  A
 
Theoremssficl 35862* The class of all subsets of a class has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  z 
 C_  B }   =>    |-  A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A
 
Theoremssuncl 35863* The class of all subsets of a class is closed under binary union. (Contributed by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  z 
 C_  B }   =>    |-  A. x  e.  A  A. y  e.  A  ( x  u.  y )  e.  A
 
Theoremssdifcl 35864* The class of all subsets of a class is closed under set difference. (Contributed by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  z 
 C_  B }   =>    |-  A. x  e.  A  A. y  e.  A  ( x  \  y )  e.  A
 
Theoremsssymdifcl 35865* The class of all subsets of a class is closed under symmetric difference. (Contributed by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  z 
 C_  B }   =>    |-  A. x  e.  A  A. y  e.  A  ( ( x 
 \  y )  u.  ( y  \  x ) )  e.  A
 
Theoremfiinfi 35866* If two classes have the finite intersection property, then so does their intersection. (Contributed by Richard Penner, 1-Jan-2020.)
 |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A )   &    |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( x  i^i  y )  e.  B )   &    |-  ( ph  ->  C  =  ( A  i^i  B ) )   =>    |-  ( ph  ->  A. x  e.  C  A. y  e.  C  ( x  i^i  y )  e.  C )
 
21.25.2  Additional statements on relations and subclasses
 
Theoremal3im 35867 Version of ax-4 1678 for a nested implication. (Contributed by RP, 13-Apr-2020.)
 |-  ( A. x ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) 
 ->  ( A. x ph  ->  ( A. x ps  ->  ( A. x ch  ->  A. x th )
 ) ) )
 
Theoremintima0 35868* Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)
 |-  |^|_ a  e.  A  ( a " B )  =  |^| { x  |  E. a  e.  A  x  =  ( a " B ) }
 
Theoremelimaint 35869* Element of image of intersection. (Contributed by RP, 13-Apr-2020.)
 |-  (
 y  e.  ( |^| A
 " B )  <->  E. b  e.  B  A. a  e.  A  <. b ,  y >.  e.  a
 )
 
Theoremcsbcog 35870 Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.)
 |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  o.  C )  =  ( [_ A  /  x ]_ B  o.  [_ A  /  x ]_ C ) )
 
Theoremcnviun 35871* Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
 |-  `' U_ x  e.  A  B  =  U_ x  e.  A  `' B
 
Theoremimaiun1 35872* The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
 |-  ( U_ x  e.  A  B " C )  = 
 U_ x  e.  A  ( B " C )
 
Theoremcoiun1 35873* Composition with an indexed union. Proof analgous to that of coiun 5365. (Contributed by RP, 20-Jun-2020.)
 |-  ( U_ x  e.  C  A  o.  B )  = 
 U_ x  e.  C  ( A  o.  B )
 
Theoremelintima 35874* Element of intersection of images. (Contributed by RP, 13-Apr-2020.)
 |-  (
 y  e.  |^| { x  |  E. a  e.  A  x  =  ( a " B ) }  <->  A. a  e.  A  E. b  e.  B  <. b ,  y >.  e.  a )
 
Theoremintimass 35875* The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
 |-  ( |^| A " B ) 
 C_  |^| { x  |  E. a  e.  A  x  =  ( a " B ) }
 
Theoremintimass2 35876* The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
 |-  ( |^| A " B ) 
 C_  |^|_ x  e.  A  ( x " B )
 
Theoremintimag 35877* Requirement for the image under the intersection of relations to equal the intersection of the images of those relations. (Contributed by RP, 13-Apr-2020.)
 |-  ( A. y ( A. a  e.  A  E. b  e.  B  <. b ,  y >.  e.  a  ->  E. b  e.  B  A. a  e.  A  <. b ,  y >.  e.  a )  ->  ( |^| A " B )  =  |^| { x  |  E. a  e.  A  x  =  ( a " B ) } )
 
Theoremintimasn 35878* Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
 |-  ( B  e.  V  ->  (
 |^| A " { B } )  =  |^| { x  |  E. a  e.  A  x  =  ( a " { B } ) } )
 
Theoremintimasn2 35879* Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
 |-  ( B  e.  V  ->  (
 |^| A " { B } )  =  |^|_ x  e.  A  ( x
 " { B }
 ) )
 
Theoremss2iundf 35880* Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
 |-  F/ x ph   &    |-  F/ y ph   &    |-  F/_ y Y   &    |-  F/_ y A   &    |-  F/_ y B   &    |-  F/_ x C   &    |-  F/_ y C   &    |-  F/_ x D   &    |-  F/_ y G   &    |-  ( ( ph  /\  x  e.  A )  ->  Y  e.  C )   &    |-  ( ( ph  /\  x  e.  A  /\  y  =  Y )  ->  D  =  G )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  C_  G )   =>    |-  ( ph  ->  U_ x  e.  A  B  C_  U_ y  e.  C  D )
 
Theoremss2iundv 35881* Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
 |-  (
 ( ph  /\  x  e.  A )  ->  Y  e.  C )   &    |-  ( ( ph  /\  x  e.  A  /\  y  =  Y )  ->  D  =  G )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  C_  G )   =>    |-  ( ph  ->  U_ x  e.  A  B  C_  U_ y  e.  C  D )
 
Theoremcbviuneq12df 35882* Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.)
 |-  F/ x ph   &    |-  F/ y ph   &    |-  F/_ x X   &    |-  F/_ y Y   &    |-  F/_ x A   &    |-  F/_ y A   &    |-  F/_ y B   &    |-  F/_ x C   &    |-  F/_ y C   &    |-  F/_ x D   &    |-  F/_ x F   &    |-  F/_ y G   &    |-  ( ( ph  /\  y  e.  C )  ->  X  e.  A )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  Y  e.  C )   &    |-  ( ( ph  /\  y  e.  C  /\  x  =  X )  ->  B  =  F )   &    |-  ( ( ph  /\  x  e.  A  /\  y  =  Y )  ->  D  =  G )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  =  G )   &    |-  ( ( ph  /\  y  e.  C ) 
 ->  D  =  F )   =>    |-  ( ph  ->  U_ x  e.  A  B  =  U_ y  e.  C  D )
 
Theoremcbviuneq12dv 35883* Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.)
 |-  (
 ( ph  /\  y  e.  C )  ->  X  e.  A )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  Y  e.  C )   &    |-  ( ( ph  /\  y  e.  C  /\  x  =  X )  ->  B  =  F )   &    |-  ( ( ph  /\  x  e.  A  /\  y  =  Y )  ->  D  =  G )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  =  G )   &    |-  ( ( ph  /\  y  e.  C ) 
 ->  D  =  F )   =>    |-  ( ph  ->  U_ x  e.  A  B  =  U_ y  e.  C  D )
 
Theoremconrel1d 35884 Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  `' A  =  (/) )   =>    |-  ( ph  ->  ( A  o.  B )  =  (/) )
 
Theoremconrel2d 35885 Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  `' A  =  (/) )   =>    |-  ( ph  ->  ( B  o.  A )  =  (/) )
 
21.25.2.1  Transitive relations (not to be confused with transitive classes).
 
Theoremtrrelind 35886 The intersection of transitive relations is a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  ( R  o.  R )  C_  R )   &    |-  ( ph  ->  ( S  o.  S )  C_  S )   &    |-  ( ph  ->  T  =  ( R  i^i  S ) )   =>    |-  ( ph  ->  ( T  o.  T )  C_  T )
 
Theoremxpintrreld 35887 The intersection of a transitive relation with a cross product is a transitve relation. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  ( R  o.  R )  C_  R )   &    |-  ( ph  ->  S  =  ( R  i^i  ( A  X.  B ) ) )   =>    |-  ( ph  ->  ( S  o.  S )  C_  S )
 
Theoremrestrreld 35888 The restriction of a transitive relation is a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  ( R  o.  R )  C_  R )   &    |-  ( ph  ->  S  =  ( R  |`  A ) )   =>    |-  ( ph  ->  ( S  o.  S )  C_  S )
 
Theoremtrrelsuperreldg 35889 Concrete construction of a superclass of relation  R which is a transitive relation. (Contributed by Richard Penner, 25-Dec-2019.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  S  =  ( dom  R  X.  ran  R ) )   =>    |-  ( ph  ->  ( R  C_  S  /\  ( S  o.  S )  C_  S ) )
 
Theoremtrficl 35890* The class of all transitive relations has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  ( z  o.  z ) 
 C_  z }   =>    |-  A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A
 
Theoremcnvtrrel 35891 The converse of a transitive relation is a transitive relation. (Contributed by Richard Penner, 25-Dec-2019.)
 |-  (
 ( S  o.  S )  C_  S  <->  ( `' S  o.  `' S )  C_  `' S )
 
Theoremtrrelsuperrel2dg 35892 Concrete construction of a superclass of relation  R which is a transitive relation. (Contributed by RP, 20-Jul-2020.)
 |-  ( ph  ->  S  =  ( R  u.  ( dom 
 R  X.  ran  R ) ) )   =>    |-  ( ph  ->  ( R  C_  S  /\  ( S  o.  S )  C_  S ) )
 
21.25.2.2  Reflexive closures
 
Syntaxcrcl 35893 Extend class notation with reflexive closure.
 class  r*
 
Definitiondf-rcl 35894* Reflexive closure of a relation. This is the smallest superset which has the reflexive property. (Contributed by RP, 5-Jun-2020.)
 |-  r*  =  ( x  e.  _V  |->  |^| { z  |  ( x  C_  z  /\  (  _I  |`  ( dom  z  u.  ran  z
 ) )  C_  z
 ) } )
 
Theoremdfrcl2 35895 Reflexive closure of a relation as union with restricted identity relation. (Contributed by RP, 6-Jun-2020.)
 |-  r*  =  ( x  e.  _V  |->  ( (  _I  |`  ( dom  x  u.  ran 
 x ) )  u.  x ) )
 
Theoremdfrcl3 35896 Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.)
 |-  r*  =  ( x  e.  _V  |->  ( ( x ^r  0 )  u.  ( x ^r  1 ) ) )
 
Theoremdfrcl4 35897* Reflexive closure of a relation as indexed union of powers of the relation. (Contributed by RP, 8-Jun-2020.)
 |-  r*  =  ( r  e.  _V  |->  U_ n  e.  {
 0 ,  1 }  ( r ^r  n ) )
 
21.25.2.3  Finite relationship composition.

In order for theorems on the transitive closure of a relation to be grouped together before the concept of continuity, we really need an analogue of ^r that works on finite ordinals or finite sets instead of natural numbers.

 
Theoremrelexp2 35898 A set operated on by the relation exponent to the second power is equal to the composition of the set with itself. (Contributed by RP, 1-Jun-2020.)
 |-  ( R  e.  V  ->  ( R ^r  2 )  =  ( R  o.  R ) )
 
Theoremrelexpnul 35899 If the domain and range of powers of a relation are disjoint then the relation raised to the sum of those exponents is empty. (Contributed by RP, 1-Jun-2020.)
 |-  (
 ( ( R  e.  V  /\  Rel  R )  /\  ( N  e.  NN0  /\  M  e.  NN0 )
 )  ->  ( ( dom  ( R ^r  N )  i^i  ran  ( R ^r  M ) )  =  (/)  <->  ( R ^r  ( N  +  M ) )  =  (/) ) )
 
Theoremeliunov2 35900* Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the the element is a member of that operator value. Generalized from dfrtrclrec2 13099. (Contributed by RP, 1-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r  .^  n )
 )   =>    |-  ( ( R  e.  U  /\  N  e.  V )  ->  ( X  e.  ( C `  R )  <->  E. n  e.  N  X  e.  ( R  .^  n ) ) )
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