P. 65 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6401-6500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	trsucss 6401	A member of the successor of a transitive class is a subclass of it. Lemma 1.13 of [Schloeder] p. 2. (Contributed by NM, 4-Oct-2003.)
⊢ (Tr 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐵 ⊆ 𝐴))
Theorem	ordsssuc 6402	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 6403	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 6404	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 6405	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 6406	There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. Lemma 1.15 of [Schloeder] p. 2. (Contributed by NM, 21-Jun-1998.) (Proof shortened by JJ, 24-Sep-2021.)
⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
Theorem	onnbtwn 6407	There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41. Lemma 1.15 of [Schloeder] p. 2. (Contributed by NM, 9-Jun-1994.)
⊢ (𝐴 ∈ On → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
Theorem	sucssel 6408	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 6409	Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
Theorem	orduniss 6410	An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴)
Theorem	ordtri2or 6411	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 6412	A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
Theorem	ordtri2or3 6413	A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6412. (Contributed by David Moews, 1-May-2017.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)))
Theorem	ordelinel 6414	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 6415	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 6416	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 6417	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	onunel 6418	The union of two ordinals is in a third iff both of the first two are. (Contributed by Scott Fenton, 10-Sep-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
Theorem	ordunisssuc 6419	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 6420	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 6421	The union of two ordinals is an ordinal. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
Theorem	ontr 6422	An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.) Put in closed form. (Resised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ On → Tr 𝐴)
Theorem	onunisuc 6423	An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6432. (Revised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
Theorem	onordi 6424	An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ Ord 𝐴
Theorem	onirri 6425	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 6426	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 6427	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 6428	An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴)
Theorem	onssnel2i 6429	An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵)
Theorem	onelini 6430	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 6431	An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)
Theorem	onunisuci 6432	An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ ∪ suc 𝐴 = 𝐴
Theorem	onsseli 6433	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 6434	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 6435	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 6436	An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.)
⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Theorem	snsn0non 6437	The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7810). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6438. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ ¬ {{∅}} ∈ On
Theorem	onxpdisj 6438	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 6437. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (On ∩ (V × V)) = ∅
Theorem	onnev 6439	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
2.3.15  Definite description binder (inverted iota)
Syntax	cio 6440	Extend class notation with Russell's definition description binder (inverted iota).
class (℩𝑥𝜑)
Theorem	iotajust 6441*	Soundness justification theorem for df-iota 6442. (Contributed by Andrew Salmon, 29-Jun-2011.)
⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
Definition	df-iota 6442*	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 6460); otherwise, it evaluates to the empty set (see iotanul 6466). 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 7326 (or iotacl 6472 for unbounded iota), as demonstrated in the proof of supub 9368. This can be easier than applying riotasbc 7328 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 6443*	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 6444	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 6445*	Deduction version of nfiotaw 6446. Version of nfiotad 6447 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2370. (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))
Theorem	nfiotaw 6446*	Bound-variable hypothesis builder for the ℩ class. Version of nfiota 6448 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 23-Aug-2011.) Avoid ax-13 2370. (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥(℩𝑦𝜑)
Theorem	nfiotad 6447	Deduction version of nfiota 6448. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker nfiotadw 6445 when possible. (Contributed by NM, 18-Feb-2013.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))
Theorem	nfiota 6448	Bound-variable hypothesis builder for the ℩ class. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker nfiotaw 6446 when possible. (Contributed by NM, 23-Aug-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥(℩𝑦𝜑)
Theorem	cbviotaw 6449*	Change bound variables in a description binder. Version of cbviota 6451 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Andrew Salmon, 1-Aug-2011.) Avoid ax-13 2370. (Revised by GG, 26-Jan-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)
Theorem	cbviotavw 6450*	Change bound variables in a description binder. Version of cbviotav 6452 with a disjoint variable condition, which requires fewer axioms . (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by GG, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)
Theorem	cbviota 6451	Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker cbviotaw 6449 when possible. (Contributed by Andrew Salmon, 1-Aug-2011.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)
Theorem	cbviotav 6452*	Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker cbviotavw 6450 when possible. (Contributed by Andrew Salmon, 1-Aug-2011.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)
Theorem	sb8iota 6453	Variable substitution in description binder. Compare sb8eu 2593. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 18-Mar-2013.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)
Theorem	iotaeq 6454	Equality theorem for descriptions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by Andrew Salmon, 30-Jun-2011.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))
Theorem	iotabi 6455	Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))
Theorem	uniabio 6456*	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	iotaval2 6457*	Version of iotaval 6460 using df-iota 6442 instead of dfiota2 6443. (Contributed by SN, 6-Nov-2024.)
⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
Theorem	iotauni2 6458*	Version of iotauni 6463 using df-iota 6442 instead of dfiota2 6443. (Contributed by SN, 6-Nov-2024.)
⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
Theorem	iotanul2 6459*	Version of iotanul 6466 using df-iota 6442 instead of dfiota2 6443. (Contributed by SN, 6-Nov-2024.)
⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
Theorem	iotaval 6460*	Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) Remove dependency on ax-10 2142, ax-11 2158, ax-12 2178. (Revised by SN, 23-Nov-2024.)
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
Theorem	iotassuni 6461	The ℩ class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.) Remove dependency on ax-10 2142, ax-11 2158, ax-12 2178. (Revised by SN, 6-Nov-2024.)
⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}
Theorem	iotaex 6462	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.) Remove dependency on ax-10 2142, ax-11 2158, ax-12 2178. (Revised by SN, 6-Nov-2024.)
⊢ (℩𝑥𝜑) ∈ V
Theorem	iotauni 6463	Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
Theorem	iotaint 6464	Equivalence between two different forms of ℩. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
Theorem	iota1 6465	Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
Theorem	iotanul 6466	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	iota4 6467	Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
Theorem	iota4an 6468	Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑)
Theorem	iota5 6469*	A method for computing iota. (Contributed by NM, 17-Sep-2013.)
⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)
Theorem	iotabidv 6470*	Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))
Theorem	iotabii 6471	Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑥𝜓)
Theorem	iotacl 6472	Membership law for descriptions. This can be useful for expanding an unbounded iota-based definition (see df-iota 6442). If you have a bounded iota-based definition, riotacl2 7326 may be useful. (Contributed by Andrew Salmon, 1-Aug-2011.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})
Theorem	iota2df 6473	A condition that allows 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 6474*	A condition that allows 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 6475*	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 6476*	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 6477	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 6478	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 6479*	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)
2.3.16  Functions
Syntax	wfun 6480	Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.)
wff Fun 𝐴
Syntax	wfn 6481	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 6482	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 6483	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 6484	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 6485	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 6486	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 6487	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 6488	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 15995). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5176 with the maps-to notation (see df-mpt 5177 and df-mpo 7358). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6489), a function with a given domain and codomain (df-f 6490), a one-to-one function (df-f1 6491), an onto function (df-fo 6492), or a one-to-one onto function (df-f1o 6493). For alternate definitions, see dffun2 6496, dffun3 6498, dffun4 6499, dffun5 6500, dffun6 6497, dffun7 6513, dffun8 6514, and dffun9 6515. (Contributed by NM, 1-Aug-1994.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))
Definition	df-fn 6489	Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 6658, dffn3 6668, dffn4 6746, and dffn5 6885. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
Definition	df-f 6490	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 7037, dff3 7038, and dff4 7039. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
Definition	df-f1 6491	Define a one-to-one function. For equivalent definitions see dff12 6723 and dff13 7195. 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 18012. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
Definition	df-fo 6492	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 6744, dffo3 7040, dffo4 7041, and dffo5 7042. 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 18013. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
Definition	df-f1o 6493	Define a one-to-one onto function. For equivalent definitions see dff1o2 6773, dff1o3 6774, dff1o4 6776, and dff1o5 6777. 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 18016. Therefore, two sets are called "isomorphic" if there is a bijection between them. According to isof1oidb 7265, 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 8889. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
Definition	df-fv 6494*	Define the value of a function, (𝐹‘𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 16077 after we define cosine in df-cos 15995). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5177 and df-mpo 7358), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 30405). Note that df-ov 7356 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 6859 and fvprc 6818). 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 6922, dffv3 6822, fv2 6821, and fv3 6844 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6914 and funfv2 6915. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6881. (Contributed by NM, 1-Aug-1994.) Revised to use ℩. Original version is now Theorem dffv4 6823. (Revised by Scott Fenton, 6-Oct-2017.)
⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
Definition	df-isom 6495*	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 6496*	Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2142, ax-12 2178. (Revised by SN, 19-Dec-2024.) Avoid ax-11 2158. (Revised by BTernaryTau, 29-Dec-2024.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Theorem	dffun6 6497*	Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) Avoid ax-10 2142, ax-12 2178. (Revised by SN, 19-Dec-2024.)
⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
Theorem	dffun3 6498*	Alternate definition of function. (Contributed by NM, 29-Dec-1996.) (Proof shortened by SN, 19-Dec-2024.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))
Theorem	dffun4 6499*	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 6500*	Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)))