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Theorem List for Metamath Proof Explorer - 41501-41600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremreuan 41501* Introduction of a conjunct into restricted uniqueness quantifier, analogous to euan 2559. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
𝑥𝜑       (∃!𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝐴 𝜓))

20.35.1.4  Analogs to Existential uniqueness (double quantification)

Theorem2reurex 41502* Double restricted quantification with existential uniqueness, analogous to 2euex 2573. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
(∃!𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)

Theorem2reurmo 41503* Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2574. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
(∃!𝑥𝐴 ∃*𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)

Theorem2reu2rex 41504* Double restricted existential uniqueness, analogous to 2eu2ex 2575. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)

Theorem2rmoswap 41505* A condition allowing swap of restricted "at most one" and restricted existential quantifiers, analogous to 2moswap 2576. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑦𝐵𝑥𝐴 𝜑))

Theorem2rexreu 41506* Double restricted existential uniqueness implies double restricted uniqueness quantification, analogous to 2exeu 2578. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)

Theorem2reu1 41507* Double restricted existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one, analogous to 2eu1 2582. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑)))

Theorem2reu2 41508* Double restricted existential uniqueness, analogous to 2eu2 2583. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
(∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃!𝑥𝐴𝑦𝐵 𝜑))

Theorem2reu3 41509* Double restricted existential uniqueness, analogous to 2eu3 2584. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
(∀𝑥𝐴𝑦𝐵 (∃*𝑥𝐴 𝜑 ∨ ∃*𝑦𝐵 𝜑) → ((∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑)))

Theorem2reu4a 41510* Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2585 with the additional requirement that the restricting classes are not empty (which is not necessary as shown in 2reu4 41511). (Contributed by Alexander van der Vekens, 1-Jul-2017.)
((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))))

Theorem2reu4 41511* Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2585. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

Theorem2reu7 41512* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2588. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))

Theorem2reu8 41513* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2589. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥𝐴∃!𝑦𝐵 using 2reu7 41512. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))

20.35.2  Alternative definitions of function and operation values

The current definition of the value (𝐹𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 5934) assures that this value is always a set, see fex 6530. This is because this definition can be applied to any classes 𝐹 and 𝐴, and evaluates to the empty set when it is not meaningful (as shown by ndmfv 6256 and fvprc 6223).

Although it is very convenient for many theorems on functions and their proofs, there are some cases in which from (𝐹𝐴) = ∅ alone it cannot be decided/derived whether (𝐹𝐴) is meaningful (𝐹 is actually a function which is defined for 𝐴 and really has the function value at 𝐴) or not. Therefore, additional assumptions are required, such as ∅ ∉ ran 𝐹, ∅ ∈ ran 𝐹 or Fun 𝐹𝐴 ∈ dom 𝐹 (see, for example, ndmfvrcl 6257).

To avoid such an ambiguity, an alternative definition (𝐹'''𝐴) (see df-afv 41518) would be possible which evaluates to the universal class ((𝐹'''𝐴) = V) if it is not meaningful (see afvnfundmuv 41540, ndmafv 41541, afvprc 41545 and nfunsnafv 41543), and which corresponds to the current definition ((𝐹𝐴) = (𝐹'''𝐴)) if it is (see afvfundmfveq 41539). That means (𝐹'''𝐴) = V → (𝐹𝐴) = ∅ (see afvpcfv0 41547), but (𝐹𝐴) = ∅ → (𝐹'''𝐴) = V is not generally valid.

In the theory of partial functions, it is a common case that 𝐹 is not defined at 𝐴, which also would result in (𝐹'''𝐴) = V. In this context we say (𝐹'''𝐴) "is not defined" instead of "is not meaningful".

With this definition the following intuitive equivalence holds: (𝐹'''𝐴) ∈ V <-> "(𝐹'''𝐴) is meaningful/defined".

An interesting question would be if (𝐹𝐴) could be replaced by (𝐹'''𝐴) in most of the theorems based on function values. If we look at the (currently 19) proofs using the definition df-fv 5934 of (𝐹𝐴), we see that analogues for the following 8 theorems can be proven using the alternative definition: fveq1 6228-> afveq1 41535, fveq2 6229-> afveq2 41536, nffv 6236-> nfafv 41537, csbfv12 6269-> csbafv12g , fvres 6245-> afvres 41573, rlimdm 14326-> rlimdmafv 41578, tz6.12-1 6248-> tz6.12-1-afv 41575, fveu 6221-> afveu 41554.

Three theorems proved by directly using df-fv 5934 are within a mathbox (fvsb 38973) or not used (isumclim3 14534, avril1 27449).

However, the remaining 8 theorems proved by directly using df-fv 5934 are used more or less often:

* fvex 6239: used in about 1750 proofs.

* tz6.12-1 6248: root theorem of many theorems which have not a strict analogue, and which are used many times: fvprc 6223 (used in about 127 proofs), tz6.12i 6252 (used - indirectly via fvbr0 6253 and fvrn0 6254- in 18 proofs, and in fvclss 6540 used in fvclex 7180 used in fvresex 7181, which is not used!), dcomex 9307 (used in 4 proofs), ndmfv 6256 (used in 86 proofs) and nfunsn 6263 (used by dffv2 6310 which is not used).

* fv2 6224: only used by elfv 6227, which is only used by fv3 6244, which is not used.

* dffv3 6225: used by dffv4 6226 (the previous "df-fv"), which now is only used in deprecated (usage discouraged) theorems or within mathboxes (csbfv12gALTVD 39449), by shftval 13858 (itself used in 9 proofs), by dffv5 32156 (mathbox) and by fvco2 6312, which has the analogue afvco2 41577.

* fvopab5 6349: used only by ajval 27845 (not used) and by adjval 28877 ( used - indirectly - in 9 proofs).

* zsum 14493: used (via isum 14494, sum0 14496 and fsumsers 14503) in more than 90 proofs.

* isumshft 14615: used in pserdv2 24229 and (via logtayl 24451) 4 other proofs.

* ovtpos 7412: used in 14 proofs.

As a result of this analysis we can say that the current definition of a function value is crucial for Metamath and cannot be exchanged easily with an alternative definition. While fv2 6224, dffv3 6225, fvopab5 6349, zsum 14493, isumshft 14615 and ovtpos 7412 are not critical or are, hopefully, also valid for the alternative definition, fvex 6239 and tz6.12-1 6248 (and the theorems based on them) are essential for the current definition of function values.

With the same arguments, an alternative definition of operation values ((𝐴𝑂𝐵)) could be meaningful to avoid ambiguities, see df-aov 41519.

Syntaxwdfat 41514 Extend the definition of a wff to include the "defined at" predicate. (Read: (The Function) 𝐹 is defined at (the argument) 𝐴). In a previous version, the token "def@" was used. However, since the @ is used (informally) as a replacement for \$ in commented out sections that may be deleted some day. While there is no violation of any standard to use the @ in a token, it could make the search for such commented-out sections slightly more difficult. (See remark of Norman Megill at https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4).
wff 𝐹 defAt 𝐴

Syntaxcafv 41515 Extend the definition of a class to include the value of a function. (Read: The value of 𝐹 at 𝐴, or "𝐹 of 𝐴."). In a previous version, the symbol " ' " was used. However, since the similarity with the symbol used for the current definition of a function's value (see df-fv 5934), which, by the way, was intended to visualize that in many cases and " ' " are exchangeable, makes reading the theorems, especially those which uses both definitions as dfafv2 41533, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 5934 and df-ima 5156. And not three backticks ( three times ) since that would be annoying to escape in a comment. (See remark of Norman Megill and Gerard Lang at https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4).
class (𝐹'''𝐴)

Syntaxcaov 41516 Extend class notation to include the value of an operation 𝐹 (such as +) for two arguments 𝐴 and 𝐵. Note that the syntax is simply three class symbols in a row surrounded by a pair of parentheses in contrast to the current definition, see df-ov 6693.
class ((𝐴𝐹𝐵))

Definitiondf-dfat 41517 Definition of the predicate that determines if some class 𝐹 is defined as function for an argument 𝐴 or, in other words, if the function value for some class 𝐹 for an argument 𝐴 is defined. We say that 𝐹 is defined at 𝐴 if a 𝐹 is a function restricted to the member 𝐴 of its domain. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

Definitiondf-afv 41518* Alternative definition of the value of a function, (𝐹'''𝐴), also known as function application. In contrast to (𝐹𝐴) = ∅ (see df-fv 5934 and ndmfv 6256), (𝐹'''𝐴) = V if F is not defined for A! (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹'''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), V)

Definitiondf-aov 41519 Define the value of an operation. In contrast to df-ov 6693, the alternative definition for a function value (see df-afv 41518) is used. By this, the value of the operation applied to two arguments is the universal class if the operation is not defined for these two arguments. There are still no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation 𝐹 and its arguments 𝐴 and 𝐵- will be useful for proving meaningful theorems. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)

20.35.2.1  Restricted quantification (extension)

Theoremralbinrald 41520* Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)
(𝜑𝑋𝐴)    &   (𝑥𝐴𝑥 = 𝑋)    &   (𝑥 = 𝑋 → (𝜓𝜃))       (𝜑 → (∀𝑥𝐴 𝜓𝜃))

20.35.2.2  The universal class (extension)

Theoremnvelim 41521 If a class is the universal class it doesn't belong to any class, generalisation of nvel 4830. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐴 = V → ¬ 𝐴𝐵)

20.35.2.3  Introduce the Axiom of Power Sets (extension)

Theoremalneu 41522 If a statement holds for all sets, there is not a unique set for which the statement holds. (Contributed by Alexander van der Vekens, 28-Nov-2017.)
(∀𝑥𝜑 → ¬ ∃!𝑥𝜑)

Theoremeu2ndop1stv 41523* If there is a unique second component in an ordered pair contained in a given set, the first component must be a set. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
(∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V)

20.35.2.4  Relations (extension)

Theoremeldmressn 41524 Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
(𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴)

20.35.2.5  Functions (extension)

Theoremfveqvfvv 41525 If a function's value at an argument is the universal class (which can never be the case because of fvex 6239), the function's value at this argument is any set (especially the empty set). In short "If a function's value is a proper class, it is a set", which sounds strange/contradictory, but which is a consequence of that a contradiction implies anything (see pm2.21i 116). (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐹𝐴) = V → (𝐹𝐴) = 𝐵)

Theoremfunresfunco 41526 Composition of two functions, generalization of funco 5966. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))

Theoremfnresfnco 41527 Composition of two functions, similar to fnco 6037. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
(((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → (𝐹𝐺) Fn 𝐵)

Theoremfuncoressn 41528 A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))

Theoremfunressnfv 41529 A restriction to a singleton with a function value is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
(((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun (𝐹 ↾ {(𝐺𝑋)}))

20.35.2.6  Predicate "defined at"

Theoremdfateq12d 41530 Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))

Theoremnfdfat 41531 Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g., for Fun/Rel, dom, C_, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐹    &   𝑥𝐴       𝑥 𝐹 defAt 𝐴

Theoremdfdfat2 41532* Alternate definition of the predicate "defined at" not using the Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))

20.35.2.7  Alternative definition of the value of a function

Theoremdfafv2 41533 Alternative definition of (𝐹'''𝐴) using (𝐹𝐴) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)

Theoremafveq12d 41534 Equality deduction for function value, analogous to fveq12d 6235. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))

Theoremafveq1 41535 Equality theorem for function value, analogous to fveq1 6228. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹 = 𝐺 → (𝐹'''𝐴) = (𝐺'''𝐴))

Theoremafveq2 41536 Equality theorem for function value, analogous to fveq1 6228. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴 = 𝐵 → (𝐹'''𝐴) = (𝐹'''𝐵))

Theoremnfafv 41537 Bound-variable hypothesis builder for function value, analogous to nffv 6236. To prove a deduction version of this analogous to nffvd 6238 is not easily possible because a deduction version of nfdfat 41531 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹'''𝐴)

Theoremcsbafv12g 41538 Move class substitution in and out of a function value, analogous to csbfv12 6269, with a direct proof proposed by Mario Carneiro, analogous to csbov123 6727. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))

Theoremafvfundmfveq 41539 If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafvnfundmuv 41540 If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)

Theoremndmafv 41541 The value of a class outside its domain is the universe, compare with ndmfv 6256. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)

Theoremafvvdm 41542 If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ dom 𝐹)

Theoremnfunsnafv 41543 If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6263. (Contributed by Alexander van der Vekens, 25-May-2017.)
(¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)

Theoremafvvfunressn 41544 If the function value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵 → Fun (𝐹 ↾ {𝐴}))

Theoremafvprc 41545 A function's value at a proper class is the universe, compare with fvprc 6223. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ V → (𝐹'''𝐴) = V)

Theoremafvvv 41546 If a function's value at an argument is a set, the argument is also a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ V)

Theoremafvpcfv0 41547 If the value of the alternative function at an argument is the universe, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)

Theoremafvnufveq 41548 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafvvfveq 41549 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵 → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafv0fv0 41550 If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅)

Theoremafvfvn0fveq 41551 If the function's value at an argument is not the empty set, it equals the value of the alternative function at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹𝐴) ≠ ∅ → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafv0nbfvbi 41552 The function's value at an argument is an element of a set if and only if the value of the alternative function at this argument is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
(∅ ∉ 𝐵 → ((𝐹'''𝐴) ∈ 𝐵 ↔ (𝐹𝐴) ∈ 𝐵))

Theoremafvfv0bi 41553 The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))

Theoremafveu 41554* The value of a function at a unique point, analogous to fveu 6221. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})

Theoremfnbrafvb 41555 Equivalence of function value and binary relation, analogous to fnbrfvb 6274. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶𝐵𝐹𝐶))

Theoremfnopafvb 41556 Equivalence of function value and ordered pair membership, analogous to fnopfvb 6275. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))

Theoremfunbrafvb 41557 Equivalence of function value and binary relation, analogous to funbrfvb 6276. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵𝐴𝐹𝐵))

Theoremfunopafvb 41558 Equivalence of function value and ordered pair membership, analogous to funopfvb 6277. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))

Theoremfunbrafv 41559 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6272. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))

Theoremfunbrafv2b 41560 Function value in terms of a binary relation, analogous to funbrfv2b 6279. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵)))

Theoremdfafn5a 41561* Representation of a function in terms of its values, analogous to dffn5 6280 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))

Theoremdfafn5b 41562* Representation of a function in terms of its values, analogous to dffn5 6280 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)
(∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥))))

Theoremfnrnafv 41563* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6281. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})

Theoremafvelrnb 41564* A member of a function's range is a value of the function, analogous to fvelrnb 6282 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))

Theoremafvelrnb0 41565* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6282. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
(𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))

Theoremdfaimafn 41566* Alternate definition of the image of a function, analogous to dfimafn 6284. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})

Theoremdfaimafn2 41567* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6285. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})

Theoremafvelima 41568* Function value in an image, analogous to fvelima 6287. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → ∃𝑥𝐵 (𝐹'''𝑥) = 𝐴)

Theoremafvelrn 41569 A function's value belongs to its range, analogous to fvelrn 6392. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹)

Theoremfnafvelrn 41570 A function's value belongs to its range, analogous to fnfvelrn 6396. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)

Theoremfafvelrn 41571 A function's value belongs to its codomain, analogous to ffvelrn 6397. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ 𝐵)

Theoremffnafv 41572* A function maps to a class to which all values belong, analogous to ffnfv 6428. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))

Theoremafvres 41573 The value of a restricted function, analogous to fvres 6245. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴𝐵 → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))

Theoremtz6.12-afv 41574* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6249. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)

Theoremtz6.12-1-afv 41575* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6248. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦)

Theoremdmfcoafv 41576 Domains of a function composition, analogous to dmfco 6311. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))

Theoremafvco2 41577 Value of a function composition, analogous to fvco2 6312. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))

Theoremrlimdmafv 41578 Two ways to express that a function has a limit, analogous to rlimdm 14326. (Contributed by Alexander van der Vekens, 27-Nov-2017.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)       (𝜑 → (𝐹 ∈ dom ⇝𝑟𝐹𝑟 ( ⇝𝑟 '''𝐹)))

20.35.2.8  Alternative definition of the value of an operation

Theoremaoveq123d 41579 Equality deduction for operation value, analogous to oveq123d 6711. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

Theoremnfaov 41580 Bound-variable hypothesis builder for operation value, analogous to nfov 6716. To prove a deduction version of this analogous to nfovd 6715 is not quickly possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of alternative operation values is based on are not available (see nfafv 41537). (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐵       𝑥 ((𝐴𝐹𝐵))

Theoremcsbaovg 41581 Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐴𝐷𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )

Theoremaovfundmoveq 41582 If a class is a function restricted to an ordered pair of its domain, then the value of the operation on this pair is equal for both definitions. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐹 defAt ⟨𝐴, 𝐵⟩ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Theoremaovnfundmuv 41583 If an ordered pair is not in the domain of a class or the class is not a function restricted to the ordered pair, then the operation value for this pair is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹 defAt ⟨𝐴, 𝐵⟩ → ((𝐴𝐹𝐵)) = V)

Theoremndmaov 41584 The value of an operation outside its domain, analogous to ndmafv 41541. (Contributed by Alexander van der Vekens, 26-May-2017.)
(¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)

Theoremndmaovg 41585 The value of an operation outside its domain, analogous to ndmovg 6859. (Contributed by Alexander van der Vekens, 26-May-2017.)
((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → ((𝐴𝐹𝐵)) = V)

Theoremaovvdm 41586 If the operation value of a class for an ordered pair is a set, the ordered pair is contained in the domain of the class. (Contributed by Alexander van der Vekens, 26-May-2017.)
( ((𝐴𝐹𝐵)) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)

Theoremnfunsnaov 41587 If the restriction of a class to a singleton is not a function, its operation value is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
(¬ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}) → ((𝐴𝐹𝐵)) = V)

Theoremaovvfunressn 41588 If the operation value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)
( ((𝐴𝐹𝐵)) ∈ 𝐶 → Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}))

Theoremaovprc 41589 The value of an operation when the one of the arguments is a proper class, analogous to ovprc 6723. (Contributed by Alexander van der Vekens, 26-May-2017.)
Rel dom 𝐹       (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)

Theoremaovrcl 41590 Reverse closure for an operation value, analogous to afvvv 41546. In contrast to ovrcl 6726, elementhood of the operation's value in a set is required, not containing an element. (Contributed by Alexander van der Vekens, 26-May-2017.)
Rel dom 𝐹       ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theoremaovpcov0 41591 If the alternative value of the operation on an ordered pair is the universal class, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
( ((𝐴𝐹𝐵)) = V → (𝐴𝐹𝐵) = ∅)

Theoremaovnuoveq 41592 The alternative value of the operation on an ordered pair equals the operation's value at this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
( ((𝐴𝐹𝐵)) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Theoremaovvoveq 41593 The alternative value of the operation on an ordered pair equals the operation's value on this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
( ((𝐴𝐹𝐵)) ∈ 𝐶 → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Theoremaov0ov0 41594 If the alternative value of the operation on an ordered pair is the empty set, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
( ((𝐴𝐹𝐵)) = ∅ → (𝐴𝐹𝐵) = ∅)

Theoremaovovn0oveq 41595 If the operation's value at an argument is not the empty set, it equals the value of the alternative operation at this argument. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐴𝐹𝐵) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Theoremaov0nbovbi 41596 The operation's value on an ordered pair is an element of a set if and only if the alternative value of the operation on this ordered pair is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
(∅ ∉ 𝐶 → ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶))

Theoremaovov0bi 41597 The operation's value on an ordered pair is the empty set if and only if the alternative value of the operation on this ordered pair is either the empty set or the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐴𝐹𝐵) = ∅ ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))

Theoremrspceaov 41598* A frequently used special case of rspc2ev 3355 for operation values, analogous to rspceov 6732. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐶𝐴𝐷𝐵𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥𝐴𝑦𝐵 𝑆 = ((𝑥𝐹𝑦)) )

Theoremfnotaovb 41599 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6275. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ( ((𝐶𝐹𝐷)) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))

Theoremffnaov 41600* An operation maps to a class to which all values belong, analogous to ffnov 6806. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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