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	iota4 5301	Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
Theorem	iota4an 5302	Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑)
Theorem	iota5 5303*	A method for computing iota. (Contributed by NM, 17-Sep-2013.)
⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)
Theorem	iotabidv 5304*	Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))
Theorem	iotabii 5305	Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑥𝜓)
Theorem	iotacl 5306	Membership law for descriptions. This can useful for expanding an unbounded iota-based definition (see df-iota 5281). (Contributed by Andrew Salmon, 1-Aug-2011.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})
Theorem	iota2df 5307	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 5308*	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 5309*	An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)
⊢ (𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
Theorem	eliotaeu 5310	An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
⊢ (𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)
Theorem	iota2 5311*	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 5312	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 5313*	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 5314*	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 5315	Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.)
wff Fun 𝐴
Syntax	wfn 5316	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 5317	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 5318	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 5319	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 5320	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 5321	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 5322	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 5323	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 5353). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4145 with the maps-to notation (see df-mpt 4147). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5324), a function with a given domain and codomain (df-f 5325), a one-to-one function (df-f1 5326), an onto function (df-fo 5327), or a one-to-one onto function (df-f1o 5328). For alternate definitions, see dffun2 5331, dffun4 5332, dffun6 5335, dffun7 5348, dffun8 5349, and dffun9 5350. (Contributed by NM, 1-Aug-1994.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))
Definition	df-fn 5324	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 5325	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 5326	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 5327	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 5328	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 5329*	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 4147), 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 5330*	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 5331*	Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Theorem	dffun4 5332*	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 5333*	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 5334*	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 5335*	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 5336*	A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
Theorem	dffun4f 5337*	Definition of function like dffun4 5332 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 5338	A function is a relation. (Contributed by NM, 1-Aug-1994.)
⊢ (Fun 𝐴 → Rel 𝐴)
Theorem	0nelfun 5339	A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)
⊢ (Fun 𝑅 → ∅ ∉ 𝑅)
Theorem	funss 5340	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 5341	Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))
Theorem	funeqi 5342	Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (Fun 𝐴 ↔ Fun 𝐵)
Theorem	funeqd 5343	Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Fun 𝐴 ↔ Fun 𝐵))
Theorem	nffun 5344	Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ Ⅎ𝑥Fun 𝐹
Theorem	sbcfung 5345	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 5346*	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 5347*	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 5348*	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 5349 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 5349*	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5348. (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 5350*	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 5351	An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
Theorem	funfnd 5352	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 5353	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 5354	The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
⊢ ¬ Fun V
Theorem	funopg 5355	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 5356*	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 5357*	A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
Theorem	funopab4 5358*	A class of ordered pairs of values in the form used by df-mpt 4147 is a function. (Contributed by NM, 17-Feb-2013.)
⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)}
Theorem	funmpt 5359	A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	funmpt2 5360	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 5361	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 5362	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 5363	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 5364	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 5365	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 5366*	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 5367	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 5368	A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)
⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴))
Theorem	funcnvsn 5369	The converse singleton of an ordered pair is a function. This is equivalent to funsn 5372 via cnvsn 5214, 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 5370	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 5371	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 5372	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 5373	A function based on the singleton of an ordered pair. Unlike funsng 5370, 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 5374	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 5375	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 5376	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 5377	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 5378	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 5379	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 5380	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 5381	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 5382	The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
⊢ Fun ∅
Theorem	funcnvcnv 5383	The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
⊢ (Fun 𝐴 → Fun ◡◡𝐴)
Theorem	funcnv2 5384*	A simpler equivalence for single-rooted (see funcnv 5385). (Contributed by NM, 9-Aug-2004.)
⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
Theorem	funcnv 5385*	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 5384 for a simpler version. (Contributed by NM, 13-Aug-2004.)
⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
Theorem	funcnv3 5386*	A condition showing a class is single-rooted. (See funcnv 5385). (Contributed by NM, 26-May-2006.)
⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)
Theorem	funcnveq 5387*	Another way of expressing that a class is single-rooted. Counterpart to dffun2 5331. (Contributed by Jim Kingdon, 24-Dec-2018.)
⊢ (Fun ◡𝐴 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))
Theorem	fun2cnv 5388*	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 5389	A single-valued relation is a function. (See fun2cnv 5388 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 5390*	Single-rootedness (see funcnv 5385) 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 5391*	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 5392*	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 5393*	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5385 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
⊢ (∀𝑓 ∈ 𝐴 (Fun ◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ◡∪ 𝐴)
Theorem	fun11uni 5394*	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 5395	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 5396	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 5397	The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ (𝐹 “ 𝐴)))
Theorem	cnvresid 5398	Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
Theorem	funcnvres2 5399	The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)
⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴)))
Theorem	funimacnv 5400	The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹))