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Theorem List for Metamath Proof Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsucidg 5801 Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
(𝐴𝑉𝐴 ∈ suc 𝐴)

Theoremsucid 5802 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

Theoremnsuceq0 5803 No successor is empty. (Contributed by NM, 3-Apr-1995.)
suc 𝐴 ≠ ∅

Theoremeqelsuc 5804 A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
𝐴 ∈ V       (𝐴 = 𝐵𝐴 ∈ suc 𝐵)

Theoremiunsuc 5805* 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 = 𝐴𝐵 = 𝐶)        𝑥 ∈ suc 𝐴𝐵 = ( 𝑥𝐴 𝐵𝐶)

Theoremsuctr 5806 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)

TheoremsuctrOLD 5807 Obsolete proof of suctr 5806 as of 24-Sep-2021. (Contributed by Alan Sare, 11-Apr-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)

Theoremtrsuc 5808 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 𝐵𝐴) → 𝐵𝐴)

Theoremtrsucss 5809 A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
(Tr 𝐴 → (𝐵 ∈ suc 𝐴𝐵𝐴))

Theoremordsssuc 5810 A subset of an ordinal belongs to its successor. (Contributed by NM, 28-Nov-2003.)
((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵𝐴 ∈ suc 𝐵))

Theoremonsssuc 5811 A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))

Theoremordsssuc2 5812 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 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))

Theoremonmindif 5813 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))

Theoremordnbtwn 5814 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 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))

TheoremordnbtwnOLD 5815 Obsolete proof of ordnbtwn 5814 as of 24-Sep-2021. (Contributed by NM, 21-Jun-1998.) (New usage is discouraged.) (Proof modification is discouraged.)
(Ord 𝐴 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))

Theoremonnbtwn 5816 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 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))

Theoremsucssel 5817 A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)
(𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))

Theoremorddif 5818 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
(Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))

Theoremorduniss 5819 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
(Ord 𝐴 𝐴𝐴)

Theoremordtri2or 5820 A trichotomy law for ordinal classes. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))

Theoremordtri2or2 5821 A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))

Theoremordtri2or3 5822 A consequence of total ordering for ordinal classes. Similar to ordtri2or2 5821. (Contributed by David Moews, 1-May-2017.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))

Theoremordelinel 5823 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) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

TheoremordelinelOLD 5824 Obsolete proof of ordelinel 5823 as of 24-Sep-2021. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

Theoremordssun 5825 Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.)
((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶)))

Theoremordequn 5826 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 = (𝐵𝐶) → (𝐴 = 𝐵𝐴 = 𝐶)))

Theoremordun 5827 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 (𝐴𝐵))

Theoremordunisssuc 5828 A subclass relationship for union and successor of ordinal classes. (Contributed by NM, 28-Nov-2003.)
((𝐴 ⊆ On ∧ Ord 𝐵) → ( 𝐴𝐵𝐴 ⊆ suc 𝐵))

Theoremsuc11 5829 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 𝐵𝐴 = 𝐵))

Theoremonordi 5830 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       Ord 𝐴

Theoremontrci 5831 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       Tr 𝐴

Theoremonirri 5832 An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On        ¬ 𝐴𝐴

Theoremoneli 5833 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       (𝐵𝐴𝐵 ∈ On)

Theoremonelssi 5834 A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
𝐴 ∈ On       (𝐵𝐴𝐵𝐴)

Theoremonssneli 5835 An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
𝐴 ∈ On       (𝐴𝐵 → ¬ 𝐵𝐴)

Theoremonssnel2i 5836 An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴 → ¬ 𝐴𝐵)

Theoremonelini 5837 An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴𝐵 = (𝐵𝐴))

Theoremoneluni 5838 An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴 → (𝐴𝐵) = 𝐴)

Theoremonunisuci 5839 An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
𝐴 ∈ On        suc 𝐴 = 𝐴

Theoremonsseli 5840 Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995.)
𝐴 ∈ On    &   𝐵 ∈ On       (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Theoremonun2i 5841 The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.)
𝐴 ∈ On    &   𝐵 ∈ On       (𝐴𝐵) ∈ On

Theoremunizlim 5842 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)))

Theoremon0eqel 5843 An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.)
(𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

Theoremsnsn0non 5844 The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7066). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 5845. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
¬ {{∅}} ∈ On

Theoremonxpdisj 5845 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 5844. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(On ∩ (V × V)) = ∅

Theoremonnev 5846 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

2.3.14  Definite description binder (inverted iota)

Syntaxcio 5847 Extend class notation with Russell's definition description binder (inverted iota).
class (℩𝑥𝜑)

Theoremiotajust 5848* Soundness justification theorem for df-iota 5849. (Contributed by Andrew Salmon, 29-Jun-2011.)
{𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}

Definitiondf-iota 5849* 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 5860); otherwise, it evaluates to the empty set (see iotanul 5864). 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 6621 (or iotacl 5872 for unbounded iota), as demonstrated in the proof of supub 8362. This can be easier than applying riotasbc 6623 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 ∣ {𝑥𝜑} = {𝑦}}

Theoremdfiota2 5850* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
(℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}

Theoremnfiota1 5851 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥(℩𝑥𝜑)

Theoremnfiotad 5852 Deduction version of nfiota 5853. (Contributed by NM, 18-Feb-2013.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥(℩𝑦𝜓))

Theoremnfiota 5853 Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.)
𝑥𝜑       𝑥(℩𝑦𝜑)

Theoremcbviota 5854 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝑦𝜑    &   𝑥𝜓       (℩𝑥𝜑) = (℩𝑦𝜓)

Theoremcbviotav 5855* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))       (℩𝑥𝜑) = (℩𝑦𝜓)

Theoremsb8iota 5856 Variable substitution in description binder. Compare sb8eu 2502. (Contributed by NM, 18-Mar-2013.)
𝑦𝜑       (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)

Theoremiotaeq 5857 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
(∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))

Theoremiotabi 5858 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
(∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))

Theoremuniabio 5859* 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(𝜑𝑥 = 𝑦) → {𝑥𝜑} = 𝑦)

Theoremiotaval 5860* Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
(∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)

Theoremiotauni 5861 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})

Theoremiotaint 5862 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})

Theoremiota1 5863 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))

Theoremiotanul 5864 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) = ∅)

Theoremiotassuni 5865 The class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.)
(℩𝑥𝜑) ⊆ {𝑥𝜑}

Theoremiotaex 5866 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

Theoremiota4 5867 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)

Theoremiota4an 5868 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑)

Theoremiota5 5869* A method for computing iota. (Contributed by NM, 17-Sep-2013.)
((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))       ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)

Theoremiotabidv 5870* Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))

Theoremiotabii 5871 Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝜑𝜓)       (℩𝑥𝜑) = (℩𝑥𝜓)

Theoremiotacl 5872 Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 5849). If you have a bounded iota-based definition, riotacl2 6621 may be useful.

(Contributed by Andrew Salmon, 1-Aug-2011.)

(∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})

Theoremiota2df 5873 A condition that allows us to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.)
(𝜑𝐵𝑉)    &   (𝜑 → ∃!𝑥𝜓)    &   ((𝜑𝑥 = 𝐵) → (𝜓𝜒))    &   𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑𝑥𝐵)       (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Theoremiota2d 5874* A condition that allows us to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.)
(𝜑𝐵𝑉)    &   (𝜑 → ∃!𝑥𝜓)    &   ((𝜑𝑥 = 𝐵) → (𝜓𝜒))       (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Theoremiota2 5875* The unique element such that 𝜑. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))

Theoremsniota 5876 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})

Theoremdfiota4 5877 The operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017.) (Proof shortened by JJ, 28-Oct-2021.)
(℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)

Theoremdfiota4OLD 5878 Obsolete proof of dfiota4 5877 as of 28-Oct-2021. (Contributed by Scott Fenton, 6-Oct-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)

Theoremcsbiota 5879* 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)

2.3.15  Functions

Syntaxwfun 5880 Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.)
wff Fun 𝐴

Syntaxwfn 5881 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

Syntaxwf 5882 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: 𝐹 maps 𝐴 into 𝐵.)
wff 𝐹:𝐴𝐵

Syntaxwf1 5883 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:𝐴1-1𝐵

Syntaxwfo 5884 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:𝐴onto𝐵

Syntaxwf1o 5885 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:𝐴1-1-onto𝐵

Syntaxcfv 5886 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 (𝐹𝐴)

Syntaxwiso 5887 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)

Definitiondf-fun 5888 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 14795). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4727 with the maps-to notation (see df-mpt 4728 and df-mpt2 6652). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5889), a function with a given domain and codomain (df-f 5890), a one-to-one function (df-f1 5891), an onto function (df-fo 5892), or a one-to-one onto function (df-f1o 5893). For alternate definitions, see dffun2 5896, dffun3 5897, dffun4 5898, dffun5 5899, dffun6 5901, dffun7 5913, dffun8 5914, and dffun9 5915. (Contributed by NM, 1-Aug-1994.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))

Definitiondf-fn 5889 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 6045, dffn3 6052, dffn4 6119, and dffn5 6239. (Contributed by NM, 1-Aug-1994.)
(𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))

Definitiondf-f 5890 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. 𝐹:𝐴𝐵 can be read as "𝐹 is a function from 𝐴 to 𝐵". For alternate definitions, see dff2 6369, dff3 6370, and dff4 6371. (Contributed by NM, 1-Aug-1994.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))

Definitiondf-f1 5891 Define a one-to-one function. For equivalent definitions see dff12 6098 and dff13 6509. 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:𝐴1-1𝐵 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 16731. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))

Definitiondf-fo 5892 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 6117, dffo3 6372, dffo4 6373, and dffo5 6374.

An onto function is also called a "surjection" or a "surjective function", 𝐹:𝐴onto𝐵 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 16732. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))

Definitiondf-f1o 5893 Define a one-to-one onto function. For equivalent definitions see dff1o2 6140, dff1o3 6141, dff1o4 6143, and dff1o5 6144. 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:𝐴1-1-onto𝐵 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 16735. Therefore, two sets are called "isomorphic" if there is a bijection between them. According to isof1oidb 6571, 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 7961. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))

Definitiondf-fv 5894* Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 14874 after we define cosine in df-cos 14795). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 4728 and df-mpt2 6652), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 27284). Note that df-ov 6650 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 6216 and fvprc 6183). 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 6269, dffv3 6185, fv2 6184, and fv3 6204 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6263 and funfv2 6264. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6237. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 6186. (Revised by Scott Fenton, 6-Oct-2017.)
(𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)

Definitiondf-isom 5895* 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:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))

Theoremdffun2 5896* Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))

Theoremdffun3 5897* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))

Theoremdffun4 5898* Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧)))

Theoremdffun5 5899* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))

Theoremdffun6f 5900* 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.)
𝑥𝐴    &   𝑦𝐴       (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))

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