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Theorem List for Metamath Proof Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiotajust 6301* Soundness justification theorem for df-iota 6302. (Contributed by Andrew Salmon, 29-Jun-2011.)
{𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
 
Definitiondf-iota 6302* 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 6317); otherwise, it evaluates to the empty set (see iotanul 6321). 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 riotacl2 7123 (or iotacl 6329 for unbounded iota), as demonstrated in the proof of supub 8920. This can be easier than applying riotasbc 7125 or 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 ∣ {𝑥𝜑} = {𝑦}}
 
Theoremdfiota2 6303* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
(℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
 
Theoremnfiota1 6304 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥(℩𝑥𝜑)
 
Theoremnfiotadw 6305* Deduction version of nfiotaw 6306. Version of nfiotad 6307 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by NM, 18-Feb-2013.) (Revised by Gino Giotto, 26-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥(℩𝑦𝜓))
 
Theoremnfiotaw 6306* Bound-variable hypothesis builder for the class. Version of nfiota 6308 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by NM, 23-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.)
𝑥𝜑       𝑥(℩𝑦𝜑)
 
Theoremnfiotad 6307 Deduction version of nfiota 6308. Usage of this theorem is discouraged because it depends on ax-13 2392. Use the weaker nfiotadw 6305 when possible. (Contributed by NM, 18-Feb-2013.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥(℩𝑦𝜓))
 
Theoremnfiota 6308 Bound-variable hypothesis builder for the class. Usage of this theorem is discouraged because it depends on ax-13 2392. Use the weaker nfiotaw 6306 when possible. (Contributed by NM, 23-Aug-2011.) (New usage is discouraged.)
𝑥𝜑       𝑥(℩𝑦𝜑)
 
Theoremcbviotaw 6309* Change bound variables in a description binder. Version of cbviota 6311 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝑦𝜑    &   𝑥𝜓       (℩𝑥𝜑) = (℩𝑦𝜓)
 
Theoremcbviotavw 6310* Change bound variables in a description binder. Version of cbviotav 6312 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (℩𝑥𝜑) = (℩𝑦𝜓)
 
Theoremcbviota 6311 Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2392. Use the weaker cbviotaw 6309 when possible. (Contributed by Andrew Salmon, 1-Aug-2011.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝑦𝜑    &   𝑥𝜓       (℩𝑥𝜑) = (℩𝑦𝜓)
 
Theoremcbviotav 6312* Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2392. Use the weaker cbviotavw 6310 when possible. (Contributed by Andrew Salmon, 1-Aug-2011.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (℩𝑥𝜑) = (℩𝑦𝜓)
 
Theoremsb8iota 6313 Variable substitution in description binder. Compare sb8eu 2687. Usage of this theorem is discouraged because it depends on ax-13 2392. (Contributed by NM, 18-Mar-2013.) (New usage is discouraged.)
𝑦𝜑       (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)
 
Theoremiotaeq 6314 Equality theorem for descriptions. Usage of this theorem is discouraged because it depends on ax-13 2392. (Contributed by Andrew Salmon, 30-Jun-2011.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))
 
Theoremiotabi 6315 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
(∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))
 
Theoremuniabio 6316* 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 6317* 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 6318 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
 
Theoremiotaint 6319 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
 
Theoremiota1 6320 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
 
Theoremiotanul 6321 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 6322 The class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.)
(℩𝑥𝜑) ⊆ {𝑥𝜑}
 
Theoremiotaex 6323 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 6324 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
 
Theoremiota4an 6325 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑)
 
Theoremiota5 6326* A method for computing iota. (Contributed by NM, 17-Sep-2013.)
((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))       ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
 
Theoremiotabidv 6327* Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))
 
Theoremiotabii 6328 Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝜑𝜓)       (℩𝑥𝜑) = (℩𝑥𝜓)
 
Theoremiotacl 6329 Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6302). If you have a bounded iota-based definition, riotacl2 7123 may be useful.

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

(∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})
 
Theoremiota2df 6330 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 6331* 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 6332* 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 6333* 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 6334 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
 
Theoremdfiota4 6335 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 6336* 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 6337 Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.)
wff Fun 𝐴
 
Syntaxwfn 6338 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 6339 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 6340 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 6341 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 6342 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 6343 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 6344 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 6345 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 15424). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5132 with the maps-to notation (see df-mpt 5133 and df-mpo 7154). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6346), a function with a given domain and codomain (df-f 6347), a one-to-one function (df-f1 6348), an onto function (df-fo 6349), or a one-to-one onto function (df-f1o 6350). For alternate definitions, see dffun2 6353, dffun3 6354, dffun4 6355, dffun5 6356, dffun6 6358, dffun7 6370, dffun8 6371, and dffun9 6372. (Contributed by NM, 1-Aug-1994.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
 
Definitiondf-fn 6346 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 6505, dffn3 6515, dffn4 6587, and dffn5 6715. (Contributed by NM, 1-Aug-1994.)
(𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
 
Definitiondf-f 6347 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 6856, dff3 6857, and dff4 6858. (Contributed by NM, 1-Aug-1994.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
 
Definitiondf-f1 6348 Define a one-to-one function. For equivalent definitions see dff12 6564 and dff13 7005. 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 17347. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
 
Definitiondf-fo 6349 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 6585, dffo3 6859, dffo4 6860, and dffo5 6861.

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 17348. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
 
Definitiondf-f1o 6350 Define a one-to-one onto function. For equivalent definitions see dff1o2 6611, dff1o3 6612, dff1o4 6614, and dff1o5 6615. 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 17351. Therefore, two sets are called "isomorphic" if there is a bijection between them. According to isof1oidb 7070, 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 8514. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
 
Definitiondf-fv 6351* Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 15503 after we define cosine in df-cos 15424). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5133 and df-mpo 7154), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 28231). Note that df-ov 7152 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 6691 and fvprc 6654). 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 6747, dffv3 6657, fv2 6656, and fv3 6679 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6741 and funfv2 6742. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6712. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 6658. (Revised by Scott Fenton, 6-Oct-2017.)
(𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
 
Definitiondf-isom 6352* 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 6353* Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
 
Theoremdffun3 6354* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
 
Theoremdffun4 6355* 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 6356* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
 
Theoremdffun6f 6357* 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 6358* Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)
(Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
 
Theoremfunmo 6359* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
(Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
 
Theoremfunrel 6360 A function is a relation. (Contributed by NM, 1-Aug-1994.)
(Fun 𝐴 → Rel 𝐴)
 
Theorem0nelfun 6361 A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)
(Fun 𝑅 → ∅ ∉ 𝑅)
 
Theoremfunss 6362 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 6363 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
(𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))
 
Theoremfuneqi 6364 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐴 = 𝐵       (Fun 𝐴 ↔ Fun 𝐵)
 
Theoremfuneqd 6365 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Fun 𝐴 ↔ Fun 𝐵))
 
Theoremnffun 6366 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
𝑥𝐹       𝑥Fun 𝐹
 
Theoremsbcfung 6367 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 6368* 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 6369* There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)
((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
 
Theoremdffun7 6370* 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 6371 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 6371* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 6370. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))
 
Theoremdffun9 6372* 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 6373 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 6374 A function is a function on its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑 → Fun 𝐴)       (𝜑𝐴 Fn dom 𝐴)
 
Theoremfuni 6375 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6464. (Contributed by NM, 30-Apr-1998.)
Fun I
 
Theoremnfunv 6376 The universal class is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
¬ Fun V
 
Theoremfunopg 6377 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 6393, as relsnopg 5663 is to relop 5708. (New usage is discouraged.)
((𝐴𝑉𝐵𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)
 
Theoremfunopab 6378* 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 6379* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
 
Theoremfunopab4 6380* A class of ordered pairs of values in the form used by df-mpt 5133 is a function. (Contributed by NM, 17-Feb-2013.)
Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)}
 
Theoremfunmpt 6381 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
Fun (𝑥𝐴𝐵)
 
Theoremfunmpt2 6382 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 6383 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 6384 Composition of two functions, generalization of funco 6383. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
 
Theoremfunres 6385 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 6386 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 6387 Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)
((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
 
Theoremfunun 6388 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 6389* 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 6390 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 6391 A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)
(Fun 𝐹 → Fun (𝐹𝐴))
 
Theoremfuncnvsn 6392 The converse singleton of an ordered pair is a function. This is equivalent to funsn 6395 via cnvsn 6070, 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 6393 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 6394 Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
 
Theoremfunsn 6395 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 6396 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 6397 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 6398 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 6399 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 6400 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}
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