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Theorem List for Metamath Proof Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremunizlim 6301 An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
(Ord 𝐴 → (𝐴 = 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
 
Theoremon0eqel 6302 An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.)
(𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
 
Theoremsnsn0non 6303 The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7572). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6304. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
¬ {{∅}} ∈ On
 
Theoremonxpdisj 6304 Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 6303. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(On ∩ (V × V)) = ∅
 
Theoremonnev 6305 The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.)
On ≠ V
 
2.3.14  Definite description binder (inverted iota)
 
Syntaxcio 6306 Extend class notation with Russell's definition description binder (inverted iota).
class (℩𝑥𝜑)
 
Theoremiotajust 6307* Soundness justification theorem for df-iota 6308. (Contributed by Andrew Salmon, 29-Jun-2011.)
{𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
 
Definitiondf-iota 6308* 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 6323); otherwise, it evaluates to the empty set (see iotanul 6327). 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 7119 (or iotacl 6335 for unbounded iota), as demonstrated in the proof of supub 8912. This can be easier than applying riotasbc 7121 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 6309* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
(℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
 
Theoremnfiota1 6310 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥(℩𝑥𝜑)
 
Theoremnfiotadw 6311* Version of nfiotad 6313 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥(℩𝑦𝜓))
 
Theoremnfiotaw 6312* Version of nfiota 6314 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
𝑥𝜑       𝑥(℩𝑦𝜑)
 
Theoremnfiotad 6313 Deduction version of nfiota 6314. (Contributed by NM, 18-Feb-2013.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥(℩𝑦𝜓))
 
Theoremnfiota 6314 Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.)
𝑥𝜑       𝑥(℩𝑦𝜑)
 
Theoremcbviotaw 6315* Version of cbviota 6317 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝑦𝜑    &   𝑥𝜓       (℩𝑥𝜑) = (℩𝑦𝜓)
 
Theoremcbviotavw 6316* Version of cbviotav 6318 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (℩𝑥𝜑) = (℩𝑦𝜓)
 
Theoremcbviota 6317 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝑦𝜑    &   𝑥𝜓       (℩𝑥𝜑) = (℩𝑦𝜓)
 
Theoremcbviotav 6318* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))       (℩𝑥𝜑) = (℩𝑦𝜓)
 
Theoremsb8iota 6319 Variable substitution in description binder. Compare sb8eu 2682. (Contributed by NM, 18-Mar-2013.)
𝑦𝜑       (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)
 
Theoremiotaeq 6320 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
(∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))
 
Theoremiotabi 6321 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
(∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))
 
Theoremuniabio 6322* 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 6323* 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 6324 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
 
Theoremiotaint 6325 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
 
Theoremiota1 6326 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
 
Theoremiotanul 6327 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 6328 The class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.)
(℩𝑥𝜑) ⊆ {𝑥𝜑}
 
Theoremiotaex 6329 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 6330 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
 
Theoremiota4an 6331 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑)
 
Theoremiota5 6332* A method for computing iota. (Contributed by NM, 17-Sep-2013.)
((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))       ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
 
Theoremiotabidv 6333* Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))
 
Theoremiotabii 6334 Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝜑𝜓)       (℩𝑥𝜑) = (℩𝑥𝜓)
 
Theoremiotacl 6335 Membership law for descriptions.

This can be useful for expanding an unbounded iota-based definition (see df-iota 6308). If you have a bounded iota-based definition, riotacl2 7119 may be useful.

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

(∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})
 
Theoremiota2df 6336 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 6337* 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 6338* 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 6339* 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 6340 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})
 
Theoremdfiota4 6341 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 6342* 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 6343 Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.)
wff Fun 𝐴
 
Syntaxwfn 6344 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 6345 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 6346 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 6347 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 6348 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 6349 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 6350 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 6351 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 15414). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5138 with the maps-to notation (see df-mpt 5139 and df-mpo 7150). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6352), a function with a given domain and codomain (df-f 6353), a one-to-one function (df-f1 6354), an onto function (df-fo 6355), or a one-to-one onto function (df-f1o 6356). For alternate definitions, see dffun2 6359, dffun3 6360, dffun4 6361, dffun5 6362, dffun6 6364, dffun7 6376, dffun8 6377, and dffun9 6378. (Contributed by NM, 1-Aug-1994.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
 
Definitiondf-fn 6352 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 6510, dffn3 6519, dffn4 6590, and dffn5 6718. (Contributed by NM, 1-Aug-1994.)
(𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
 
Definitiondf-f 6353 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 6858, dff3 6859, and dff4 6860. (Contributed by NM, 1-Aug-1994.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
 
Definitiondf-f1 6354 Define a one-to-one function. For equivalent definitions see dff12 6568 and dff13 7004. 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 17337. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
 
Definitiondf-fo 6355 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 6588, dffo3 6861, dffo4 6862, and dffo5 6863.

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 17338. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
 
Definitiondf-f1o 6356 Define a one-to-one onto function. For equivalent definitions see dff1o2 6614, dff1o3 6615, dff1o4 6617, and dff1o5 6618. 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 17341. Therefore, two sets are called "isomorphic" if there is a bijection between them. According to isof1oidb 7066, 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 8507. (Contributed by NM, 1-Aug-1994.)

(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
 
Definitiondf-fv 6357* Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 15493 after we define cosine in df-cos 15414). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5139 and df-mpo 7150), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 28150). Note that df-ov 7148 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 6694 and fvprc 6657). 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 6750, dffv3 6660, fv2 6659, and fv3 6682 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6744 and funfv2 6745. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6715. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 6661. (Revised by Scott Fenton, 6-Oct-2017.)
(𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
 
Definitiondf-isom 6358* 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 6359* Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
 
Theoremdffun3 6360* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
 
Theoremdffun4 6361* 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 6362* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
 
Theoremdffun6f 6363* 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 6364* Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)
(Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
 
Theoremfunmo 6365* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
(Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
 
Theoremfunrel 6366 A function is a relation. (Contributed by NM, 1-Aug-1994.)
(Fun 𝐴 → Rel 𝐴)
 
Theorem0nelfun 6367 A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)
(Fun 𝑅 → ∅ ∉ 𝑅)
 
Theoremfunss 6368 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 6369 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
(𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))
 
Theoremfuneqi 6370 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐴 = 𝐵       (Fun 𝐴 ↔ Fun 𝐵)
 
Theoremfuneqd 6371 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Fun 𝐴 ↔ Fun 𝐵))
 
Theoremnffun 6372 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
𝑥𝐹       𝑥Fun 𝐹
 
Theoremsbcfung 6373 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 6374* 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 6375* There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)
((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
 
Theoremdffun7 6376* 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 6377 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 6377* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 6376. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))
 
Theoremdffun9 6378* 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 6379 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 6380 A function is a function on its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑 → Fun 𝐴)       (𝜑𝐴 Fn dom 𝐴)
 
Theoremfuni 6381 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6469. (Contributed by NM, 30-Apr-1998.)
Fun I
 
Theoremnfunv 6382 The universal class is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
¬ Fun V
 
Theoremfunopg 6383 A Kuratowski ordered pair of sets is a function only if its components are equal. Compare with funsng 6399. (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 6384* 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 6385* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
 
Theoremfunopab4 6386* A class of ordered pairs of values in the form used by df-mpt 5139 is a function. (Contributed by NM, 17-Feb-2013.)
Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)}
 
Theoremfunmpt 6387 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
Fun (𝑥𝐴𝐵)
 
Theoremfunmpt2 6388 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 6389 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 6390 Composition of two functions, generalization of funco 6389. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))
 
Theoremfunres 6391 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 6392 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 6393 Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)
((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
 
Theoremfunun 6394 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 6395* 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 6396 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 6397 A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)
(Fun 𝐹 → Fun (𝐹𝐴))
 
Theoremfuncnvsn 6398 The converse singleton of an ordered pair is a function. This is equivalent to funsn 6401 via cnvsn 6077, 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 6399 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 6400 Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
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