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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | relcnvfld 4901 | 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 4902 | Alternate definition of domain df-dm 4401 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴) | ||
Theorem | unixpm 4903* | 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 4904 | The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.) |
⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅) | ||
Theorem | cnvexg 4905 | 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 4906 | 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 4907 | 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 4908 | Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) | ||
Theorem | cnviinm 4909* | 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 4910* | 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 4911* | 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 4912 | The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V) | ||
Theorem | coex 4913 | The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∘ 𝐵) ∈ V | ||
Theorem | xpcom 4914* | Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.) |
⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶)) | ||
Syntax | cio 4915 | Extend class notation with Russell's definition description binder (inverted iota). |
class (℩𝑥𝜑) | ||
Theorem | iotajust 4916* | Soundness justification theorem for df-iota 4917. (Contributed by Andrew Salmon, 29-Jun-2011.) |
⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | ||
Definition | df-iota 4917* |
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 4928);
otherwise, it evaluates to the empty set (see iotanul 4932). 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 4940 (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 4918* | 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 4919 | 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 4920* | Deduction version of nfiotaxy 4921. (Contributed by Jim Kingdon, 21-Dec-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) | ||
Theorem | nfiotaxy 4921* | Bound-variable hypothesis builder for the ℩ class. (Contributed by NM, 23-Aug-2011.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥(℩𝑦𝜑) | ||
Theorem | cbviota 4922 | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | cbviotav 4923* | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | sb8iota 4924 | Variable substitution in description binder. Compare sb8eu 1956. (Contributed by NM, 18-Mar-2013.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | iotaeq 4925 | Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑)) | ||
Theorem | iotabi 4926 | Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓)) | ||
Theorem | uniabio 4927* | 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 4928* | 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 4929 | Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) | ||
Theorem | iotaint 4930 | Equivalence between two different forms of ℩. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) | ||
Theorem | iota1 4931 | Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | ||
Theorem | iotanul 4932 | 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 4933 | 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 4934* | 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 4935 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑) | ||
Theorem | iota4an 4936 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑) | ||
Theorem | iota5 4937* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴) | ||
Theorem | iotabidv 4938* | Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒)) | ||
Theorem | iotabii 4939 | Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑥𝜓) | ||
Theorem | iotacl 4940 |
Membership law for descriptions.
This can useful for expanding an unbounded iota-based definition (see df-iota 4917). (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | iota2df 4941 | 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 4942* | 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 4943* | 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 4944 | 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 4945* | Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)) | ||
Syntax | wfun 4946 | Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.) |
wff Fun 𝐴 | ||
Syntax | wfn 4947 | 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 4948 | 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 4949 | 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 4950 | 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 4951 | 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 4952 | 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 4953 | 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 4954 | 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 4982). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3859 with the maps-to notation (see df-mpt 3861). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4955), a function with a given domain and codomain (df-f 4956), a one-to-one function (df-f1 4957), an onto function (df-fo 4958), or a one-to-one onto function (df-f1o 4959). For alternate definitions, see dffun2 4962, dffun4 4963, dffun6 4966, dffun7 4978, dffun8 4979, and dffun9 4980. (Contributed by NM, 1-Aug-1994.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) | ||
Definition | df-fn 4955 | 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 4956 | 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 4957 | 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 4958 | 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 4959 | 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 4960* | 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 3861), 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 4961* | 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 4962* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))) | ||
Theorem | dffun4 4963* | 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 4964* | A way of proving a relation is a function, analogous to mo2r 1995. (Contributed by Jim Kingdon, 27-May-2020.) |
⊢ ((Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧)) → Fun 𝐴) | ||
Theorem | dffun6f 4965* | 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 4966* | 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 4967* | A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) |
⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) | ||
Theorem | dffun4f 4968* | Definition of function like dffun4 4963 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 4969 | A function is a relation. (Contributed by NM, 1-Aug-1994.) |
⊢ (Fun 𝐴 → Rel 𝐴) | ||
Theorem | funss 4970 | Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) |
⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴)) | ||
Theorem | funeq 4971 | Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵)) | ||
Theorem | funeqi 4972 | Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (Fun 𝐴 ↔ Fun 𝐵) | ||
Theorem | funeqd 4973 | Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (Fun 𝐴 ↔ Fun 𝐵)) | ||
Theorem | nffun 4974 | Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ Ⅎ𝑥Fun 𝐹 | ||
Theorem | sbcfung 4975 | Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹)) | ||
Theorem | funeu 4976* | There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦) | ||
Theorem | funeu2 4977* | There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.) |
⊢ ((Fun 𝐹 ∧ 〈𝐴, 𝐵〉 ∈ 𝐹) → ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) | ||
Theorem | dffun7 4978* | Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one." However, dffun8 4979 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦)) | ||
Theorem | dffun8 4979* | Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 4978. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦)) | ||
Theorem | dffun9 4980* | Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)) | ||
Theorem | funfn 4981 | An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.) |
⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴) | ||
Theorem | funi 4982 | The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.) |
⊢ Fun I | ||
Theorem | nfunv 4983 | The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.) |
⊢ ¬ Fun V | ||
Theorem | funopg 4984 | A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ Fun 〈𝐴, 𝐵〉) → 𝐴 = 𝐵) | ||
Theorem | funopab 4985* | A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.) |
⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑) | ||
Theorem | funopabeq 4986* | A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.) |
⊢ Fun {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐴} | ||
Theorem | funopab4 4987* | A class of ordered pairs of values in the form used by df-mpt 3861 is a function. (Contributed by NM, 17-Feb-2013.) |
⊢ Fun {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐴)} | ||
Theorem | funmpt 4988 | A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.) |
⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | funmpt2 4989 | Functionality of a class given by a "maps to" notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ Fun 𝐹 | ||
Theorem | funco 4990 | The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | ||
Theorem | funres 4991 | A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.) |
⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴)) | ||
Theorem | funssres 4992 | The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.) |
⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺) | ||
Theorem | fun2ssres 4993 | Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.) |
⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) | ||
Theorem | funun 4994 | The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.) |
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺)) | ||
Theorem | funcnvsn 4995 | The converse singleton of an ordered pair is a function. This is equivalent to funsn 4998 via cnvsn 4853, but stating it this way allows us to skip the sethood assumptions on 𝐴 and 𝐵. (Contributed by NM, 30-Apr-2015.) |
⊢ Fun ◡{〈𝐴, 𝐵〉} | ||
Theorem | funsng 4996 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {〈𝐴, 𝐵〉}) | ||
Theorem | fnsng 4997 | Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉} Fn {𝐴}) | ||
Theorem | funsn 4998 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ Fun {〈𝐴, 𝐵〉} | ||
Theorem | funinsn 4999 | A function based on the singleton of an ordered pair. Unlike funsng 4996, this holds even if 𝐴 or 𝐵 is a proper class. (Contributed by Jim Kingdon, 17-Apr-2022.) |
⊢ Fun ({〈𝐴, 𝐵〉} ∩ (𝑉 × 𝑊)) | ||
Theorem | funprg 5000 | A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
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