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	euiotaex 5301	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 5302*	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 5303	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 5304	Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
Theorem	iota4an 5305	Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑)
Theorem	iota5 5306*	A method for computing iota. (Contributed by NM, 17-Sep-2013.)
⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)
Theorem	iotabidv 5307*	Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))
Theorem	iotabii 5308	Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑥𝜓)
Theorem	iotacl 5309	Membership law for descriptions. This can useful for expanding an unbounded iota-based definition (see df-iota 5284). (Contributed by Andrew Salmon, 1-Aug-2011.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})
Theorem	iota2df 5310	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 5311*	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 5312*	An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)
⊢ (𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
Theorem	eliotaeu 5313	An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
⊢ (𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)
Theorem	iota2 5314*	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 5315	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 5316*	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 5317*	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 5318	Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.)
wff Fun 𝐴
Syntax	wfn 5319	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 5320	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 5321	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 5322	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 5323	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 5324	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 5325	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 5326	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 5356). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4148 with the maps-to notation (see df-mpt 4150). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5327), a function with a given domain and codomain (df-f 5328), a one-to-one function (df-f1 5329), an onto function (df-fo 5330), or a one-to-one onto function (df-f1o 5331). For alternate definitions, see dffun2 5334, dffun4 5335, dffun6 5338, dffun7 5351, dffun8 5352, and dffun9 5353. (Contributed by NM, 1-Aug-1994.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))
Definition	df-fn 5327	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 5328	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 5329	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 5330	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 5331	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 5332*	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 4150), 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 5333*	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 5334*	Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Theorem	dffun4 5335*	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 5336*	A way of proving a relation is a function, analogous to mo2r 2130. (Contributed by Jim Kingdon, 27-May-2020.)
⊢ ((Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)) → Fun 𝐴)
Theorem	dffun6f 5337*	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 5338*	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 5339*	A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
Theorem	dffun4f 5340*	Definition of function like dffun4 5335 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 5341	A function is a relation. (Contributed by NM, 1-Aug-1994.)
⊢ (Fun 𝐴 → Rel 𝐴)
Theorem	0nelfun 5342	A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)
⊢ (Fun 𝑅 → ∅ ∉ 𝑅)
Theorem	funss 5343	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 5344	Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))
Theorem	funeqi 5345	Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (Fun 𝐴 ↔ Fun 𝐵)
Theorem	funeqd 5346	Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Fun 𝐴 ↔ Fun 𝐵))
Theorem	nffun 5347	Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ Ⅎ𝑥Fun 𝐹
Theorem	sbcfung 5348	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 5349*	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 5350*	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 5351*	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 5352 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 5352*	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5351. (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 5353*	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 5354	An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
Theorem	funfnd 5355	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 5356	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 5357	The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
⊢ ¬ Fun V
Theorem	funopg 5358	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 5359*	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 5360*	A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
Theorem	funopab4 5361*	A class of ordered pairs of values in the form used by df-mpt 4150 is a function. (Contributed by NM, 17-Feb-2013.)
⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)}
Theorem	funmpt 5362	A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	funmpt2 5363	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 5364	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 5365	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	funssres 5366	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 5367	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 5368	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 5369*	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 5370	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 5371	A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)
⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴))
Theorem	funcnvsn 5372	The converse singleton of an ordered pair is a function. This is equivalent to funsn 5375 via cnvsn 5217, 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⟩}
Theorem	funsng 5373	A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {⟨𝐴, 𝐵⟩})
Theorem	fnsng 5374	Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
Theorem	funsn 5375	A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ Fun {⟨𝐴, 𝐵⟩}
Theorem	funinsn 5376	A function based on the singleton of an ordered pair. Unlike funsng 5373, this holds even if 𝐴 or 𝐵 is a proper class. (Contributed by Jim Kingdon, 17-Apr-2022.)
⊢ Fun ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))
Theorem	funprg 5377	A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
Theorem	funtpg 5378	A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → Fun {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩})
Theorem	funpr 5379	A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (𝐴 ≠ 𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
Theorem	funtp 5380	A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ 𝐸 ∈ V    &   ⊢ 𝐹 ∈ V    ⇒   ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → Fun {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
Theorem	fnsn 5381	Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}
Theorem	fnprg 5382	Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} Fn {𝐴, 𝐵})
Theorem	fntpg 5383	Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} Fn {𝑋, 𝑌, 𝑍})
Theorem	fntp 5384	A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ 𝐸 ∈ V    &   ⊢ 𝐹 ∈ V    ⇒   ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} Fn {𝐴, 𝐵, 𝐶})
Theorem	fun0 5385	The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
⊢ Fun ∅
Theorem	funcnvcnv 5386	The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
⊢ (Fun 𝐴 → Fun ◡◡𝐴)
Theorem	funcnv2 5387*	A simpler equivalence for single-rooted (see funcnv 5388). (Contributed by NM, 9-Aug-2004.)
⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
Theorem	funcnv 5388*	The converse of a class is a function iff the class is single-rooted, which means that for any 𝑦 in the range of 𝐴 there is at most one 𝑥 such that 𝑥𝐴𝑦. Definition of single-rooted in [Enderton] p. 43. See funcnv2 5387 for a simpler version. (Contributed by NM, 13-Aug-2004.)
⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
Theorem	funcnv3 5389*	A condition showing a class is single-rooted. (See funcnv 5388). (Contributed by NM, 26-May-2006.)
⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)
Theorem	funcnveq 5390*	Another way of expressing that a class is single-rooted. Counterpart to dffun2 5334. (Contributed by Jim Kingdon, 24-Dec-2018.)
⊢ (Fun ◡𝐴 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))
Theorem	fun2cnv 5391*	The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that 𝐴 is not necessarily a function. (Contributed by NM, 13-Aug-2004.)
⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
Theorem	svrelfun 5392	A single-valued relation is a function. (See fun2cnv 5391 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun ◡◡𝐴))
Theorem	fncnv 5393*	Single-rootedness (see funcnv 5388) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)
⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)
Theorem	fun11 5394*	Two ways of stating that 𝐴 is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.)
⊢ ((Fun ◡◡𝐴 ∧ Fun ◡𝐴) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤)))
Theorem	fununi 5395*	The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ∪ 𝐴)
Theorem	funcnvuni 5396*	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5388 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ◡∪ 𝐴)
Theorem	fun11uni 5397*	The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)
⊢ (∀𝑓 ∈ 𝐴 ((Fun 𝑓 ∧ Fun ◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun ∪ 𝐴 ∧ Fun ◡∪ 𝐴))
Theorem	funin 5398	The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (Fun 𝐹 → Fun (𝐹 ∩ 𝐺))
Theorem	funres11 5399	The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴))
Theorem	funcnvres 5400	The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ (𝐹 “ 𝐴)))