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Theorem List for Metamath Proof Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsb8iota 6401 Variable substitution in description binder. Compare sb8eu 2602. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 18-Mar-2013.) (New usage is discouraged.)
𝑦𝜑       (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)
 
Theoremiotaeq 6402 Equality theorem for descriptions. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Andrew Salmon, 30-Jun-2011.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))
 
Theoremiotabi 6403 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
(∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))
 
Theoremuniabio 6404* 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 6405* 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 6406 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
 
Theoremiotaint 6407 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
 
Theoremiota1 6408 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
 
Theoremiotanul 6409 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 6410 The class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.)
(℩𝑥𝜑) ⊆ {𝑥𝜑}
 
Theoremiotaex 6411 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 6412 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
 
Theoremiota4an 6413 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑)
 
Theoremiota5 6414* A method for computing iota. (Contributed by NM, 17-Sep-2013.)
((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))       ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
 
Theoremiotabidv 6415* Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))
 
Theoremiotabii 6416 Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝜑𝜓)       (℩𝑥𝜑) = (℩𝑥𝜓)
 
Theoremiotacl 6417 Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6389). If you have a bounded iota-based definition, riotacl2 7243 may be useful.

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

(∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})
 
Theoremiota2df 6418 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 6419* 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 6420* 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))
 
Theoremiotan0 6421* Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is not the empty set (that means that "the unique element such that 𝜑 " exists). (Contributed by AV, 30-Jan-2024.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝑉𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
 
Theoremsniota 6422 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
 
Theoremdfiota4 6423 The operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017.) (Proof shortened by JJ, 28-Oct-2021.)
(℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)
 
Theoremcsbiota 6424* 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)
 
2.3.16  Functions
 
Syntaxwfun 6425 Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.)
wff Fun 𝐴
 
Syntaxwfn 6426 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 6427 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 6428 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 6429 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 6430 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 6431 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 6432 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 6433 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 15770). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5162 with the maps-to notation (see df-mpt 5163 and df-mpo 7274). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6434), a function with a given domain and codomain (df-f 6435), a one-to-one function (df-f1 6436), an onto function (df-fo 6437), or a one-to-one onto function (df-f1o 6438). For alternate definitions, see dffun2 6441, dffun3 6442, dffun4 6443, dffun5 6444, dffun6 6446, dffun7 6458, dffun8 6459, and dffun9 6460. (Contributed by NM, 1-Aug-1994.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
 
Definitiondf-fn 6434 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 6599, dffn3 6610, dffn4 6691, and dffn5 6823. (Contributed by NM, 1-Aug-1994.)
(𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
 
Definitiondf-f 6435 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 6970, dff3 6971, and dff4 6972. (Contributed by NM, 1-Aug-1994.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
 
Definitiondf-f1 6436 Define a one-to-one function. For equivalent definitions see dff12 6666 and dff13 7123. 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 17792. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
 
Definitiondf-fo 6437 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 6689, dffo3 6973, dffo4 6974, and dffo5 6975.

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 17793. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
 
Definitiondf-f1o 6438 Define a one-to-one onto function. For equivalent definitions see dff1o2 6718, dff1o3 6719, dff1o4 6721, and dff1o5 6722. 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 17796. Therefore, two sets are called "isomorphic" if there is a bijection between them. According to isof1oidb 7189, 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 8718. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
 
Definitiondf-fv 6439* Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 15849 after we define cosine in df-cos 15770). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5163 and df-mpo 7274), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 28795). Note that df-ov 7272 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 6799 and fvprc 6761). 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 6858, dffv3 6765, fv2 6764, and fv3 6787 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6850 and funfv2 6851. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6820. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now Theorem dffv4 6766. (Revised by Scott Fenton, 6-Oct-2017.)
(𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
 
Definitiondf-isom 6440* 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 6441* Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
 
Theoremdffun3 6442* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
 
Theoremdffun4 6443* 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 6444* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
 
Theoremdffun6f 6445* 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 6446* Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)
(Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
 
Theoremfunmo 6447* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
(Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
 
Theoremfunrel 6448 A function is a relation. (Contributed by NM, 1-Aug-1994.)
(Fun 𝐴 → Rel 𝐴)
 
Theorem0nelfun 6449 A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)
(Fun 𝑅 → ∅ ∉ 𝑅)
 
Theoremfunss 6450 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 6451 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
(𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))
 
Theoremfuneqi 6452 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐴 = 𝐵       (Fun 𝐴 ↔ Fun 𝐵)
 
Theoremfuneqd 6453 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Fun 𝐴 ↔ Fun 𝐵))
 
Theoremnffun 6454 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
𝑥𝐹       𝑥Fun 𝐹
 
Theoremsbcfung 6455 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 6456* 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 6457* There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)
((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
 
Theoremdffun7 6458* 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 6459 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))
 
Theoremdffun8 6459* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 6458. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))
 
Theoremdffun9 6460* 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 6461 A class is a function if and only if it is a function on its domain. (Contributed by NM, 13-Aug-2004.)
(Fun 𝐴𝐴 Fn dom 𝐴)
 
Theoremfunfnd 6462 A function is a function on its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑 → Fun 𝐴)       (𝜑𝐴 Fn dom 𝐴)
 
Theoremfuni 6463 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6557. (Contributed by NM, 30-Apr-1998.)
Fun I
 
Theoremnfunv 6464 The universal class is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
¬ Fun V
 
Theoremfunopg 6465 A Kuratowski ordered pair of sets is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6482, as relsnopg 5711 is to relop 5757. (New usage is discouraged.)
((𝐴𝑉𝐵𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)
 
Theoremfunopab 6466* 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 6467* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
 
Theoremfunopab4 6468* A class of ordered pairs of values in the form used by df-mpt 5163 is a function. (Contributed by NM, 17-Feb-2013.)
Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)}
 
Theoremfunmpt 6469 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
Fun (𝑥𝐴𝐵)
 
Theoremfunmpt2 6470 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 6471 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 6472 Composition of two functions, generalization of funco 6471. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
 
Theoremfunres 6473 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))
 
Theoremfunresd 6474 A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑 → Fun 𝐹)       (𝜑 → Fun (𝐹𝐴))
 
Theoremfunssres 6475 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 6476 Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)
((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
 
Theoremfunun 6477 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 6478* 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 6479 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 6480 A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)
(Fun 𝐹 → Fun (𝐹𝐴))
 
Theoremfuncnvsn 6481 The converse singleton of an ordered pair is a function. This is equivalent to funsn 6484 via cnvsn 6127, 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 6482 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 6483 Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
 
Theoremfunsn 6484 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 6485 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 6486 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 6487 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 6488 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 6489 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 6490 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 6491 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 6492 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 6493 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 6494 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 6495 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 6496 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
Fun ∅
 
Theoremfuncnv0 6497 The converse of the empty set is a function. (Contributed by AV, 7-Jan-2021.)
Fun
 
Theoremfuncnvcnv 6498 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
(Fun 𝐴 → Fun 𝐴)
 
Theoremfuncnv2 6499* A simpler equivalence for single-rooted (see funcnv 6500). (Contributed by NM, 9-Aug-2004.)
(Fun 𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
 
Theoremfuncnv 6500* 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 6499 for a simpler version. (Contributed by NM, 13-Aug-2004.)
(Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
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