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Theorem List for Intuitionistic Logic Explorer - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnvcnv2 4701 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
A = (A ↾ V)

Theoremcnvcnvss 4702 The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)
AA

Theoremcnveqb 4703 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
((Rel A Rel B) → (A = BA = B))

Theoremcnveq0 4704 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
(Rel A → (A = ∅ ↔ A = ∅))

Theoremdfrel3 4705 Alternate definition of relation. (Contributed by NM, 14-May-2008.)
(Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)

Theoremdmresv 4706 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
dom (A ↾ V) = dom A

Theoremrnresv 4707 The range of a universal restriction. (Contributed by NM, 14-May-2008.)
ran (A ↾ V) = ran A

Theoremdfrn4 4708 Range defined in terms of image. (Contributed by NM, 14-May-2008.)
ran A = (A “ V)

Theoremcsbrng 4709 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(A 𝑉A / xran B = ran A / xB)

Theoremrescnvcnv 4710 The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(AB) = (AB)

Theoremcnvcnvres 4711 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
(AB) = (AB)

Theoremimacnvcnv 4712 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
(AB) = (AB)

Theoremdmsnm 4713* The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
(A (V × V) ↔ x x dom {A})

Theoremrnsnm 4714* The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
(A (V × V) ↔ x x ran {A})

Theoremdmsn0 4715 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
dom {∅} = ∅

Theoremcnvsn0 4716 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
{∅} = ∅

Theoremdmsn0el 4717 The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)
(∅ A → dom {A} = ∅)

Theoremrelsn2m 4718* A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)
A V       (Rel {A} ↔ x x dom {A})

Theoremdmsnopg 4719 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)
(B 𝑉 → dom {⟨A, B⟩} = {A})

Theoremdmpropg 4720 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
((B 𝑉 𝐷 𝑊) → dom {⟨A, B⟩, ⟨𝐶, 𝐷⟩} = {A, 𝐶})

Theoremdmsnop 4721 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
B V       dom {⟨A, B⟩} = {A}

Theoremdmprop 4722 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)
B V    &   𝐷 V       dom {⟨A, B⟩, ⟨𝐶, 𝐷⟩} = {A, 𝐶}

Theoremdmtpop 4723 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)
B V    &   𝐷 V    &   𝐹 V       dom {⟨A, B⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {A, 𝐶, 𝐸}

Theoremcnvcnvsn 4724 Double converse of a singleton of an ordered pair. (Unlike cnvsn 4730, this does not need any sethood assumptions on A and B.) (Contributed by Mario Carneiro, 26-Apr-2015.)
{⟨A, B⟩} = {⟨B, A⟩}

Theoremdmsnsnsng 4725 The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)
(A V → dom {{{A}}} = {A})

Theoremrnsnopg 4726 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
(A 𝑉 → ran {⟨A, B⟩} = {B})

Theoremrnpropg 4727 The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)
((A 𝑉 B 𝑊) → ran {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} = {𝐶, 𝐷})

Theoremrnsnop 4728 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
A V       ran {⟨A, B⟩} = {B}

Theoremop1sta 4729 Extract the first member of an ordered pair. (See op2nda 4732 to extract the second member and op1stb 4159 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.)
A V    &   B V        dom {⟨A, B⟩} = A

Theoremcnvsn 4730 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
A V    &   B V       {⟨A, B⟩} = {⟨B, A⟩}

Theoremop2ndb 4731 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4159 to extract the first member and op2nda 4732 for an alternate version.) (Contributed by NM, 25-Nov-2003.)
A V    &   B V        {⟨A, B⟩} = B

Theoremop2nda 4732 Extract the second member of an ordered pair. (See op1sta 4729 to extract the first member and op2ndb 4731 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
A V    &   B V        ran {⟨A, B⟩} = B

Theoremcnvsng 4733 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
((A 𝑉 B 𝑊) → {⟨A, B⟩} = {⟨B, A⟩})

Theoremopswapg 4734 Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)
((A 𝑉 B 𝑊) → {⟨A, B⟩} = ⟨B, A⟩)

Theoremelxp4 4735 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4736. (Contributed by NM, 17-Feb-2004.)
(A (B × 𝐶) ↔ (A = ⟨ dom {A}, ran {A}⟩ ( dom {A} B ran {A} 𝐶)))

Theoremelxp5 4736 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4735 when the double intersection does not create class existence problems (caused by int0 3603). (Contributed by NM, 1-Aug-2004.)
(A (B × 𝐶) ↔ (A = ⟨ A, ran {A}⟩ ( A B ran {A} 𝐶)))

Theoremcnvresima 4737 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
((𝐹A) “ B) = ((𝐹B) ∩ A)

Theoremresdm2 4738 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
(A ↾ dom A) = A

Theoremresdmres 4739 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
(A ↾ dom (AB)) = (AB)

Theoremimadmres 4740 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
(A “ dom (AB)) = (AB)

Theoremmptpreima 4741* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐹 = (x AB)       (𝐹𝐶) = {x AB 𝐶}

Theoremmptiniseg 4742* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐹 = (x AB)       (𝐶 𝑉 → (𝐹 “ {𝐶}) = {x AB = 𝐶})

Theoremdmmpt 4743 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)
𝐹 = (x AB)       dom 𝐹 = {x AB V}

Theoremdmmptss 4744* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
𝐹 = (x AB)       dom 𝐹A

Theoremdmmptg 4745* The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
(x A B 𝑉 → dom (x AB) = A)

Theoremrelco 4746 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
Rel (AB)

Theoremdfco2 4747* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
(AB) = x V ((B “ {x}) × (A “ {x}))

Theoremdfco2a 4748* Generalization of dfco2 4747, where 𝐶 can have any value between dom A ∩ ran B and V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((dom A ∩ ran B) ⊆ 𝐶 → (AB) = x 𝐶 ((B “ {x}) × (A “ {x})))

Theoremcoundi 4749 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(A ∘ (B𝐶)) = ((AB) ∪ (A𝐶))

Theoremcoundir 4750 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((AB) ∘ 𝐶) = ((A𝐶) ∪ (B𝐶))

Theoremcores 4751 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(ran B𝐶 → ((A𝐶) ∘ B) = (AB))

Theoremresco 4752 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
((AB) ↾ 𝐶) = (A ∘ (B𝐶))

Theoremimaco 4753 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
((AB) “ 𝐶) = (A “ (B𝐶))

Theoremrnco 4754 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
ran (AB) = ran (A ↾ ran B)

Theoremrnco2 4755 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
ran (AB) = (A “ ran B)

Theoremdmco 4756 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
dom (AB) = (B “ dom A)

Theoremcoiun 4757* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
(A x 𝐶 B) = x 𝐶 (AB)

Theoremcocnvcnv1 4758 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
(AB) = (AB)

Theoremcocnvcnv2 4759 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
(AB) = (AB)

Theoremcores2 4760 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
(dom A𝐶 → (A(B𝐶)) = (AB))

Theoremco02 4761 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
(A ∘ ∅) = ∅

Theoremco01 4762 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
(∅ ∘ A) = ∅

Theoremcoi1 4763 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
(Rel A → (A ∘ I ) = A)

Theoremcoi2 4764 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
(Rel A → ( I ∘ A) = A)

Theoremcoires1 4765 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
(A ∘ ( I ↾ B)) = (AB)

Theoremcoass 4766 Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
((AB) ∘ 𝐶) = (A ∘ (B𝐶))

Theoremrelcnvtr 4767 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
(Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))

Theoremrelssdmrn 4768 A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
(Rel AA ⊆ (dom A × ran A))

Theoremcnvssrndm 4769 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
A ⊆ (ran A × dom A)

Theoremcossxp 4770 Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
(AB) ⊆ (dom B × ran A)

Theoremrelrelss 4771 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
((Rel A Rel dom A) ↔ A ⊆ ((V × V) × V))

Theoremunielrel 4772 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
((Rel 𝑅 A 𝑅) → A 𝑅)

Theoremrelfld 4773 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
(Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))

Theoremrelresfld 4774 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
(Rel 𝑅 → (𝑅 𝑅) = 𝑅)

Theoremrelcoi2 4775 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
(Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)

Theoremrelcoi1 4776 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)
(Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = 𝑅)

Theoremunidmrn 4777 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
A = (dom A ∪ ran A)

Theoremrelcnvfld 4778 if 𝑅 is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
(Rel 𝑅 𝑅 = 𝑅)

Theoremdfdm2 4779 Alternate definition of domain df-dm 4282 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
dom A = (AA)

Theoremunixpm 4780* The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)
(x x (A × B) → (A × B) = (AB))

Theoremunixp0im 4781 The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)
((A × B) = ∅ → (A × B) = ∅)

Theoremcnvexg 4782 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
(A 𝑉A V)

Theoremcnvex 4783 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
A V       A V

Theoremrelcnvexb 4784 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
(Rel 𝑅 → (𝑅 V ↔ 𝑅 V))

Theoremressn 4785 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(A ↾ {B}) = ({B} × (A “ {B}))

Theoremcnviinm 4786* The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
(y y A x A B = x A B)

Theoremcnvpom 4787* The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
(x x A → (𝑅 Po A𝑅 Po A))

Theoremcnvsom 4788* The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)
(x x A → (𝑅 Or A𝑅 Or A))

Theoremcoexg 4789 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
((A 𝑉 B 𝑊) → (AB) V)

Theoremcoex 4790 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
A V    &   B V       (AB) V

Theoremxpcom 4791* Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
(x x B → ((B × 𝐶) ∘ (A × B)) = (A × 𝐶))

2.6.7  Definite description binder (inverted iota)

Syntaxcio 4792 Extend class notation with Russell's definition description binder (inverted iota).
class (℩xφ)

Theoremiotajust 4793* Soundness justification theorem for df-iota 4794. (Contributed by Andrew Salmon, 29-Jun-2011.)
{y ∣ {xφ} = {y}} = {z ∣ {xφ} = {z}}

Definitiondf-iota 4794* Define Russell's definition description binder, which can be read as "the unique x such that φ," where φ ordinarily contains x as a free variable. Our definition is meaningful only when there is exactly one x such that φ is true (see iotaval 4805); otherwise, it evaluates to the empty set (see iotanul 4809). Russell used the inverted iota symbol to represent the binder.

Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use iotacl 4817 (for unbounded iota). This can be easier than applying a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF.

(Contributed by Andrew Salmon, 30-Jun-2011.)

(℩xφ) = {y ∣ {xφ} = {y}}

Theoremdfiota2 4795* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
(℩xφ) = {yx(φx = y)}

Theoremnfiota1 4796 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
x(℩xφ)

Theoremnfiotadxy 4797* Deduction version of nfiotaxy 4798. (Contributed by Jim Kingdon, 21-Dec-2018.)
yφ    &   (φ → Ⅎxψ)       (φx(℩yψ))

Theoremnfiotaxy 4798* Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.)
xφ       x(℩yφ)

Theoremcbviota 4799 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(x = y → (φψ))    &   yφ    &   xψ       (℩xφ) = (℩yψ)

Theoremcbviotav 4800* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(x = y → (φψ))       (℩xφ) = (℩yψ)

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