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Theorem List for Metamath Proof Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiotaval 6101* 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 6102 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})

Theoremiotaint 6103 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})

Theoremiota1 6104 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))

Theoremiotanul 6105 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 6106 The class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.)
(℩𝑥𝜑) ⊆ {𝑥𝜑}

Theoremiotaex 6107 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 6108 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)

Theoremiota4an 6109 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑)

Theoremiota5 6110* A method for computing iota. (Contributed by NM, 17-Sep-2013.)
((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))       ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)

Theoremiotabidv 6111* Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))

Theoremiotabii 6112 Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝜑𝜓)       (℩𝑥𝜑) = (℩𝑥𝜓)

Theoremiotacl 6113 Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6090). If you have a bounded iota-based definition, riotacl2 6884 may be useful.

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

(∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})

Theoremiota2df 6114 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 6115* 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 6116* 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 6117 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})

Theoremdfiota4 6118 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 6119 Obsolete proof of dfiota4 6118 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 6120* 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 6121 Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.)
wff Fun 𝐴

Syntaxwfn 6122 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 6123 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 6124 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 6125 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 6126 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 6127 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 6128 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 6129 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 15180). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4954 with the maps-to notation (see df-mpt 4955 and df-mpt2 6915). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6130), a function with a given domain and codomain (df-f 6131), a one-to-one function (df-f1 6132), an onto function (df-fo 6133), or a one-to-one onto function (df-f1o 6134). For alternate definitions, see dffun2 6137, dffun3 6138, dffun4 6139, dffun5 6140, dffun6 6142, dffun7 6154, dffun8 6155, and dffun9 6156. (Contributed by NM, 1-Aug-1994.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))

Definitiondf-fn 6130 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 6284, dffn3 6293, dffn4 6363, and dffn5 6492. (Contributed by NM, 1-Aug-1994.)
(𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))

Definitiondf-f 6131 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 6625, dff3 6626, and dff4 6627. (Contributed by NM, 1-Aug-1994.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))

Definitiondf-f1 6132 Define a one-to-one function. For equivalent definitions see dff12 6341 and dff13 6772. 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 17096. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))

Definitiondf-fo 6133 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 6361, dffo3 6628, dffo4 6629, and dffo5 6630.

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 17097. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))

Definitiondf-f1o 6134 Define a one-to-one onto function. For equivalent definitions see dff1o2 6387, dff1o3 6388, dff1o4 6390, and dff1o5 6391. 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 17100. Therefore, two sets are called "isomorphic" if there is a bijection between them. According to isof1oidb 6834, 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 8237. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))

Definitiondf-fv 6135* Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 15259 after we define cosine in df-cos 15180). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 4955 and df-mpt2 6915), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 27854). Note that df-ov 6913 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 6467 and fvprc 6430). 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 6522, dffv3 6433, fv2 6432, and fv3 6455 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6516 and funfv2 6517. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6489. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 6434. (Revised by Scott Fenton, 6-Oct-2017.)
(𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)

Definitiondf-isom 6136* 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 6137* Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))

Theoremdffun3 6138* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))

Theoremdffun4 6139* 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 6140* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))

Theoremdffun6f 6141* 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))

Theoremdffun6 6142* Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)
(Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))

Theoremfunmo 6143* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
(Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)

Theoremfunrel 6144 A function is a relation. (Contributed by NM, 1-Aug-1994.)
(Fun 𝐴 → Rel 𝐴)

Theorem0nelfun 6145 A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)
(Fun 𝑅 → ∅ ∉ 𝑅)

Theoremfunss 6146 Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
(𝐴𝐵 → (Fun 𝐵 → Fun 𝐴))

Theoremfuneq 6147 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
(𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))

Theoremfuneqi 6148 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐴 = 𝐵       (Fun 𝐴 ↔ Fun 𝐵)

Theoremfuneqd 6149 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Fun 𝐴 ↔ Fun 𝐵))

Theoremnffun 6150 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
𝑥𝐹       𝑥Fun 𝐹

Theoremsbcfung 6151 Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun 𝐴 / 𝑥𝐹))

Theoremfuneu 6152* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((Fun 𝐹𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)

Theoremfuneu2 6153* There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)
((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)

Theoremdffun7 6154* 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 6155 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))

Theoremdffun8 6155* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 6154. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))

Theoremdffun9 6156* 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))

Theoremfunfn 6157 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
(Fun 𝐴𝐴 Fn dom 𝐴)

Theoremfunfnd 6158 A function is a function over its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑 → Fun 𝐴)       (𝜑𝐴 Fn dom 𝐴)

Theoremfuni 6159 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)
Fun I

Theoremnfunv 6160 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
¬ Fun V

Theoremfunopg 6161 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.) (Avoid depending on this detail.)
((𝐴𝑉𝐵𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)

Theoremfunopab 6162* 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)

Theoremfunopabeq 6163* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}

Theoremfunopab4 6164* A class of ordered pairs of values in the form used by df-mpt 4955 is a function. (Contributed by NM, 17-Feb-2013.)
Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)}

Theoremfunmpt 6165 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
Fun (𝑥𝐴𝐵)

Theoremfunmpt2 6166 Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)
𝐹 = (𝑥𝐴𝐵)       Fun 𝐹

Theoremfunco 6167 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 (𝐹𝐺))

Theoremfunresfunco 6168 Composition of two functions, generalization of funco 6167. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))

Theoremfunres 6169 A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)
(Fun 𝐹 → Fun (𝐹𝐴))

Theoremfunssres 6170 The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.)
((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)

Theoremfun2ssres 6171 Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)
((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))

Theoremfunun 6172 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 (𝐹𝐺))

Theoremfununmo 6173* 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 (𝐹𝐺) → ∃*𝑦 𝑥𝐹𝑦)

Theoremfununfun 6174 If the union of classes is a function, the classes itselves are functions. (Contributed by AV, 18-Jul-2019.)
(Fun (𝐹𝐺) → (Fun 𝐹 ∧ Fun 𝐺))

Theoremfundif 6175 A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)
(Fun 𝐹 → Fun (𝐹𝐴))

Theoremfuncnvsn 6176 The converse singleton of an ordered pair is a function. This is equivalent to funsn 6179 via cnvsn 5864, 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⟩}

Theoremfunsng 6177 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)
((𝐴𝑉𝐵𝑊) → Fun {⟨𝐴, 𝐵⟩})

Theoremfnsng 6178 Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})

Theoremfunsn 6179 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⟩}

Theoremfunprg 6180 A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})

Theoremfuntpg 6181 A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Proof shortened by JJ, 14-Jul-2021.)
(((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → Fun {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩})

Theoremfunpr 6182 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (𝐴𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})

Theoremfuntp 6183 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       ((𝐴𝐵𝐴𝐶𝐵𝐶) → Fun {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})

Theoremfnsn 6184 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}

Theoremfnprg 6185 Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} Fn {𝐴, 𝐵})

Theoremfntpg 6186 Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
(((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} Fn {𝑋, 𝑌, 𝑍})

Theoremfntp 6187 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       ((𝐴𝐵𝐴𝐶𝐵𝐶) → {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} Fn {𝐴, 𝐵, 𝐶})

Theoremfuncnvpr 6188 The converse pair of ordered pairs is a function if the second members are different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.)
((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})

Theoremfuncnvtp 6189 The converse triple of ordered pairs is a function if the second members are pairwise different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.)
(((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})

Theoremfuncnvqp 6190 The converse quadruple of ordered pairs is a function if the second members are pairwise different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.) (Proof shortened by JJ, 14-Jul-2021.)
((((𝐴𝑈𝐶𝑉) ∧ (𝐸𝑊𝐺𝑇)) ∧ ((𝐵𝐷𝐵𝐹𝐵𝐻) ∧ (𝐷𝐹𝐷𝐻) ∧ 𝐹𝐻)) → Fun ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩}))

Theoremfun0 6191 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
Fun ∅

Theoremfuncnv0 6192 The converse of the empty set is a function. (Contributed by AV, 7-Jan-2021.)
Fun

Theoremfuncnvcnv 6193 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
(Fun 𝐴 → Fun 𝐴)

Theoremfuncnv2 6194* A simpler equivalence for single-rooted (see funcnv 6195). (Contributed by NM, 9-Aug-2004.)
(Fun 𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)

Theoremfuncnv 6195* 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 6194 for a simpler version. (Contributed by NM, 13-Aug-2004.)
(Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)

Theoremfuncnv3 6196* A condition showing a class is single-rooted. (See funcnv 6195). (Contributed by NM, 26-May-2006.)
(Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)

Theoremfun2cnv 6197* 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)

Theoremsvrelfun 6198 A single-valued relation is a function. (See fun2cnv 6197 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun 𝐴))

Theoremfncnv 6199* Single-rootedness (see funcnv 6195) of a class cut down by a Cartesian product. (Contributed by NM, 5-Mar-2007.)
((𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)

Theoremfun11 6200* 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) ↔ ∀𝑥𝑦𝑧𝑤((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))

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