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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | iota1 6301 | Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | ||
Theorem | iotanul 6302 | 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) = ∅) | ||
Theorem | iotassuni 6303 | The ℩ class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑} | ||
Theorem | iotaex 6304 | 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 | ||
Theorem | iota4 6305 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑) | ||
Theorem | iota4an 6306 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑) | ||
Theorem | iota5 6307* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴) | ||
Theorem | iotabidv 6308* | Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒)) | ||
Theorem | iotabii 6309 | Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (℩𝑥𝜑) = (℩𝑥𝜓) | ||
Theorem | iotacl 6310 |
Membership law for descriptions.
This can be useful for expanding an unbounded iota-based definition (see df-iota 6283). If you have a bounded iota-based definition, riotacl2 7109 may be useful. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | iota2df 6311 | 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 6312* | 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 | iota2 6313* | 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 | iotan0 6314* | 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓) | ||
Theorem | sniota 6315 | 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 | dfiota4 6316 | The ℩ operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017.) (Proof shortened by JJ, 28-Oct-2021.) |
⊢ (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) | ||
Theorem | csbiota 6317* | 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) | ||
Syntax | wfun 6318 | Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.) |
wff Fun 𝐴 | ||
Syntax | wfn 6319 | 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 6320 | 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 6321 | 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 6322 | 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 6323 | 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 6324 | 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 6325 | 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 6326 | 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 15416). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5110 with the maps-to notation (see df-mpt 5111 and df-mpo 7140). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6327), a function with a given domain and codomain (df-f 6328), a one-to-one function (df-f1 6329), an onto function (df-fo 6330), or a one-to-one onto function (df-f1o 6331). For alternate definitions, see dffun2 6334, dffun3 6335, dffun4 6336, dffun5 6337, dffun6 6339, dffun7 6351, dffun8 6352, and dffun9 6353. (Contributed by NM, 1-Aug-1994.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) | ||
Definition | df-fn 6327 | Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 6489, dffn3 6499, dffn4 6571, and dffn5 6699. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵)) | ||
Definition | df-f 6328 | Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. 𝐹:𝐴⟶𝐵 can be read as "𝐹 is a function from 𝐴 to 𝐵". For alternate definitions, see dff2 6842, dff3 6843, and dff4 6844. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | ||
Definition | df-f1 6329 |
Define a one-to-one function. For equivalent definitions see dff12 6548
and dff13 6991. 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:𝐴–1-1→𝐵 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 17339. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | ||
Definition | df-fo 6330 |
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 6569, dffo3 6845, dffo4 6846, and dffo5 6847.
An onto function is also called a "surjection" or a "surjective function", 𝐹:𝐴–onto→𝐵 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 17340. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | ||
Definition | df-f1o 6331 |
Define a one-to-one onto function. For equivalent definitions see
dff1o2 6595, dff1o3 6596, dff1o4 6598, and dff1o5 6599. 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:𝐴–1-1-onto→𝐵 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 17343. Therefore, two sets are called "isomorphic" if there is a bijection between them. According to isof1oidb 7056, 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 8501. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | ||
Definition | df-fv 6332* | Define the value of a function, (𝐹‘𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 15495 after we define cosine in df-cos 15416). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5111 and df-mpo 7140), but this is not required. For example, 𝐹 = {〈2, 6〉, 〈3, 9〉} → (𝐹‘3) = 9 (ex-fv 28228). Note that df-ov 7138 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 6675 and fvprc 6638). 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 6733, dffv3 6641, fv2 6640, and fv3 6663 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6725 and funfv2 6726. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6696. (Contributed by NM, 1-Aug-1994.) Revised to use ℩. Original version is now theorem dffv4 6642. (Revised by Scott Fenton, 6-Oct-2017.) |
⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | ||
Definition | df-isom 6333* | 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 6334* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))) | ||
Theorem | dffun3 6335* | Alternate definition of function. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) | ||
Theorem | dffun4 6336* | 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 | dffun5 6337* | Alternate definition of function. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧))) | ||
Theorem | dffun6f 6338* | 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 6339* | 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 6340* | A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) |
⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) | ||
Theorem | funrel 6341 | A function is a relation. (Contributed by NM, 1-Aug-1994.) |
⊢ (Fun 𝐴 → Rel 𝐴) | ||
Theorem | 0nelfun 6342 | A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.) |
⊢ (Fun 𝑅 → ∅ ∉ 𝑅) | ||
Theorem | funss 6343 | 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 6344 | Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵)) | ||
Theorem | funeqi 6345 | Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (Fun 𝐴 ↔ Fun 𝐵) | ||
Theorem | funeqd 6346 | Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (Fun 𝐴 ↔ Fun 𝐵)) | ||
Theorem | nffun 6347 | Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ Ⅎ𝑥Fun 𝐹 | ||
Theorem | sbcfung 6348 | 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 6349* | 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 6350* | 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 6351* | 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 6352 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)) | ||
Theorem | dffun8 6352* | Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 6351. (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 6353* | 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 6354 | A class is a function if and only if it is a function on its domain. (Contributed by NM, 13-Aug-2004.) |
⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴) | ||
Theorem | funfnd 6355 | A function is a function on its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → Fun 𝐴) ⇒ ⊢ (𝜑 → 𝐴 Fn dom 𝐴) | ||
Theorem | funi 6356 | The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6447. (Contributed by NM, 30-Apr-1998.) |
⊢ Fun I | ||
Theorem | nfunv 6357 | The universal class is not a function. (Contributed by Raph Levien, 27-Jan-2004.) |
⊢ ¬ Fun V | ||
Theorem | funopg 6358 | 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 6375, as relsnopg 5640 is to relop 5685. (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ Fun 〈𝐴, 𝐵〉) → 𝐴 = 𝐵) | ||
Theorem | funopab 6359* | 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 6360* | A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.) |
⊢ Fun {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐴} | ||
Theorem | funopab4 6361* | A class of ordered pairs of values in the form used by df-mpt 5111 is a function. (Contributed by NM, 17-Feb-2013.) |
⊢ Fun {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐴)} | ||
Theorem | funmpt 6362 | A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.) |
⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | funmpt2 6363 | 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 6364 | 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 | funresfunco 6365 | Composition of two functions, generalization of funco 6364. (Contributed by Alexander van der Vekens, 25-Jul-2017.) |
⊢ ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | ||
Theorem | funres 6366 | 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 | funresd 6367 | A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴)) | ||
Theorem | funssres 6368 | 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 6369 | 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 6370 | 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 6371* | 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 6372 | 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 6373 | A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.) |
⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴)) | ||
Theorem | funcnvsn 6374 | The converse singleton of an ordered pair is a function. This is equivalent to funsn 6377 via cnvsn 6050, 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 6375 | 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 6376 | 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 6377 | 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 | funprg 6378 | 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.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) | ||
Theorem | funtpg 6379 | 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.) |
⊢ (((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 6380 | 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 6381 | 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 6382 | 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 6383 | 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 6384 | 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 6385 | 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 | funcnvpr 6386 | 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.) |
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → Fun ◡{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉}) | ||
Theorem | funcnvtp 6387 | 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.) |
⊢ (((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹)) → Fun ◡{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}) | ||
Theorem | funcnvqp 6388 | 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.) |
⊢ ((((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉) ∧ (𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻) ∧ (𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻) ∧ 𝐹 ≠ 𝐻)) → Fun ◡({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉})) | ||
Theorem | fun0 6389 | The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.) |
⊢ Fun ∅ | ||
Theorem | funcnv0 6390 | The converse of the empty set is a function. (Contributed by AV, 7-Jan-2021.) |
⊢ Fun ◡∅ | ||
Theorem | funcnvcnv 6391 | The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.) |
⊢ (Fun 𝐴 → Fun ◡◡𝐴) | ||
Theorem | funcnv2 6392* | A simpler equivalence for single-rooted (see funcnv 6393). (Contributed by NM, 9-Aug-2004.) |
⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) | ||
Theorem | funcnv 6393* | 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 6392 for a simpler version. (Contributed by NM, 13-Aug-2004.) |
⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦) | ||
Theorem | funcnv3 6394* | A condition showing a class is single-rooted. (See funcnv 6393). (Contributed by NM, 26-May-2006.) |
⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦) | ||
Theorem | fun2cnv 6395* | 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 6396 | A single-valued relation is a function. (See fun2cnv 6395 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 6397* | Single-rootedness (see funcnv 6393) of a class cut down by a Cartesian product. (Contributed by NM, 5-Mar-2007.) |
⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦) | ||
Theorem | fun11 6398* | 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 6399* | 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 | funin 6400 | 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)) |
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