P. 54 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5301-5400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	relcnvexb 5301	A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V))
Theorem	ressn 5302	Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
Theorem	cnviinm 5303*	The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
⊢ (∃𝑦 𝑦 ∈ 𝐴 → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
Theorem	cnvpom 5304*	The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴))
Theorem	cnvsom 5305*	The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴))
Theorem	coexg 5306	The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V)
Theorem	coex 5307	The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∘ 𝐵) ∈ V
Theorem	xpcom 5308*	Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
2.6.7  Definite description binder (inverted iota)
Syntax	cio 5309	Extend class notation with Russell's definition description binder (inverted iota).
class (℩𝑥𝜑)
Theorem	iotajust 5310*	Soundness justification theorem for df-iota 5311. (Contributed by Andrew Salmon, 29-Jun-2011.)
⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
Definition	df-iota 5311*	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 5323); otherwise, it evaluates to the empty set (see iotanul 5327). 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 5336 (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 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
Theorem	dfiota2 5312*	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 5313	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 5314*	Bound-variable hypothesis builder for the ℩ class. (Contributed by Jim Kingdon, 21-Dec-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))
Theorem	nfiotaw 5315*	Bound-variable hypothesis builder for the ℩ class. (Contributed by NM, 23-Aug-2011.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥(℩𝑦𝜑)
Theorem	cbviota 5316	Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)
Theorem	cbviotavw 5317*	Change bound variables in a description binder. Version of cbviotav 5318 with a disjoint variable condition. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by GG, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)
Theorem	cbviotav 5318*	Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)
Theorem	sb8iota 5319	Variable substitution in description binder. Compare sb8eu 2093. (Contributed by NM, 18-Mar-2013.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)
Theorem	iotaeq 5320	Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))
Theorem	iotabi 5321	Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))
Theorem	uniabio 5322*	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 5323*	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 5324	Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
Theorem	iotaint 5325	Equivalence between two different forms of ℩. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
Theorem	iota1 5326	Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
Theorem	iotanul 5327	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	euiotaex 5328	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)
Theorem	iotass 5329*	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(𝜑 → 𝑥 ⊆ 𝐴) → (℩𝑥𝜑) ⊆ 𝐴)
Theorem	iotaexab 5330	Existence of the ℩ class when all the possible values are contained in a set. (Contributed by Jim Kingdon, 27-May-2025.)
⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → (℩𝑥𝜑) ∈ V)
Theorem	iota4 5331	Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
Theorem	iota4an 5332	Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑)
Theorem	iota5 5333*	A method for computing iota. (Contributed by NM, 17-Sep-2013.)
⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)
Theorem	iotabidv 5334*	Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))
Theorem	iotabii 5335	Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑥𝜓)
Theorem	iotacl 5336	Membership law for descriptions. This can useful for expanding an unbounded iota-based definition (see df-iota 5311). (Contributed by Andrew Salmon, 1-Aug-2011.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})
Theorem	iota2df 5337	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 5338*	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	eliota 5339*	An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)
⊢ (𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
Theorem	eliotaeu 5340	An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
⊢ (𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)
Theorem	iota2 5341*	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	sniota 5342	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	iotam 5343*	Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is inhabited (that means that "the unique element such that 𝜑 " exists). (Contributed by AV, 30-Jan-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
Theorem	csbiotag 5344*	Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
2.6.8  Functions
Syntax	wfun 5345	Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.)
wff Fun 𝐴
Syntax	wfn 5346	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 5347	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 5348	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 5349	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 5350	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 5351	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 5352	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 5353	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 5383). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4170 with the maps-to notation (see df-mpt 4172). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5354), a function with a given domain and codomain (df-f 5355), a one-to-one function (df-f1 5356), an onto function (df-fo 5357), or a one-to-one onto function (df-f1o 5358). For alternate definitions, see dffun2 5361, dffun4 5362, dffun6 5365, dffun7 5378, dffun8 5379, and dffun9 5380. (Contributed by NM, 1-Aug-1994.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))
Definition	df-fn 5354	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))
Definition	df-f 5355	Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
Definition	df-f1 5356	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:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
Definition	df-fo 5357	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:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
Definition	df-f1o 5358	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:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
Definition	df-fv 5359*	Define the value of a function, (𝐹‘𝐴), also known as function application. For example, ( I ‘∅) = ∅. Typically, function 𝐹 is defined using maps-to notation (see df-mpt 4172), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9. We will later 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. 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. (Contributed by NM, 1-Aug-1994.) Revised to use ℩. (Revised by Scott Fenton, 6-Oct-2017.)
⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
Definition	df-isom 5360*	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 5361*	Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Theorem	dffun4 5362*	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	dffun5r 5363*	A way of proving a relation is a function, analogous to mo2r 2133. (Contributed by Jim Kingdon, 27-May-2020.)
⊢ ((Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)) → Fun 𝐴)
Theorem	dffun6f 5364*	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 5365*	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 5366*	A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
Theorem	dffun4f 5367*	Definition of function like dffun4 5362 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑧𝐴    ⇒   ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧)))
Theorem	funrel 5368	A function is a relation. (Contributed by NM, 1-Aug-1994.)
⊢ (Fun 𝐴 → Rel 𝐴)
Theorem	0nelfun 5369	A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)
⊢ (Fun 𝑅 → ∅ ∉ 𝑅)
Theorem	funss 5370	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 5371	Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))
Theorem	funeqi 5372	Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (Fun 𝐴 ↔ Fun 𝐵)
Theorem	funeqd 5373	Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Fun 𝐴 ↔ Fun 𝐵))
Theorem	nffun 5374	Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ Ⅎ𝑥Fun 𝐹
Theorem	sbcfung 5375	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 5376*	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 5377*	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 5378*	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 5379 shows that it does not matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
Theorem	dffun8 5379*	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5378. (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 5380*	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 5381	An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
Theorem	funfnd 5382	A function is a function over its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → Fun 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 Fn dom 𝐴)
Theorem	funi 5383	The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)
⊢ Fun I
Theorem	nfunv 5384	The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
⊢ ¬ Fun V
Theorem	funopg 5385	A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)
Theorem	funopab 5386*	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 5387*	A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
Theorem	funopab4 5388*	A class of ordered pairs of values in the form used by df-mpt 4172 is a function. (Contributed by NM, 17-Feb-2013.)
⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)}
Theorem	funmpt 5389	A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	funmpt2 5390	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 5391	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	funres 5392	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	funresd 5393	A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))
Theorem	funssres 5394	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 5395	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 5396	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))
Theorem	fununmo 5397*	If the union of classes is a function, there is at most one element in relation to an arbitrary element regarding one of these classes. (Contributed by AV, 18-Jul-2019.)
⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦)
Theorem	fununfun 5398	If the union of classes is a function, the classes itselves are functions. (Contributed by AV, 18-Jul-2019.)
⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺))
Theorem	fundif 5399	A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)
⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴))
Theorem	funcnvsn 5400	The converse singleton of an ordered pair is a function. This is equivalent to funsn 5403 via cnvsn 5244, but stating it this way allows us to skip the sethood assumptions on 𝐴 and 𝐵. (Contributed by NM, 30-Apr-2015.)
⊢ Fun ◡{⟨𝐴, 𝐵⟩}