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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rescnvcnv 5001 | The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (^{◡}^{◡}𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | ||
Theorem | cnvcnvres 5002 | The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.) |
⊢ ^{◡}^{◡}(𝐴 ↾ 𝐵) = (^{◡}^{◡}𝐴 ↾ 𝐵) | ||
Theorem | imacnvcnv 5003 | The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.) |
⊢ (^{◡}^{◡}𝐴 “ 𝐵) = (𝐴 “ 𝐵) | ||
Theorem | dmsnm 5004* | 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}) | ||
Theorem | rnsnm 5005* | 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}) | ||
Theorem | dmsn0 5006 | The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.) |
⊢ dom {∅} = ∅ | ||
Theorem | cnvsn0 5007 | The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ ^{◡}{∅} = ∅ | ||
Theorem | dmsn0el 5008 | 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} = ∅) | ||
Theorem | relsn2m 5009* | 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}) | ||
Theorem | dmsnopg 5010 | 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}) | ||
Theorem | dmpropg 5011 | 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}) | ||
Theorem | dmsnop 5012 | 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} | ||
Theorem | dmprop 5013 | 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} | ||
Theorem | dmtpop 5014 | 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} | ||
Theorem | cnvcnvsn 5015 | Double converse of a singleton of an ordered pair. (Unlike cnvsn 5021, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ^{◡}^{◡}{⟨𝐴, 𝐵⟩} = ^{◡}{⟨𝐵, 𝐴⟩} | ||
Theorem | dmsnsnsng 5016 | The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.) |
⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴}) | ||
Theorem | rnsnopg 5017 | 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}) | ||
Theorem | rnpropg 5018 | 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}) | ||
Theorem | rnsnop 5019 | 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} | ||
Theorem | op1sta 5020 | Extract the first member of an ordered pair. (See op2nda 5023 to extract the second member and op1stb 4399 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴 | ||
Theorem | cnvsn 5021 | 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⟩} | ||
Theorem | op2ndb 5022 | Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4399 to extract the first member and op2nda 5023 for an alternate version.) (Contributed by NM, 25-Nov-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∩ ∩ ∩ ^{◡}{⟨𝐴, 𝐵⟩} = 𝐵 | ||
Theorem | op2nda 5023 | Extract the second member of an ordered pair. (See op1sta 5020 to extract the first member and op2ndb 5022 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 | ||
Theorem | cnvsng 5024 | Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ^{◡}{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}) | ||
Theorem | opswapg 5025 | Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ ^{◡}{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩) | ||
Theorem | elxp4 5026 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5027. (Contributed by NM, 17-Feb-2004.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) | ||
Theorem | elxp5 5027 | Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5026 when the double intersection does not create class existence problems (caused by int0 3785). (Contributed by NM, 1-Aug-2004.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) | ||
Theorem | cnvresima 5028 | An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) |
⊢ (^{◡}(𝐹 ↾ 𝐴) “ 𝐵) = ((^{◡}𝐹 “ 𝐵) ∩ 𝐴) | ||
Theorem | resdm2 5029 | A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.) |
⊢ (𝐴 ↾ dom 𝐴) = ^{◡}^{◡}𝐴 | ||
Theorem | resdmres 5030 | Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵) | ||
Theorem | imadmres 5031 | The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵) | ||
Theorem | mptpreima 5032* | The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (^{◡}𝐹 “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} | ||
Theorem | mptiniseg 5033* | Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝐶 ∈ 𝑉 → (^{◡}𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶}) | ||
Theorem | dmmpt 5034 | The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} | ||
Theorem | dmmptss 5035* | The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ dom 𝐹 ⊆ 𝐴 | ||
Theorem | dmmptg 5036* | 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 ∈ 𝐴 ↦ 𝐵) = 𝐴) | ||
Theorem | relco 5037 | A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) |
⊢ Rel (𝐴 ∘ 𝐵) | ||
Theorem | dfco2 5038* | Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.) |
⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((^{◡}𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) | ||
Theorem | dfco2a 5039* | Generalization of dfco2 5038, 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 → (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 ((^{◡}𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))) | ||
Theorem | coundi 5040 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶)) | ||
Theorem | coundir 5041 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶)) | ||
Theorem | cores 5042 | Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵)) | ||
Theorem | resco 5043 | Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.) |
⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶)) | ||
Theorem | imaco 5044 | Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) |
⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶)) | ||
Theorem | rnco 5045 | The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) |
⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) | ||
Theorem | rnco2 5046 | The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.) |
⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵) | ||
Theorem | dmco 5047 | The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.) |
⊢ dom (𝐴 ∘ 𝐵) = (^{◡}𝐵 “ dom 𝐴) | ||
Theorem | coiun 5048* | Composition with an indexed union. (Contributed by NM, 21-Dec-2008.) |
⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) | ||
Theorem | cocnvcnv1 5049 | A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.) |
⊢ (^{◡}^{◡}𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) | ||
Theorem | cocnvcnv2 5050 | A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.) |
⊢ (𝐴 ∘ ^{◡}^{◡}𝐵) = (𝐴 ∘ 𝐵) | ||
Theorem | cores2 5051 | Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.) |
⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ^{◡}(^{◡}𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵)) | ||
Theorem | co02 5052 | Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.) |
⊢ (𝐴 ∘ ∅) = ∅ | ||
Theorem | co01 5053 | Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
⊢ (∅ ∘ 𝐴) = ∅ | ||
Theorem | coi1 5054 | 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) | ||
Theorem | coi2 5055 | 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) | ||
Theorem | coires1 5056 | Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.) |
⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵) | ||
Theorem | coass 5057 | 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.) |
⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶)) | ||
Theorem | relcnvtr 5058 | A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) |
⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (^{◡}𝑅 ∘ ^{◡}𝑅) ⊆ ^{◡}𝑅)) | ||
Theorem | relssdmrn 5059 | 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 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) | ||
Theorem | cnvssrndm 5060 | 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) | ||
Theorem | cossxp 5061 | Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) | ||
Theorem | cossxp2 5062 | The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.) |
⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) & ⊢ (𝜑 → 𝑆 ⊆ (𝐵 × 𝐶)) ⇒ ⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ (𝐴 × 𝐶)) | ||
Theorem | cocnvres 5063 | Restricting a relation and a converse relation when they are composed together (Contributed by BJ, 10-Jul-2022.) |
⊢ (𝑆 ∘ ^{◡}𝑅) = ((𝑆 ↾ dom 𝑅) ∘ ^{◡}(𝑅 ↾ dom 𝑆)) | ||
Theorem | cocnvss 5064 | Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.) |
⊢ (𝑆 ∘ ^{◡}𝑅) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅)) | ||
Theorem | relrelss 5065 | 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)) | ||
Theorem | unielrel 5066 | The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.) |
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∪ 𝐴 ∈ ∪ 𝑅) | ||
Theorem | relfld 5067 | The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.) |
⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) | ||
Theorem | relresfld 5068 | Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.) |
⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅) | ||
Theorem | relcoi2 5069 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.) |
⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = 𝑅) | ||
Theorem | relcoi1 5070 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) |
⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅) | ||
Theorem | unidmrn 5071 | The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.) |
⊢ ∪ ∪ ^{◡}𝐴 = (dom 𝐴 ∪ ran 𝐴) | ||
Theorem | relcnvfld 5072 | 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) | ||
Theorem | dfdm2 5073 | Alternate definition of domain df-dm 4549 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
⊢ dom 𝐴 = ∪ ∪ (^{◡}𝐴 ∘ 𝐴) | ||
Theorem | unixpm 5074* | 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) = (𝐴 ∪ 𝐵)) | ||
Theorem | unixp0im 5075 | The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.) |
⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅) | ||
Theorem | cnvexg 5076 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.) |
⊢ (𝐴 ∈ 𝑉 → ^{◡}𝐴 ∈ V) | ||
Theorem | cnvex 5077 | 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 | ||
Theorem | relcnvexb 5078 | A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.) |
⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ^{◡}𝑅 ∈ V)) | ||
Theorem | ressn 5079 | Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) | ||
Theorem | cnviinm 5080* | 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) | ||
Theorem | cnvpom 5081* | The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.) |
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑅 Po 𝐴 ↔ ^{◡}𝑅 Po 𝐴)) | ||
Theorem | cnvsom 5082* | The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.) |
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑅 Or 𝐴 ↔ ^{◡}𝑅 Or 𝐴)) | ||
Theorem | coexg 5083 | The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V) | ||
Theorem | coex 5084 | The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∘ 𝐵) ∈ V | ||
Theorem | xpcom 5085* | Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.) |
⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶)) | ||
Syntax | cio 5086 | Extend class notation with Russell's definition description binder (inverted iota). |
class (℩𝑥𝜑) | ||
Theorem | iotajust 5087* | Soundness justification theorem for df-iota 5088. (Contributed by Andrew Salmon, 29-Jun-2011.) |
⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | ||
Definition | df-iota 5088* |
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 5099);
otherwise, it evaluates to the empty set (see iotanul 5103). 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 5111 (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 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | ||
Theorem | dfiota2 5089* | Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) |
⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} | ||
Theorem | nfiota1 5090 | Bound-variable hypothesis builder for the ℩ class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥(℩𝑥𝜑) | ||
Theorem | nfiotadw 5091* | Bound-variable hypothesis builder for the ℩ class. (Contributed by Jim Kingdon, 21-Dec-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) | ||
Theorem | nfiotaw 5092* | Bound-variable hypothesis builder for the ℩ class. (Contributed by NM, 23-Aug-2011.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥(℩𝑦𝜑) | ||
Theorem | cbviota 5093 | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | cbviotav 5094* | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | sb8iota 5095 | Variable substitution in description binder. Compare sb8eu 2012. (Contributed by NM, 18-Mar-2013.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | iotaeq 5096 | Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑)) | ||
Theorem | iotabi 5097 | Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓)) | ||
Theorem | uniabio 5098* | Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = 𝑦) | ||
Theorem | iotaval 5099* | Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | ||
Theorem | iotauni 5100 | Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
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