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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | coundir 4901 | 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 4902 | 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 4903 | Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.) |
⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶)) | ||
Theorem | imaco 4904 | Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) |
⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶)) | ||
Theorem | rnco 4905 | The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) |
⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) | ||
Theorem | rnco2 4906 | The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.) |
⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵) | ||
Theorem | dmco 4907 | 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 4908* | Composition with an indexed union. (Contributed by NM, 21-Dec-2008.) |
⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) | ||
Theorem | cocnvcnv1 4909 | 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 4910 | 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 4911 | 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 4912 | Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.) |
⊢ (𝐴 ∘ ∅) = ∅ | ||
Theorem | co01 4913 | Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
⊢ (∅ ∘ 𝐴) = ∅ | ||
Theorem | coi1 4914 | 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 4915 | 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 4916 | 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 4917 | 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 4918 | 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 4919 | 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 4920 | 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 4921 | 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 4922 | 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 4923 | 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 4924 | 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 4925 | 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 4926 | 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 4927 | The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.) |
⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) | ||
Theorem | relresfld 4928 | Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.) |
⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅) | ||
Theorem | relcoi2 4929 | 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 4930 | 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 4931 | 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 4932 | 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 4933 | Alternate definition of domain df-dm 4423 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
⊢ dom 𝐴 = ∪ ∪ (^{◡}𝐴 ∘ 𝐴) | ||
Theorem | unixpm 4934* | 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 4935 | The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.) |
⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅) | ||
Theorem | cnvexg 4936 | 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 4937 | 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 4938 | 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 4939 | Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) | ||
Theorem | cnviinm 4940* | 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 4941* | 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 4942* | 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 4943 | The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V) | ||
Theorem | coex 4944 | The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∘ 𝐵) ∈ V | ||
Theorem | xpcom 4945* | Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.) |
⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶)) | ||
Syntax | cio 4946 | Extend class notation with Russell's definition description binder (inverted iota). |
class (℩𝑥𝜑) | ||
Theorem | iotajust 4947* | Soundness justification theorem for df-iota 4948. (Contributed by Andrew Salmon, 29-Jun-2011.) |
⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | ||
Definition | df-iota 4948* |
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 4959);
otherwise, it evaluates to the empty set (see iotanul 4963). 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 4971 (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 4949* | 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 4950 | Bound-variable hypothesis builder for the ℩ class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥(℩𝑥𝜑) | ||
Theorem | nfiotadxy 4951* | Deduction version of nfiotaxy 4952. (Contributed by Jim Kingdon, 21-Dec-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) | ||
Theorem | nfiotaxy 4952* | Bound-variable hypothesis builder for the ℩ class. (Contributed by NM, 23-Aug-2011.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥(℩𝑦𝜑) | ||
Theorem | cbviota 4953 | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | cbviotav 4954* | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | sb8iota 4955 | Variable substitution in description binder. Compare sb8eu 1958. (Contributed by NM, 18-Mar-2013.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | iotaeq 4956 | Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑)) | ||
Theorem | iotabi 4957 | Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓)) | ||
Theorem | uniabio 4958* | 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 4959* | 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 4960 | Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) | ||
Theorem | iotaint 4961 | Equivalence between two different forms of ℩. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) | ||
Theorem | iota1 4962 | Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | ||
Theorem | iotanul 4963 | Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one 𝑥 that satisfies 𝜑. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) | ||
Theorem | euiotaex 4964 | Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the ℩ class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) | ||
Theorem | iotass 4965* | Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.) |
⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → (℩𝑥𝜑) ⊆ 𝐴) | ||
Theorem | iota4 4966 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑) | ||
Theorem | iota4an 4967 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑) | ||
Theorem | iota5 4968* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴) | ||
Theorem | iotabidv 4969* | Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒)) | ||
Theorem | iotabii 4970 | Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑥𝜓) | ||
Theorem | iotacl 4971 |
Membership law for descriptions.
This can useful for expanding an unbounded iota-based definition (see df-iota 4948). (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | iota2df 4972 | A condition that allows us to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → ∃!𝑥𝜓) & ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) & ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵)) | ||
Theorem | iota2d 4973* | A condition that allows us to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → ∃!𝑥𝜓) & ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵)) | ||
Theorem | iota2 4974* | The unique element such that 𝜑. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴)) | ||
Theorem | sniota 4975 | A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)}) | ||
Theorem | csbiotag 4976* | Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)) | ||
Syntax | wfun 4977 | Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.) |
wff Fun 𝐴 | ||
Syntax | wfn 4978 | Extend the definition of a wff to include the function predicate with a domain. (Read: 𝐴 is a function on 𝐵.) |
wff 𝐴 Fn 𝐵 | ||
Syntax | wf 4979 | Extend the definition of a wff to include the function predicate with domain and codomain. (Read: 𝐹 maps 𝐴 into 𝐵.) |
wff 𝐹:𝐴⟶𝐵 | ||
Syntax | wf1 4980 | Extend the definition of a wff to include one-to-one functions. (Read: 𝐹 maps 𝐴 one-to-one into 𝐵.) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27. |
wff 𝐹:𝐴–1-1→𝐵 | ||
Syntax | wfo 4981 | Extend the definition of a wff to include onto functions. (Read: 𝐹 maps 𝐴 onto 𝐵.) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27. |
wff 𝐹:𝐴–onto→𝐵 | ||
Syntax | wf1o 4982 | Extend the definition of a wff to include one-to-one onto functions. (Read: 𝐹 maps 𝐴 one-to-one onto 𝐵.) The notation ("1-1" above the arrow and "onto" below the arrow) is from Definition 6.15(6) of [TakeutiZaring] p. 27. |
wff 𝐹:𝐴–1-1-onto→𝐵 | ||
Syntax | cfv 4983 | Extend the definition of a class to include the value of a function. (Read: The value of 𝐹 at 𝐴, or "𝐹 of 𝐴.") |
class (𝐹‘𝐴) | ||
Syntax | wiso 4984 | Extend the definition of a wff to include the isomorphism property. (Read: 𝐻 is an 𝑅, 𝑆 isomorphism of 𝐴 onto 𝐵.) |
wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) | ||
Definition | df-fun 4985 | Define predicate that determines if some class 𝐴 is a function. Definition 10.1 of [Quine] p. 65. For example, the expression Fun I is true (funi 5013). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3876 with the maps-to notation (see df-mpt 3878). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4986), a function with a given domain and codomain (df-f 4987), a one-to-one function (df-f1 4988), an onto function (df-fo 4989), or a one-to-one onto function (df-f1o 4990). For alternate definitions, see dffun2 4993, dffun4 4994, dffun6 4997, dffun7 5009, dffun8 5010, and dffun9 5011. (Contributed by NM, 1-Aug-1994.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ^{◡}𝐴) ⊆ I )) | ||
Definition | df-fn 4986 | Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵)) | ||
Definition | df-f 4987 | Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | ||
Definition | df-f1 4988 | Define a one-to-one function. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow). (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^{◡}𝐹)) | ||
Definition | df-fo 4989 | Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | ||
Definition | df-f1o 4990 | Define a one-to-one onto function. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow). (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | ||
Definition | df-fv 4991* | Define the value of a function, (𝐹‘𝐴), also known as function application. For example, ( I ‘∅) = ∅. Typically, function 𝐹 is defined using maps-to notation (see df-mpt 3878), but this is not required. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } -> ( F ‘ 3 ) = 9 . We will later define two-argument functions using ordered pairs as (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩). This particular definition is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful. The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar 𝐹(𝐴) notation for a function's value at 𝐴, i.e. "𝐹 of 𝐴," but without context-dependent notational ambiguity. (Contributed by NM, 1-Aug-1994.) Revised to use ℩. (Revised by Scott Fenton, 6-Oct-2017.) |
⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | ||
Definition | df-isom 4992* | Define the isomorphism predicate. We read this as "𝐻 is an 𝑅, 𝑆 isomorphism of 𝐴 onto 𝐵." Normally, 𝑅 and 𝑆 are ordering relations on 𝐴 and 𝐵 respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that 𝑅 and 𝑆 are subscripts. (Contributed by NM, 4-Mar-1997.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) | ||
Theorem | dffun2 4993* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))) | ||
Theorem | dffun4 4994* | Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))) | ||
Theorem | dffun5r 4995* | A way of proving a relation is a function, analogous to mo2r 1997. (Contributed by Jim Kingdon, 27-May-2020.) |
⊢ ((Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)) → Fun 𝐴) | ||
Theorem | dffun6f 4996* | Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 ⇒ ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) | ||
Theorem | dffun6 4997* | Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) |
⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) | ||
Theorem | funmo 4998* | A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) |
⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) | ||
Theorem | dffun4f 4999* | Definition of function like dffun4 4994 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑧𝐴 ⇒ ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))) | ||
Theorem | funrel 5000 | A function is a relation. (Contributed by NM, 1-Aug-1994.) |
⊢ (Fun 𝐴 → Rel 𝐴) |
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