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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sucid 6301 | A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝐴 ∈ suc 𝐴 | ||
Theorem | nsuceq0 6302 | No successor is empty. (Contributed by NM, 3-Apr-1995.) |
⊢ suc 𝐴 ≠ ∅ | ||
Theorem | eqelsuc 6303 | A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵) | ||
Theorem | iunsuc 6304* | Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) | ||
Theorem | suctr 6305 | The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) (Proof shortened by JJ, 24-Sep-2021.) |
⊢ (Tr 𝐴 → Tr suc 𝐴) | ||
Theorem | trsuc 6306 | A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) | ||
Theorem | trsucss 6307 | A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.) |
⊢ (Tr 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐵 ⊆ 𝐴)) | ||
Theorem | ordsssuc 6308 | An ordinal is a subset of another ordinal if and only if it belongs to its successor. (Contributed by NM, 28-Nov-2003.) |
⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) | ||
Theorem | onsssuc 6309 | A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) | ||
Theorem | ordsssuc2 6310 | An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) | ||
Theorem | onmindif 6311 | When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.) |
⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → 𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵)) | ||
Theorem | ordnbtwn 6312 | There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 21-Jun-1998.) (Proof shortened by JJ, 24-Sep-2021.) |
⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)) | ||
Theorem | onnbtwn 6313 | There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 9-Jun-1994.) |
⊢ (𝐴 ∈ On → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)) | ||
Theorem | sucssel 6314 | A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.) |
⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵)) | ||
Theorem | orddif 6315 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) | ||
Theorem | orduniss 6316 | An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.) |
⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴) | ||
Theorem | ordtri2or 6317 | A trichotomy law for ordinal classes. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | ||
Theorem | ordtri2or2 6318 | A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | ||
Theorem | ordtri2or3 6319 | A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6318. (Contributed by David Moews, 1-May-2017.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵))) | ||
Theorem | ordelinel 6320 | The intersection of two ordinal classes is an element of a third if and only if either one of them is. (Contributed by David Moews, 1-May-2017.) (Proof shortened by JJ, 24-Sep-2021.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∩ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶))) | ||
Theorem | ordssun 6321 | Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.) |
⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶))) | ||
Theorem | ordequn 6322 | The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.) |
⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐵 ∪ 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | ||
Theorem | ordun 6323 | The maximum (i.e. union) of two ordinals is ordinal. Exercise 12 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∪ 𝐵)) | ||
Theorem | ordunisssuc 6324 | A subclass relationship for union and successor of ordinal classes. (Contributed by NM, 28-Nov-2003.) |
⊢ ((𝐴 ⊆ On ∧ Ord 𝐵) → (∪ 𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ suc 𝐵)) | ||
Theorem | suc11 6325 | The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | onun2 6326 | The union of two ordinals is an ordinal. (Contributed by Scott Fenton, 9-Aug-2024.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On) | ||
Theorem | onordi 6327 | An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ Ord 𝐴 | ||
Theorem | ontrci 6328 | An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ Tr 𝐴 | ||
Theorem | onirri 6329 | An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ ¬ 𝐴 ∈ 𝐴 | ||
Theorem | oneli 6330 | A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On) | ||
Theorem | onelssi 6331 | A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴) | ||
Theorem | onssneli 6332 | An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴) | ||
Theorem | onssnel2i 6333 | An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵) | ||
Theorem | onelini 6334 | 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 6335 | An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴) | ||
Theorem | onunisuci 6336 | An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ ∪ suc 𝐴 = 𝐴 | ||
Theorem | onsseli 6337 | 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 6338 | 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 6339 | 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 6340 | An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.) |
⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | ||
Theorem | snsn0non 6341 | The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7657). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6342. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ ¬ {{∅}} ∈ On | ||
Theorem | onxpdisj 6342 | 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 6341. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (On ∩ (V × V)) = ∅ | ||
Theorem | onnev 6343 | 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.) (Proof shortened by Wolf Lammen, 27-May-2024.) |
⊢ On ≠ V | ||
Theorem | onnevOLD 6344 | Obsolete version of onnev 6343 as of 27-May-2024. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ On ≠ V | ||
Syntax | cio 6345 | Extend class notation with Russell's definition description binder (inverted iota). |
class (℩𝑥𝜑) | ||
Theorem | iotajust 6346* | Soundness justification theorem for df-iota 6347. (Contributed by Andrew Salmon, 29-Jun-2011.) |
⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | ||
Definition | df-iota 6347* |
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 6363);
otherwise, it evaluates to the empty set (see iotanul 6367). 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 7196 (or iotacl 6375 for unbounded iota), as demonstrated in the proof of supub 9088. This can be easier than applying riotasbc 7198 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 6348* | 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 6349 | 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 6350* | Deduction version of nfiotaw 6351. Version of nfiotad 6352 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 18-Feb-2013.) (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) | ||
Theorem | nfiotaw 6351* | Bound-variable hypothesis builder for the ℩ class. Version of nfiota 6353 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 23-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥(℩𝑦𝜑) | ||
Theorem | nfiotad 6352 | Deduction version of nfiota 6353. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfiotadw 6350 when possible. (Contributed by NM, 18-Feb-2013.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) | ||
Theorem | nfiota 6353 | Bound-variable hypothesis builder for the ℩ class. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfiotaw 6351 when possible. (Contributed by NM, 23-Aug-2011.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥(℩𝑦𝜑) | ||
Theorem | cbviotaw 6354* | Change bound variables in a description binder. Version of cbviota 6357 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | cbviotavw 6355* | Change bound variables in a description binder. Version of cbviotav 6358 with a disjoint variable condition, which requires fewer axioms . (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 30-Sep-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | cbviotavwOLD 6356* | Obsolete version of cbviotavw 6355 as of 30-Sep-2024. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | cbviota 6357 | Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbviotaw 6354 when possible. (Contributed by Andrew Salmon, 1-Aug-2011.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | cbviotav 6358* | Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbviotavw 6355 when possible. (Contributed by Andrew Salmon, 1-Aug-2011.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | ||
Theorem | sb8iota 6359 | Variable substitution in description binder. Compare sb8eu 2600. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 18-Mar-2013.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | iotaeq 6360 | Equality theorem for descriptions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by Andrew Salmon, 30-Jun-2011.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑)) | ||
Theorem | iotabi 6361 | Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓)) | ||
Theorem | uniabio 6362* | 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 6363* | 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 6364 | Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) | ||
Theorem | iotaint 6365 | Equivalence between two different forms of ℩. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) | ||
Theorem | iota1 6366 | Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | ||
Theorem | iotanul 6367 | 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 6368 | 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 6369 | 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 6370 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑) | ||
Theorem | iota4an 6371 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑) | ||
Theorem | iota5 6372* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴) | ||
Theorem | iotabidv 6373* | Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒)) | ||
Theorem | iotabii 6374 | Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑥𝜓) | ||
Theorem | iotacl 6375 |
Membership law for descriptions.
This can be useful for expanding an unbounded iota-based definition (see df-iota 6347). If you have a bounded iota-based definition, riotacl2 7196 may be useful. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | iota2df 6376 | 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 6377* | 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 6378* | 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 6379* | 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 6380 | 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 6381 | 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 6382* | 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 6383 | Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.) |
wff Fun 𝐴 | ||
Syntax | wfn 6384 | 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 6385 | 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 6386 | 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 6387 | 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 6388 | 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 6389 | 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 6390 | 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 6391 | 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 15645). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5144 with the maps-to notation (see df-mpt 5145 and df-mpo 7227). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6392), a function with a given domain and codomain (df-f 6393), a one-to-one function (df-f1 6394), an onto function (df-fo 6395), or a one-to-one onto function (df-f1o 6396). For alternate definitions, see dffun2 6399, dffun3 6400, dffun4 6401, dffun5 6402, dffun6 6404, dffun7 6416, dffun8 6417, and dffun9 6418. (Contributed by NM, 1-Aug-1994.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) | ||
Definition | df-fn 6392 | Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 6556, dffn3 6567, dffn4 6648, and dffn5 6780. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵)) | ||
Definition | df-f 6393 | 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 6927, dff3 6928, and dff4 6929. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | ||
Definition | df-f1 6394 |
Define a one-to-one function. For equivalent definitions see dff12 6623
and dff13 7076. 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 17606. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | ||
Definition | df-fo 6395 |
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 6646, dffo3 6930, dffo4 6931, and dffo5 6932.
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 17607. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | ||
Definition | df-f1o 6396 |
Define a one-to-one onto function. For equivalent definitions see
dff1o2 6675, dff1o3 6676, dff1o4 6678, and dff1o5 6679. 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 17610. Therefore, two sets are called "isomorphic" if there is a bijection between them. According to isof1oidb 7142, 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 8645. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | ||
Definition | df-fv 6397* | Define the value of a function, (𝐹‘𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 15724 after we define cosine in df-cos 15645). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5145 and df-mpo 7227), but this is not required. For example, 𝐹 = {〈2, 6〉, 〈3, 9〉} → (𝐹‘3) = 9 (ex-fv 28539). Note that df-ov 7225 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 6756 and fvprc 6718). 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 6815, dffv3 6722, fv2 6721, and fv3 6744 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6807 and funfv2 6808. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6777. (Contributed by NM, 1-Aug-1994.) Revised to use ℩. Original version is now Theorem dffv4 6723. (Revised by Scott Fenton, 6-Oct-2017.) |
⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | ||
Definition | df-isom 6398* | 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 6399* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))) | ||
Theorem | dffun3 6400* | Alternate definition of function. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) |
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