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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | onssnel2i 6301 | An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵) | ||
Theorem | onelini 6302 | An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐵 ∈ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴)) | ||
Theorem | oneluni 6303 | An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴) | ||
Theorem | onunisuci 6304 | An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ ∪ suc 𝐴 = 𝐴 | ||
Theorem | onsseli 6305 | Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995.) |
⊢ 𝐴 ∈ On & ⊢ 𝐵 ∈ On ⇒ ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | onun2i 6306 | The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) |
⊢ 𝐴 ∈ On & ⊢ 𝐵 ∈ On ⇒ ⊢ (𝐴 ∪ 𝐵) ∈ On | ||
Theorem | unizlim 6307 | An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.) |
⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴))) | ||
Theorem | on0eqel 6308 | An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.) |
⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | ||
Theorem | snsn0non 6309 | The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7584). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6310. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ ¬ {{∅}} ∈ On | ||
Theorem | onxpdisj 6310 | Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 6309. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (On ∩ (V × V)) = ∅ | ||
Theorem | onnev 6311 | The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.) |
⊢ On ≠ V | ||
Syntax | cio 6312 | Extend class notation with Russell's definition description binder (inverted iota). |
class (℩𝑥𝜑) | ||
Theorem | iotajust 6313* | Soundness justification theorem for df-iota 6314. (Contributed by Andrew Salmon, 29-Jun-2011.) |
⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | ||
Definition | df-iota 6314* |
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 6329);
otherwise, it evaluates to the empty set (see iotanul 6333). 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 riotacl2 7130 (or iotacl 6341 for unbounded iota), as demonstrated in the proof of supub 8923. This can be easier than applying riotasbc 7132 or 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 6315* | 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 6316 | 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 6317* | Deduction version of nfiotaw 6318. Version of nfiotad 6319 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 18-Feb-2013.) (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) | ||
Theorem | nfiotaw 6318* | Bound-variable hypothesis builder for the ℩ class. Version of nfiota 6320 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 23-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥(℩𝑦𝜑) | ||
Theorem | nfiotad 6319 | Deduction version of nfiota 6320. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker nfiotadw 6317 when possible. (Contributed by NM, 18-Feb-2013.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) | ||
Theorem | nfiota 6320 | Bound-variable hypothesis builder for the ℩ class. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker nfiotaw 6318 when possible. (Contributed by NM, 23-Aug-2011.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥(℩𝑦𝜑) | ||
Theorem | cbviotaw 6321* | Change bound variables in a description binder. Version of cbviota 6323 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | cbviotavw 6322* | Change bound variables in a description binder. Version of cbviotav 6324 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | cbviota 6323 | Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker cbviotaw 6321 when possible. (Contributed by Andrew Salmon, 1-Aug-2011.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | cbviotav 6324* | Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker cbviotavw 6322 when possible. (Contributed by Andrew Salmon, 1-Aug-2011.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | sb8iota 6325 | Variable substitution in description binder. Compare sb8eu 2686. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by NM, 18-Mar-2013.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | iotaeq 6326 | Equality theorem for descriptions. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by Andrew Salmon, 30-Jun-2011.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑)) | ||
Theorem | iotabi 6327 | Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓)) | ||
Theorem | uniabio 6328* | 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 6329* | 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 6330 | Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) | ||
Theorem | iotaint 6331 | Equivalence between two different forms of ℩. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) | ||
Theorem | iota1 6332 | Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | ||
Theorem | iotanul 6333 | 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 | iotassuni 6334 | The ℩ class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑} | ||
Theorem | iotaex 6335 | Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the ℩ class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (℩𝑥𝜑) ∈ V | ||
Theorem | iota4 6336 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑) | ||
Theorem | iota4an 6337 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑) | ||
Theorem | iota5 6338* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴) | ||
Theorem | iotabidv 6339* | Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒)) | ||
Theorem | iotabii 6340 | Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑥𝜓) | ||
Theorem | iotacl 6341 |
Membership law for descriptions.
This can be useful for expanding an unbounded iota-based definition (see df-iota 6314). If you have a bounded iota-based definition, riotacl2 7130 may be useful. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | iota2df 6342 | 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 6343* | 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 6344* | 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 | iotan0 6345* | Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is not the empty set (that means that "the unique element such that 𝜑 " exists). (Contributed by AV, 30-Jan-2024.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓) | ||
Theorem | sniota 6346 | 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 | dfiota4 6347 | The ℩ operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017.) (Proof shortened by JJ, 28-Oct-2021.) |
⊢ (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) | ||
Theorem | csbiota 6348* | Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) (Revised by NM, 23-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑) | ||
Syntax | wfun 6349 | Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.) |
wff Fun 𝐴 | ||
Syntax | wfn 6350 | 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 6351 | 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 6352 | 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 6353 | 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 6354 | 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 6355 | 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 6356 | 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 6357 | Define predicate that determines if some class 𝐴 is a function. Definition 10.1 of [Quine] p. 65. For example, the expression Fun cos is true once we define cosine (df-cos 15424). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5146 with the maps-to notation (see df-mpt 5147 and df-mpo 7161). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6358), a function with a given domain and codomain (df-f 6359), a one-to-one function (df-f1 6360), an onto function (df-fo 6361), or a one-to-one onto function (df-f1o 6362). For alternate definitions, see dffun2 6365, dffun3 6366, dffun4 6367, dffun5 6368, dffun6 6370, dffun7 6382, dffun8 6383, and dffun9 6384. (Contributed by NM, 1-Aug-1994.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) | ||
Definition | df-fn 6358 | Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 6516, dffn3 6525, dffn4 6596, and dffn5 6724. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵)) | ||
Definition | df-f 6359 | Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. 𝐹:𝐴⟶𝐵 can be read as "𝐹 is a function from 𝐴 to 𝐵". For alternate definitions, see dff2 6865, dff3 6866, and dff4 6867. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | ||
Definition | df-f1 6360 |
Define a one-to-one function. For equivalent definitions see dff12 6574
and dff13 7013. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We
use their notation ("1-1" above the arrow).
A one-to-one function is also called an "injection" or an "injective function", 𝐹:𝐴–1-1→𝐵 can be read as "𝐹 is an injection from 𝐴 into 𝐵". Injections are precisely the monomorphisms in the category SetCat of sets and set functions, see setcmon 17347. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | ||
Definition | df-fo 6361 |
Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27.
We use their notation ("onto" under the arrow). For alternate
definitions, see dffo2 6594, dffo3 6868, dffo4 6869, and dffo5 6870.
An onto function is also called a "surjection" or a "surjective function", 𝐹:𝐴–onto→𝐵 can be read as "𝐹 is a surjection from 𝐴 onto 𝐵". Surjections are precisely the epimorphisms in the category SetCat of sets and set functions, see setcepi 17348. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | ||
Definition | df-f1o 6362 |
Define a one-to-one onto function. For equivalent definitions see
dff1o2 6620, dff1o3 6621, dff1o4 6623, and dff1o5 6624. Compare Definition
6.15(6) of [TakeutiZaring] p. 27.
We use their notation ("1-1" above
the arrow and "onto" below the arrow).
A one-to-one onto function is also called a "bijection" or a "bijective function", 𝐹:𝐴–1-1-onto→𝐵 can be read as "𝐹 is a bijection between 𝐴 and 𝐵". Bijections are precisely the isomorphisms in the category SetCat of sets and set functions, see setciso 17351. Therefore, two sets are called "isomorphic" if there is a bijection between them. According to isof1oidb 7077, two sets are isomorphic iff there is an isomorphism Isom regarding the identity relation. In this case, the two sets are also "equinumerous", see bren 8518. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | ||
Definition | df-fv 6363* | Define the value of a function, (𝐹‘𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 15503 after we define cosine in df-cos 15424). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5147 and df-mpo 7161), but this is not required. For example, 𝐹 = {〈2, 6〉, 〈3, 9〉} → (𝐹‘3) = 9 (ex-fv 28222). Note that df-ov 7159 will 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 (as shown by ndmfv 6700 and fvprc 6663). 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. Alternate definitions are dffv2 6756, dffv3 6666, fv2 6665, and fv3 6688 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6750 and funfv2 6751. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6721. (Contributed by NM, 1-Aug-1994.) Revised to use ℩. Original version is now theorem dffv4 6667. (Revised by Scott Fenton, 6-Oct-2017.) |
⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | ||
Definition | df-isom 6364* | 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 6365* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))) | ||
Theorem | dffun3 6366* | Alternate definition of function. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) | ||
Theorem | dffun4 6367* | 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 | dffun5 6368* | Alternate definition of function. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧))) | ||
Theorem | dffun6f 6369* | 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 6370* | 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 6371* | A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) |
⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) | ||
Theorem | funrel 6372 | A function is a relation. (Contributed by NM, 1-Aug-1994.) |
⊢ (Fun 𝐴 → Rel 𝐴) | ||
Theorem | 0nelfun 6373 | A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.) |
⊢ (Fun 𝑅 → ∅ ∉ 𝑅) | ||
Theorem | funss 6374 | 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 6375 | Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵)) | ||
Theorem | funeqi 6376 | Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (Fun 𝐴 ↔ Fun 𝐵) | ||
Theorem | funeqd 6377 | Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (Fun 𝐴 ↔ Fun 𝐵)) | ||
Theorem | nffun 6378 | Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ Ⅎ𝑥Fun 𝐹 | ||
Theorem | sbcfung 6379 | 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 6380* | 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 6381* | 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 6382* | 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 6383 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 6383* | Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 6382. (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 6384* | 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 6385 | A class is a function if and only if it is a function on its domain. (Contributed by NM, 13-Aug-2004.) |
⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴) | ||
Theorem | funfnd 6386 | A function is a function on its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → Fun 𝐴) ⇒ ⊢ (𝜑 → 𝐴 Fn dom 𝐴) | ||
Theorem | funi 6387 | The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6475. (Contributed by NM, 30-Apr-1998.) |
⊢ Fun I | ||
Theorem | nfunv 6388 | The universal class is not a function. (Contributed by Raph Levien, 27-Jan-2004.) |
⊢ ¬ Fun V | ||
Theorem | funopg 6389 | A Kuratowski ordered pair of sets is a function only if its components are equal. Compare with funsng 6405. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ Fun 〈𝐴, 𝐵〉) → 𝐴 = 𝐵) | ||
Theorem | funopab 6390* | 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 6391* | A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.) |
⊢ Fun {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐴} | ||
Theorem | funopab4 6392* | A class of ordered pairs of values in the form used by df-mpt 5147 is a function. (Contributed by NM, 17-Feb-2013.) |
⊢ Fun {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐴)} | ||
Theorem | funmpt 6393 | A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.) |
⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | funmpt2 6394 | 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 6395 | 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 | funresfunco 6396 | Composition of two functions, generalization of funco 6395. (Contributed by Alexander van der Vekens, 25-Jul-2017.) |
⊢ ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | ||
Theorem | funres 6397 | 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 6398 | 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 6399 | 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 6400 | 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)) |
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