HomeHome Intuitionistic Logic Explorer
Theorem List (p. 51 of 132)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremimadmres 5001 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
(𝐴 “ dom (𝐴𝐵)) = (𝐴𝐵)
 
Theoremmptpreima 5002* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐹 = (𝑥𝐴𝐵)       (𝐹𝐶) = {𝑥𝐴𝐵𝐶}
 
Theoremmptiniseg 5003* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐹 = (𝑥𝐴𝐵)       (𝐶𝑉 → (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 = 𝐶})
 
Theoremdmmpt 5004 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)
𝐹 = (𝑥𝐴𝐵)       dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
 
Theoremdmmptss 5005* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
𝐹 = (𝑥𝐴𝐵)       dom 𝐹𝐴
 
Theoremdmmptg 5006* The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
(∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
 
Theoremrelco 5007 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
Rel (𝐴𝐵)
 
Theoremdfco2 5008* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
(𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
 
Theoremdfco2a 5009* Generalization of dfco2 5008, where 𝐶 can have any value between dom 𝐴 ∩ ran 𝐵 and V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴𝐵) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
 
Theoremcoundi 5010 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ∘ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 
Theoremcoundir 5011 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐵) ∘ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 
Theoremcores 5012 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(ran 𝐵𝐶 → ((𝐴𝐶) ∘ 𝐵) = (𝐴𝐵))
 
Theoremresco 5013 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))
 
Theoremimaco 5014 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
((𝐴𝐵) “ 𝐶) = (𝐴 “ (𝐵𝐶))
 
Theoremrnco 5015 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
 
Theoremrnco2 5016 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
ran (𝐴𝐵) = (𝐴 “ ran 𝐵)
 
Theoremdmco 5017 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
 
Theoremcoiun 5018* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
(𝐴 𝑥𝐶 𝐵) = 𝑥𝐶 (𝐴𝐵)
 
Theoremcocnvcnv1 5019 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)
 
Theoremcocnvcnv2 5020 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)
 
Theoremcores2 5021 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
(dom 𝐴𝐶 → (𝐴(𝐵𝐶)) = (𝐴𝐵))
 
Theoremco02 5022 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
(𝐴 ∘ ∅) = ∅
 
Theoremco01 5023 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
(∅ ∘ 𝐴) = ∅
 
Theoremcoi1 5024 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
(Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
 
Theoremcoi2 5025 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
(Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)
 
Theoremcoires1 5026 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
(𝐴 ∘ ( I ↾ 𝐵)) = (𝐴𝐵)
 
Theoremcoass 5027 Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
((𝐴𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵𝐶))
 
Theoremrelcnvtr 5028 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
(Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
 
Theoremrelssdmrn 5029 A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
(Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
 
Theoremcnvssrndm 5030 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐴 ⊆ (ran 𝐴 × dom 𝐴)
 
Theoremcossxp 5031 Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)
 
Theoremcossxp2 5032 The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.)
(𝜑𝑅 ⊆ (𝐴 × 𝐵))    &   (𝜑𝑆 ⊆ (𝐵 × 𝐶))       (𝜑 → (𝑆𝑅) ⊆ (𝐴 × 𝐶))
 
Theoremcocnvres 5033 Restricting a relation and a converse relation when they are composed together (Contributed by BJ, 10-Jul-2022.)
(𝑆𝑅) = ((𝑆 ↾ dom 𝑅) ∘ (𝑅 ↾ dom 𝑆))
 
Theoremcocnvss 5034 Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.)
(𝑆𝑅) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
 
Theoremrelrelss 5035 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))
 
Theoremunielrel 5036 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
((Rel 𝑅𝐴𝑅) → 𝐴 𝑅)
 
Theoremrelfld 5037 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
(Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
 
Theoremrelresfld 5038 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
(Rel 𝑅 → (𝑅 𝑅) = 𝑅)
 
Theoremrelcoi2 5039 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
(Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)
 
Theoremrelcoi1 5040 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)
(Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = 𝑅)
 
Theoremunidmrn 5041 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
𝐴 = (dom 𝐴 ∪ ran 𝐴)
 
Theoremrelcnvfld 5042 if 𝑅 is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
(Rel 𝑅 𝑅 = 𝑅)
 
Theoremdfdm2 5043 Alternate definition of domain df-dm 4519 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
dom 𝐴 = (𝐴𝐴)
 
Theoremunixpm 5044* The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)
(∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) = (𝐴𝐵))
 
Theoremunixp0im 5045 The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)
((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
 
Theoremcnvexg 5046 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
(𝐴𝑉𝐴 ∈ V)
 
Theoremcnvex 5047 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
𝐴 ∈ V       𝐴 ∈ V
 
Theoremrelcnvexb 5048 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
(Rel 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))
 
Theoremressn 5049 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
 
Theoremcnviinm 5050* The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
(∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
 
Theoremcnvpom 5051* The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
(∃𝑥 𝑥𝐴 → (𝑅 Po 𝐴𝑅 Po 𝐴))
 
Theoremcnvsom 5052* The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)
(∃𝑥 𝑥𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐴))
 
Theoremcoexg 5053 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theoremcoex 5054 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V
 
Theoremxpcom 5055* Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
(∃𝑥 𝑥𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
 
2.6.7  Definite description binder (inverted iota)
 
Syntaxcio 5056 Extend class notation with Russell's definition description binder (inverted iota).
class (℩𝑥𝜑)
 
Theoremiotajust 5057* Soundness justification theorem for df-iota 5058. (Contributed by Andrew Salmon, 29-Jun-2011.)
{𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
 
Definitiondf-iota 5058* 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 5069); otherwise, it evaluates to the empty set (see iotanul 5073). Russell used the inverted iota symbol to represent the binder.

Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use iotacl 5081 (for unbounded iota). This can be easier than applying a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF.

(Contributed by Andrew Salmon, 30-Jun-2011.)

(℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
 
Theoremdfiota2 5059* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
(℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
 
Theoremnfiota1 5060 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥(℩𝑥𝜑)
 
Theoremnfiotadxy 5061* Deduction version of nfiotaxy 5062. (Contributed by Jim Kingdon, 21-Dec-2018.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥(℩𝑦𝜓))
 
Theoremnfiotaxy 5062* Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.)
𝑥𝜑       𝑥(℩𝑦𝜑)
 
Theoremcbviota 5063 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝑦𝜑    &   𝑥𝜓       (℩𝑥𝜑) = (℩𝑦𝜓)
 
Theoremcbviotav 5064* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))       (℩𝑥𝜑) = (℩𝑦𝜓)
 
Theoremsb8iota 5065 Variable substitution in description binder. Compare sb8eu 1990. (Contributed by NM, 18-Mar-2013.)
𝑦𝜑       (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)
 
Theoremiotaeq 5066 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
(∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))
 
Theoremiotabi 5067 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
(∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))
 
Theoremuniabio 5068* 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 5069* 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 5070 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
 
Theoremiotaint 5071 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
 
Theoremiota1 5072 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
 
Theoremiotanul 5073 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) = ∅)
 
Theoremeuiotaex 5074 Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.)
(∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
 
Theoremiotass 5075* Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)
(∀𝑥(𝜑𝑥𝐴) → (℩𝑥𝜑) ⊆ 𝐴)
 
Theoremiota4 5076 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
 
Theoremiota4an 5077 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑)
 
Theoremiota5 5078* A method for computing iota. (Contributed by NM, 17-Sep-2013.)
((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))       ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
 
Theoremiotabidv 5079* Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))
 
Theoremiotabii 5080 Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝜑𝜓)       (℩𝑥𝜑) = (℩𝑥𝜓)
 
Theoremiotacl 5081 Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5058).

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

(∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})
 
Theoremiota2df 5082 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 5083* 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 5084* 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 5085 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
 
Theoremcsbiotag 5086* Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)
(𝐴𝑉𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
 
2.6.8  Functions
 
Syntaxwfun 5087 Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.)
wff Fun 𝐴
 
Syntaxwfn 5088 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 5089 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 5090 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 5091 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 5092 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 5093 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 5094 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 5095 Define predicate that determines if some class 𝐴 is a function. Definition 10.1 of [Quine] p. 65. For example, the expression Fun I is true (funi 5125). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3959 with the maps-to notation (see df-mpt 3961). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5096), a function with a given domain and codomain (df-f 5097), a one-to-one function (df-f1 5098), an onto function (df-fo 5099), or a one-to-one onto function (df-f1o 5100). For alternate definitions, see dffun2 5103, dffun4 5104, dffun6 5107, dffun7 5120, dffun8 5121, and dffun9 5122. (Contributed by NM, 1-Aug-1994.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
 
Definitiondf-fn 5096 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)
(𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
 
Definitiondf-f 5097 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
 
Definitiondf-f1 5098 Define a one-to-one function. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow). (Contributed by NM, 1-Aug-1994.)
(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
 
Definitiondf-fo 5099 Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). (Contributed by NM, 1-Aug-1994.)
(𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
 
Definitiondf-f1o 5100 Define a one-to-one onto function. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow). (Contributed by NM, 1-Aug-1994.)
(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13177
  Copyright terms: Public domain < Previous  Next >