P. 467 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 46601-46700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fnrnafv 46601*	The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6951. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)})
Theorem	afvelrnb 46602*	A member of a function's range is a value of the function, analogous to fvelrnb 6952 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 46603*	A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6952. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
Theorem	dfaimafn 46604*	Alternate definition of the image of a function, analogous to dfimafn 6954. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦})
Theorem	dfaimafn2 46605*	Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6955. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)})
Theorem	afvelima 46606*	Function value in an image, analogous to fvelima 6957. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹'''𝑥) = 𝐴)
Theorem	afvelrn 46607	A function's value belongs to its range, analogous to fvelrn 7079. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹)
Theorem	fnafvelrn 46608	A function's value belongs to its range, analogous to fnfvelrn 7083. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)
Theorem	fafvelcdm 46609	A function's value belongs to its codomain, analogous to ffvelcdm 7084. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ 𝐵)
Theorem	ffnafv 46610*	A function maps to a class to which all values belong, analogous to ffnfv 7122. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵))
Theorem	afvres 46611	The value of a restricted function, analogous to fvres 6909. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))
Theorem	tz6.12-afv 46612*	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6915. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)
Theorem	tz6.12-1-afv 46613*	Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6913. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦)
Theorem	dmfcoafv 46614	Domains of a function composition, analogous to dmfco 6987. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))
Theorem	afvco2 46615	Value of a function composition, analogous to fvco2 6988. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))
Theorem	rlimdmafv 46616	Two ways to express that a function has a limit, analogous to rlimdm 15522. (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)))
21.45.4.6  Alternative definition of the value of an operation
Theorem	aoveq123d 46617	Equality deduction for operation value, analogous to oveq123d 7434. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )
Theorem	nfaov 46618	Bound-variable hypothesis builder for operation value, analogous to nfov 7443. To prove a deduction version of this analogous to nfovd 7442 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 46575). (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵))
Theorem	csbaovg 46619	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 46620	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 46621	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 46622	The value of an operation outside its domain, analogous to ndmafv 46579. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
Theorem	ndmaovg 46623	The value of an operation outside its domain, analogous to ndmovg 7598. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V)
Theorem	aovvdm 46624	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 46625	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 46626	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 46627	The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7451. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Rel dom 𝐹    ⇒   ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)
Theorem	aovrcl 46628	Reverse closure for an operation value, analogous to afvvv 46584. In contrast to ovrcl 7454, 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 46629	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 46630	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 46631	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 46632	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 46633	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 46634	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 46635	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 46636*	A frequently used special case of rspc2ev 3616 for operation values, analogous to rspceov 7461. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) )
Theorem	fnotaovb 46637	Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6944. (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 46638*	An operation maps to a class to which all values belong, analogous to ffnov 7541. (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 46639	Closure law for an operation, analogous to fovcl 7543. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)
Theorem	aovmpt4g 46640*	Value of a function given by the maps-to notation, analogous to ovmpt4g 7562. (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 46641*	Domain of closure of an operation. In contrast to oprssdm 7596, 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 46642	The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7600 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 46643	Reverse closure law, in contrast to ndmovrcl 7601 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 46644	Any operation is commutative outside its domain, analogous to ndmovcom 7602. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    ⇒   ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )
Theorem	ndmaovass 46645	Any operation is associative outside its domain. In contrast to ndmovass 7603 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 46646	Any operation is distributive outside its domain. In contrast to ndmovdistr 7604 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)) )) )
21.45.5  Alternative definitions of function values (2) In the following, a second approach is followed to define function values alternately to df-afv 46559. The current definition of the value (𝐹‘𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6551) assures that this value is always a set, see fex 7232. 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 6925 and fvprc 6882). "(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 46558. 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 6926). To avoid such an ambiguity, an alternative definition (𝐹''''𝐴) (see df-afv2 46648) 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 46650). 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 46663, 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 46651). 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 46699), but (𝐹‘𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 46704. The alternate definition, however, corresponds to the current definition ((𝐹‘𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 46700). With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 46666. 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 6551 of (𝐹‘𝐴), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6889-> afv2eq1 46655, fveq2 6890-> afv2eq2 46656, nffv 6900-> nfafv2 46657, csbfv12 6938-> csbafv212g , rlimdm 15522-> rlimdmafv2 46697, tz6.12-1 6913-> tz6.12-1-afv2 46680, fveu 6879-> afv2eu 46677. Six theorems proved by directly using df-fv 6551 are within a mathbox (fvsb 43950, uncov 37127) or not used (rlimdmafv 46616, avril1 30312) or experimental (dfafv2 46571, dfafv22 46698). However, the remaining 11 theorems proved by directly using df-fv 6551 are used more or less often: * fvex 6903: 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 46652 resp. afv2ex 46653). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6903. * fvres 6909: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 46678). In the undefined case such a theorem cannot exist (without additional assumtions), 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 46678 can be used instead of fvres 6909. * tz6.12-2 6878 (-> tz6.12-2-afv2 46676): root theorem of many theorems which have not a strict analogue, and which are used many times: ** fvprc 6882 (-> afv2prc 46665), used in 193 proofs, ** tz6.12i 6918 (-> tz6.12i-afv2 46682), used - indirectly via fvbr0 6919 and fvrn0 6920 - in 19 proofs, and in fvclss 7245 used in fvclex 7956 used in fvresex 7957 (which is not used!) and in dcomex 10465 (used in 4 proofs), ** ndmfv 6925 (-> ndmafv2nrn ), used in 124 proofs ** nfunsn 6932 (-> nfunsnafv2 ), used by fvfundmfvn0 6933 (used in 3 proofs), and dffv2 6986 (not used) ** funpartfv 35594, setrec2lem1 48232 (mathboxes) * fv2 6885: only used by elfv 6888, which is only used by fv3 6908, which is not used. * dffv3 6886 (-> dfafv23 ): used by dffv4 6887 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 44399), by shftval 15048 (itself used in 11 proofs), by dffv5 35573 (mathbox) and by fvco2 6988 (-> afv2co2 46696). * fvopab5 7031: used only by ajval 30710 (not used) and by adjval 31739, which is used in adjval2 31740 (not used) and in adjbdln 31932 (used in 7 proofs). * zsum 15691: used (via isum 15692, sum0 15694, sumss 15697 and fsumsers 15701) in 76 proofs. * isumshft 15812: used in pserdv2 26380 (used in logtayl 26607, binomcxplemdvsum 43853) , eftlub 16080 (used in 4 proofs), binomcxplemnotnn0 43854 (used in binomcxp 43855 only) and logtayl 26607 (used in 4 proofs). * ovtpos 8240: used in 16 proofs. * zprod 15908: used in 3 proofs: iprod 15909, zprodn0 15910 and prodss 15918 * iprodclim3 15971: 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 6885, dffv3 6886, fvopab5 7031, zsum 15691, isumshft 15812, ovtpos 8240 and zprod 15908 are not critical or are, hopefully, also valid for the alternative definition, fvex 6903, fvres 6909 and tz6.12-2 6878 (and the theorems based on them) are essential for the current definition of function values.
Syntax	cafv2 46647	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 46556.
class (𝐹''''𝐴)
Definition	df-afv2 46648*	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 6551, and especially ndmfv 6925), (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 46649*	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 46650	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 46651	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 46652	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 46653	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 46654	Equality deduction for function value, analogous to fveq12d 6897. (Contributed by AV, 4-Sep-2022.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))
Theorem	afv2eq1 46655	Equality theorem for function value, analogous to fveq1 6889. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴))
Theorem	afv2eq2 46656	Equality theorem for function value, analogous to fveq2 6890. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵))
Theorem	nfafv2 46657	Bound-variable hypothesis builder for function value, analogous to nffv 6900. To prove a deduction version of this analogous to nffvd 6902 is not easily possible because a deduction version of nfdfat 46566 cannot be shown easily. (Contributed by AV, 4-Sep-2022.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(𝐹''''𝐴)
Theorem	csbafv212g 46658	Move class substitution in and out of a function value, analogous to csbfv12 6938, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7456. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵))
Theorem	fexafv2ex 46659	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 46660	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 46661	The value of a class outside its domain is not in the range, compare with ndmfv 6925. (Contributed by AV, 2-Sep-2022.)
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	funressndmafv2rn 46662	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 46663	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 46664	If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6932. (Contributed by AV, 2-Sep-2022.)
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	afv2prc 46665	A function's value at a proper class is not defined, compare with fvprc 6882. (Contributed by AV, 5-Sep-2022.)
⊢ (¬ 𝐴 ∈ V → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	dfatafv2rnb 46666	The alternate function value at a class 𝐴 is defined, i.e. in the range of the function, iff the function is defined at 𝐴. (Contributed by AV, 2-Sep-2022.)
⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
Theorem	afv2orxorb 46667	If a set is in the range of a function, the alternate function value at a class 𝐴 equals this set or is not in the range of the function iff the alternate function value at the class 𝐴 either equals this set or is not in the range of the function. If 𝐵 ∉ ran 𝐹, both disjuncts of the exclusive or can be true: (𝐹''''𝐴) = 𝐵 → (𝐹''''𝐴) ∉ ran 𝐹. (Contributed by AV, 11-Sep-2022.)
⊢ (𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 ∨ (𝐹''''𝐴) ∉ ran 𝐹) ↔ ((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))
Theorem	dmafv2rnb 46668	The alternate function value at a class 𝐴 is defined, i.e., in the range of the function, iff 𝐴 is in the domain of the function. (Contributed by AV, 3-Sep-2022.)
⊢ (Fun (𝐹 ↾ {𝐴}) → (𝐴 ∈ dom 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
Theorem	fundmafv2rnb 46669	The alternate function value at a class 𝐴 is defined, i.e., in the range of the function iff 𝐴 is in the domain of the function. (Contributed by AV, 3-Sep-2022.)
⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
Theorem	afv2elrn 46670	An alternate function value belongs to the range of the function, analogous to fvelrn 7079. (Contributed by AV, 3-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹)
Theorem	afv20defat 46671	If the alternate function value at an argument is the empty set, the function is defined at this argument. (Contributed by AV, 3-Sep-2022.)
⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)
Theorem	fnafv2elrn 46672	An alternate function value belongs to the range of the function, analogous to fnfvelrn 7083. (Contributed by AV, 2-Sep-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ ran 𝐹)
Theorem	fafv2elcdm 46673	An alternate function value belongs to the codomain of the function, analogous to ffvelcdm 7084. (Contributed by AV, 2-Sep-2022.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ 𝐵)
Theorem	fafv2elrnb 46674	An alternate function value is defined, i.e., belongs to the range of the function, iff its argument is in the domain of the function. (Contributed by AV, 3-Sep-2022.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∈ 𝐴 ↔ (𝐹''''𝐶) ∈ ran 𝐹))
Theorem	fcdmvafv2v 46675	If the codomain of a function is a set, the alternate function value is always also a set. (Contributed by AV, 4-Sep-2022.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐹''''𝐶) ∈ V)
Theorem	tz6.12-2-afv2 46676*	Function value when 𝐹 is (locally) not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27, analogous to tz6.12-2 6878. (Contributed by AV, 5-Sep-2022.)
⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	afv2eu 46677*	The value of a function at a unique point, analogous to fveu 6879. (Contributed by AV, 5-Sep-2022.)
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
Theorem	afv2res 46678	The value of a restricted function for an argument at which the function is defined. Analog to fvres 6909. (Contributed by AV, 5-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (𝐹''''𝐴))
Theorem	tz6.12-afv2 46679*	Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6915. (Contributed by AV, 5-Sep-2022.)
⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦)
Theorem	tz6.12-1-afv2 46680*	Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12-1 6913. (Contributed by AV, 5-Sep-2022.)
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
Theorem	tz6.12c-afv2 46681*	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6912. (Contributed by AV, 5-Sep-2022.)
⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
Theorem	tz6.12i-afv2 46682	Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6918. (Contributed by AV, 5-Sep-2022.)
⊢ (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵))
Theorem	funressnbrafv2 46683	The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6941. (Contributed by AV, 7-Sep-2022.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Theorem	dfatbrafv2b 46684	Equivalence of function value and binary relation, analogous to fnbrfvb 6943 or funbrfvb 6945. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 46652). (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	dfatopafv2b 46685	Equivalence of function value and ordered pair membership, analogous to fnopfvb 6944 or funopfvb 6946. (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Theorem	funbrafv2 46686	The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6941. (Contributed by AV, 6-Sep-2022.)
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Theorem	fnbrafv2b 46687	Equivalence of function value and binary relation, analogous to fnbrfvb 6943. (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))
Theorem	fnopafv2b 46688	Equivalence of function value and ordered pair membership, analogous to fnopfvb 6944. (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
Theorem	funbrafv22b 46689	Equivalence of function value and binary relation, analogous to funbrfvb 6945. (Contributed by AV, 6-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	funopafv2b 46690	Equivalence of function value and ordered pair membership, analogous to funopfvb 6946. (Contributed by AV, 6-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Theorem	dfatsnafv2 46691	Singleton of function value, analogous to fnsnfv 6970. (Contributed by AV, 7-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))
Theorem	dfafv23 46692*	A definition of function value in terms of iota, analogous to dffv3 6886. (Contributed by AV, 6-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
Theorem	dfatdmfcoafv2 46693	Domain of a function composition, analogous to dmfco 6987. (Contributed by AV, 7-Sep-2022.)
⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺''''𝐴) ∈ dom 𝐹))
Theorem	dfatcolem 46694*	Lemma for dfatco 46695. (Contributed by AV, 8-Sep-2022.)
⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ∃!𝑦 𝑋(𝐹 ∘ 𝐺)𝑦)
Theorem	dfatco 46695	The predicate "defined at" for a function composition. (Contributed by AV, 8-Sep-2022.)
⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → (𝐹 ∘ 𝐺) defAt 𝑋)
Theorem	afv2co2 46696	Value of a function composition, analogous to fvco2 6988. (Contributed by AV, 8-Sep-2022.)
⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ((𝐹 ∘ 𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋)))
Theorem	rlimdmafv2 46697	Two ways to express that a function has a limit, analogous to rlimdm 15522. (Contributed by AV, 5-Sep-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝𝑟 ↔ 𝐹 ⇝𝑟 ( ⇝𝑟 ''''𝐹)))
Theorem	dfafv22 46698	Alternate definition of (𝐹''''𝐴) using (𝐹‘𝐴) directly. (Contributed by AV, 3-Sep-2022.)
⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)
Theorem	afv2ndeffv0 46699	If the alternate function value at an argument is undefined, i.e., not in the range of the function, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.)
⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹‘𝐴) = ∅)
Theorem	dfatafv2eqfv 46700	If a function is defined at a class 𝐴, the alternate function value equals the function's value at 𝐴. (Contributed by AV, 3-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴))