P. 485 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48401-48500   *Has distinct variable group(s)

Type	Label	Description
Statement
21.52.4.3  Predicate "defined at"
Theorem	dfateq12d 48401	Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵))
Theorem	nfdfat 48402	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, ⊆, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
Theorem	dfdfat2 48403*	Alternate definition of the predicate "defined at" not using the Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
Theorem	fundmdfat 48404	A function is defined at any element of its domain. (Contributed by AV, 2-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)
Theorem	dfatprc 48405	A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.)
⊢ (¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)
Theorem	dfatelrn 48406	The value of a function 𝐹 at a set 𝐴 is in the range of the function 𝐹 if 𝐹 is defined at 𝐴. (Contributed by AV, 1-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) ∈ ran 𝐹)
21.52.4.4  Alternative definition of the value of a function
Theorem	dfafv2 48407	Alternative definition of (𝐹'''𝐴) using (𝐹‘𝐴) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.) (Revised by AV, 25-Aug-2022.)
⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
Theorem	afveq12d 48408	Equality deduction for function value, analogous to fveq12d 6915. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))
Theorem	afveq1 48409	Equality theorem for function value, analogous to fveq1 6907. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
⊢ (𝐹 = 𝐺 → (𝐹'''𝐴) = (𝐺'''𝐴))
Theorem	afveq2 48410	Equality theorem for function value, analogous to fveq1 6907. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
⊢ (𝐴 = 𝐵 → (𝐹'''𝐴) = (𝐹'''𝐵))
Theorem	nfafv 48411	Bound-variable hypothesis builder for function value, analogous to nffv 6918. To prove a deduction version of this analogous to nffvd 6920 is not easily possible because a deduction version of nfdfat 48402 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(𝐹'''𝐴)
Theorem	csbafv12g 48412	Move class substitution in and out of a function value, analogous to csbfv12 6953, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7487. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵))
Theorem	afvfundmfveq 48413	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) = (𝐹‘𝐴))
Theorem	afvnfundmuv 48414	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)
Theorem	ndmafv 48415	The value of a class outside its domain is the universe, compare with ndmfv 6940. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)
Theorem	afvvdm 48416	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) ∈ 𝐵 → 𝐴 ∈ dom 𝐹)
Theorem	nfunsnafv 48417	If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6947. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)
Theorem	afvvfunressn 48418	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}))
Theorem	afvprc 48419	A function's value at a proper class is the universe, compare with fvprc 6900. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (¬ 𝐴 ∈ V → (𝐹'''𝐴) = V)
Theorem	afvvv 48420	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) ∈ 𝐵 → 𝐴 ∈ V)
Theorem	afvpcfv0 48421	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 → (𝐹‘𝐴) = ∅)
Theorem	afvnufveq 48422	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) = (𝐹‘𝐴))
Theorem	afvvfveq 48423	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) = (𝐹‘𝐴))
Theorem	afv0fv0 48424	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) = ∅ → (𝐹‘𝐴) = ∅)
Theorem	afvfvn0fveq 48425	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.)
⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹'''𝐴) = (𝐹‘𝐴))
Theorem	afv0nbfvbi 48426	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 ↔ (𝐹‘𝐴) ∈ 𝐵))
Theorem	afvfv0bi 48427	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.)
⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))
Theorem	afveu 48428*	The value of a function at a unique point, analogous to fveu 6897. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
Theorem	fnbrafvb 48429	Equivalence of function value and binary relation, analogous to fnbrfvb 6958. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))
Theorem	fnopafvb 48430	Equivalence of function value and ordered pair membership, analogous to fnopfvb 6959. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
Theorem	funbrafvb 48431	Equivalence of function value and binary relation, analogous to funbrfvb 6961. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	funopafvb 48432	Equivalence of function value and ordered pair membership, analogous to funopfvb 6962. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Theorem	funbrafv 48433	The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6956. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))
Theorem	funbrafv2b 48434	Function value in terms of a binary relation, analogous to funbrfv2b 6965. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵)))
Theorem	dfafn5a 48435*	Representation of a function in terms of its values, analogous to dffn5 6966 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))
Theorem	dfafn5b 48436*	Representation of a function in terms of its values, analogous to dffn5 6966 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))))
Theorem	fnrnafv 48437*	The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6967. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)})
Theorem	afvelrnb 48438*	A member of a function's range is a value of the function, analogous to fvelrnb 6968 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
Theorem	afvelrnb0 48439*	A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6968. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
Theorem	dfaimafn 48440*	Alternate definition of the image of a function, analogous to dfimafn 6970. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦})
Theorem	dfaimafn2 48441*	Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6971. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)})
Theorem	afvelima 48442*	Function value in an image, analogous to fvelima 6973. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹'''𝑥) = 𝐴)
Theorem	afvelrn 48443	A function's value belongs to its range, analogous to fvelrn 7100. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹)
Theorem	fnafvelrn 48444	A function's value belongs to its range, analogous to fnfvelrn 7104. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)
Theorem	fafvelcdm 48445	A function's value belongs to its codomain, analogous to ffvelcdm 7105. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ 𝐵)
Theorem	ffnafv 48446*	A function maps to a class to which all values belong, analogous to ffnfv 7143. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵))
Theorem	afvres 48447	The value of a restricted function, analogous to fvres 6927. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))
Theorem	tz6.12-afv 48448*	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6932. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)
Theorem	tz6.12-1-afv 48449*	Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6931. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦)
Theorem	dmfcoafv 48450	Domains of a function composition, analogous to dmfco 7004. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))
Theorem	afvco2 48451	Value of a function composition, analogous to fvco2 7005. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))
Theorem	rlimdmafv 48452	Two ways to express that a function has a limit, analogous to rlimdm 15747. (Contributed by Alexander van der Vekens, 27-Nov-2017.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝𝑟 ↔ 𝐹 ⇝𝑟 ( ⇝𝑟 '''𝐹)))
21.52.4.5  Alternative definition of the value of an operation
Theorem	aoveq123d 48453	Equality deduction for operation value, analogous to oveq123d 7464. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )
Theorem	nfaov 48454	Bound-variable hypothesis builder for operation value, analogous to nfov 7473. To prove a deduction version of this analogous to nfovd 7472 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 48411). (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵))
Theorem	csbaovg 48455	Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌ ((𝐵𝐹𝐶)) = ((⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)) )
Theorem	aovfundmoveq 48456	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))
Theorem	aovnfundmuv 48457	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)
Theorem	ndmaov 48458	The value of an operation outside its domain, analogous to ndmafv 48415. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
Theorem	ndmaovg 48459	The value of an operation outside its domain, analogous to ndmovg 7628. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V)
Theorem	aovvdm 48460	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)
Theorem	nfunsnaov 48461	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)
Theorem	aovvfunressn 48462	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⟩}))
Theorem	aovprc 48463	The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7481. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Rel dom 𝐹    ⇒   ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)
Theorem	aovrcl 48464	Reverse closure for an operation value, analogous to afvvv 48420. In contrast to ovrcl 7484, 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))
Theorem	aovpcov0 48465	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) = ∅)
Theorem	aovnuoveq 48466	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))
Theorem	aovvoveq 48467	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))
Theorem	aov0ov0 48468	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) = ∅)
Theorem	aovovn0oveq 48469	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))
Theorem	aov0nbovbi 48470	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))
Theorem	aovov0bi 48471	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))
Theorem	rspceaov 48472*	A frequently used special case of rspc2ev 3592 for operation values, analogous to rspceov 7492. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) )
Theorem	fnotaovb 48473	Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6959. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ( ((𝐶𝐹𝐷)) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
Theorem	ffnaov 48474*	An operation maps to a class to which all values belong, analogous to ffnov 7569. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶))
Theorem	faovcl 48475	Closure law for an operation, analogous to fovcl 7571. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)
Theorem	aovmpt4g 48476*	Value of a function given by the maps-to notation, analogous to ovmpt4g 7591. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = 𝐶)
Theorem	aoprssdm 48477*	Domain of closure of an operation. In contrast to oprssdm 7626, no additional property for S (¬ ∅ ∈ 𝑆) is required! (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆)    ⇒   ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹
Theorem	ndmaovcl 48478	The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7630 where it is required that the domain contains the empty set (∅ ∈ 𝑆). (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    &   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)    &   ⊢ ((𝐴𝐹𝐵)) ∈ V    ⇒   ⊢ ((𝐴𝐹𝐵)) ∈ 𝑆
Theorem	ndmaovrcl 48479	Reverse closure law, in contrast to ndmovrcl 7631 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional asumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    ⇒   ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
Theorem	ndmaovcom 48480	Any operation is commutative outside its domain, analogous to ndmovcom 7632. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    ⇒   ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )
Theorem	ndmaovass 48481	Any operation is associative outside its domain. In contrast to ndmovass 7633 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    ⇒   ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) )
Theorem	ndmaovdistr 48482	Any operation is distributive outside its domain. In contrast to ndmovdistr 7634 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    &   ⊢ dom 𝐺 = (𝑆 × 𝑆)    ⇒   ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) )
21.52.5  Alternative definitions of function values (2) In the following, a second approach is followed to define function values alternately to df-afv 48396. The current definition of the value (𝐹‘𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6566) assures that this value is always a set, see fex 7255. 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 6940 and fvprc 6900). "(𝐹‘𝐴) is meaningful" means "the class 𝐹 regarded as function is defined at the argument 𝐴" in this context. This is also expressed by 𝐹 defAt 𝐴, see df-dfat 48395. In the theory of partial functions, it is a common case that 𝐹 is not defined at 𝐴. Although it is very convenient for many theorems on functions and their proofs, there are some cases in which from (𝐹‘𝐴) = ∅ alone it cannot be decided/derived whether (𝐹‘𝐴) 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, 𝐹 defAt 𝐴, or Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 (see, for example, ndmfvrcl 6941). To avoid such an ambiguity, an alternative definition (𝐹''''𝐴) (see df-afv2 48484) would be possible which evaluates to a set not belonging to the range of 𝐹 ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹) if it is not meaningful (see ndfatafv2 48486). We say "(𝐹''''𝐴) is not defined (or undefined)" if (𝐹''''𝐴) is not in the range of 𝐹 ((𝐹''''𝐴) ∉ ran 𝐹). Because of afv2ndefb 48499, this is equivalent to ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹. If (𝐹''''𝐴) is in the range of 𝐹 ((𝐹''''𝐴) ∈ ran 𝐹), we say that "(𝐹''''𝐴) is defined". If ran 𝐹 is a set, we can use the symbol Undef to express that (𝐹''''𝐴) is not defined: (𝐹''''𝐴) = (Undef‘ran 𝐹) (see ndfatafv2undef 48487). We could have used this symbol directly to define the alternate value of a function, which would have the advantage that (𝐹''''𝐴) would always be a set. But first this symbol is defined using the original function value, which would not make it possible to replace the original definition by the alternate definition, and second we would have to assume that ran 𝐹 ∈ V in most of the theorems. To summarize, that means (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹‘𝐴) = ∅ (see afv2ndeffv0 48535), but (𝐹‘𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 48540. The alternate definition, however, corresponds to the current definition ((𝐹‘𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 48536). With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 48502. An interesting question would be if (𝐹‘𝐴) could be replaced by (𝐹'''𝐴) in most of the theorems based on function values. If we look at the (currently 24) proofs using the definition df-fv 6566 of (𝐹‘𝐴), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6907-> afv2eq1 48491, fveq2 6908-> afv2eq2 48492, nffv 6918-> nfafv2 48493, csbfv12 6953-> csbafv212g , rlimdm 15747-> rlimdmafv2 48533, tz6.12-1 6931-> tz6.12-1-afv2 48516, fveu 6897-> afv2eu 48513. Six theorems proved by directly using df-fv 6566 are within a mathbox (fvsb 45625, uncov 8921) or not used (rlimdmafv 48452, avril1 31094) or experimental (dfafv2 48407, dfafv22 48534). However, the remaining 11 theorems proved by directly using df-fv 6566 are used more or less often: * fvex 6921: used in about 1600 proofs: Only if the function is defined at the argument, or the range of the function/class is a set, analog theorems can be proven (dfatafv2ex 48488 resp. afv2ex 48489). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6921. * fvres 6927: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 48514). In the undefined case such a theorem cannot exist (without additional assumptions), because the range of (𝐹 ↾ 𝐵) is mostly different from the range of 𝐹, and therefore also the "undefined" values are different. All of these 400 proofs have to be checked if afv2res 48514 can be used instead of fvres 6927. * tz6.12-2 6895 (-> tz6.12-2-afv2 48512): root theorem of many theorems which have not a strict analogue, and which are used many times: ** fvprc 6900 (-> afv2prc 48501), used in 193 proofs, ** tz6.12i 6934 (-> tz6.12i-afv2 48518), used - indirectly via fvbr0 6935 and fvrn0 6936 - in 19 proofs, and in fvclss 7268 used in fvclex 7995 used in fvresex 7996 (which is not used!) and in dcomex 10552 (used in 4 proofs), ** ndmfv 6940 (-> ndmafv2nrn ), used in 124 proofs ** nfunsn 6947 (-> nfunsnafv2 ), used by fvfundmfvn0 6948 (used in 3 proofs), and dffv2 7003 (not used) ** funpartfv 36887, setrec2lem1 10001 (mathboxes) * fv2 6903: only used by elfv 6906, which is only used by fv3 6926, which is not used. * dffv3 6904 (-> dfafv23 ): used by dffv4 6905 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 46072), by shftval 15256 (itself used in 11 proofs), by dffv5 36864 (mathbox) and by fvco2 7005 (-> afv2co2 48532). * fvopab5 7050: used only by ajval 31493 (not used) and by adjval 32522, which is used in adjval2 32523 (not used) and in adjbdln 32715 (used in 7 proofs). * zsum 15913: used (via isum 15914, sum0 15916, sumss 15919 and fsumsers 15923) in 76 proofs. * isumshft 16037: used in pserdv2 26787 (used in logtayl 27018, binomcxplemdvsum 45531) , eftlub 16306 (used in 4 proofs), binomcxplemnotnn0 45532 (used in binomcxp 45533 only) and logtayl 27018 (used in 4 proofs). * ovtpos 8286: used in 16 proofs. * zprod 16133: used in 3 proofs: iprod 16134, zprodn0 16135 and prodss 16143 * iprodclim3 16196: not used! 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 6903, dffv3 6904, fvopab5 7050, zsum 15913, isumshft 16037, ovtpos 8286 and zprod 16133 are not critical or are, hopefully, also valid for the alternative definition, fvex 6921, fvres 6927 and tz6.12-2 6895 (and the theorems based on them) are essential for the current definition of function values.
Syntax	cafv2 48483	Extend the definition of a class to include the alternate function value. Read: "the value of 𝐹 at 𝐴 " or "𝐹 of 𝐴". For using several apostrophes as a symbol see comment for cafv 48393.
class (𝐹''''𝐴)
Definition	df-afv2 48484*	Alternate definition of the value of a function, (𝐹''''𝐴), also known as function application (and called "alternate function value" in the following). In contrast to (𝐹‘𝐴) = ∅ (see comment of df-fv 6566, and especially ndmfv 6940), (𝐹''''𝐴) is guaranteed not to be in the range of 𝐹 if 𝐹 is not defined at 𝐴 (whereas ∅ can be a member of ran 𝐹). (Contributed by AV, 2-Sep-2022.)
⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
Theorem	dfatafv2iota 48485*	If a function is defined at a class 𝐴 the alternate function value at 𝐴 is the unique value assigned to 𝐴 by the function (analogously to (𝐹‘𝐴)). (Contributed by AV, 2-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
Theorem	ndfatafv2 48486	The alternate function value at a class 𝐴 if the function is not defined at this set 𝐴. (Contributed by AV, 2-Sep-2022.)
⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
Theorem	ndfatafv2undef 48487	The alternate function value at a class 𝐴 is undefined if the function, whose range is a set, is not defined at 𝐴. (Contributed by AV, 2-Sep-2022.)
⊢ ((ran 𝐹 ∈ 𝑉 ∧ ¬ 𝐹 defAt 𝐴) → (𝐹''''𝐴) = (Undef‘ran 𝐹))
Theorem	dfatafv2ex 48488	The alternate function value at a class 𝐴 is always a set if the function/class 𝐹 is defined at 𝐴. (Contributed by AV, 6-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)
Theorem	afv2ex 48489	The alternate function value is always a set if the range of the function is a set. (Contributed by AV, 2-Sep-2022.)
⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V)
Theorem	afv2eq12d 48490	Equality deduction for function value, analogous to fveq12d 6915. (Contributed by AV, 4-Sep-2022.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))
Theorem	afv2eq1 48491	Equality theorem for function value, analogous to fveq1 6907. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴))
Theorem	afv2eq2 48492	Equality theorem for function value, analogous to fveq2 6908. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵))
Theorem	nfafv2 48493	Bound-variable hypothesis builder for function value, analogous to nffv 6918. To prove a deduction version of this analogous to nffvd 6920 is not easily possible because a deduction version of nfdfat 48402 cannot be shown easily. (Contributed by AV, 4-Sep-2022.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(𝐹''''𝐴)
Theorem	csbafv212g 48494	Move class substitution in and out of a function value, analogous to csbfv12 6953, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7487. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵))
Theorem	fexafv2ex 48495	The alternate function value is always a set if the function (resp. the domain of the function) is a set. (Contributed by AV, 3-Sep-2022.)
⊢ (𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V)
Theorem	ndfatafv2nrn 48496	The alternate function value at a class 𝐴 at which the function is not defined is undefined, i.e., not in the range of the function. (Contributed by AV, 2-Sep-2022.)
⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	ndmafv2nrn 48497	The value of a class outside its domain is not in the range, compare with ndmfv 6940. (Contributed by AV, 2-Sep-2022.)
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	funressndmafv2rn 48498	The alternate function value at a class 𝐴 is defined, i.e., in the range of the function if the function is defined at 𝐴. (Contributed by AV, 2-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
Theorem	afv2ndefb 48499	Two ways to say that an alternate function value is not defined. (Contributed by AV, 5-Sep-2022.)
⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 ↔ (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	nfunsnafv2 48500	If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6947. (Contributed by AV, 2-Sep-2022.)
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)