P. 473 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47201-47300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ffnafv 47201*	A function maps to a class to which all values belong, analogous to ffnfv 7052. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵))
Theorem	afvres 47202	The value of a restricted function, analogous to fvres 6841. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))
Theorem	tz6.12-afv 47203*	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6846. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)
Theorem	tz6.12-1-afv 47204*	Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6845. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦)
Theorem	dmfcoafv 47205	Domains of a function composition, analogous to dmfco 6918. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))
Theorem	afvco2 47206	Value of a function composition, analogous to fvco2 6919. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))
Theorem	rlimdmafv 47207	Two ways to express that a function has a limit, analogous to rlimdm 15455. (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)))
21.48.4.6  Alternative definition of the value of an operation
Theorem	aoveq123d 47208	Equality deduction for operation value, analogous to oveq123d 7367. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )
Theorem	nfaov 47209	Bound-variable hypothesis builder for operation value, analogous to nfov 7376. To prove a deduction version of this analogous to nfovd 7375 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 47166). (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵))
Theorem	csbaovg 47210	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 47211	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 47212	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 47213	The value of an operation outside its domain, analogous to ndmafv 47170. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
Theorem	ndmaovg 47214	The value of an operation outside its domain, analogous to ndmovg 7529. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V)
Theorem	aovvdm 47215	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 47216	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 47217	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 47218	The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7384. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Rel dom 𝐹    ⇒   ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)
Theorem	aovrcl 47219	Reverse closure for an operation value, analogous to afvvv 47175. In contrast to ovrcl 7387, 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 47220	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 47221	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 47222	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 47223	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 47224	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 47225	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 47226	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 47227*	A frequently used special case of rspc2ev 3590 for operation values, analogous to rspceov 7395. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) )
Theorem	fnotaovb 47228	Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6873. (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 47229*	An operation maps to a class to which all values belong, analogous to ffnov 7472. (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 47230	Closure law for an operation, analogous to fovcl 7474. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)
Theorem	aovmpt4g 47231*	Value of a function given by the maps-to notation, analogous to ovmpt4g 7493. (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 47232*	Domain of closure of an operation. In contrast to oprssdm 7527, 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 47233	The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7531 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 47234	Reverse closure law, in contrast to ndmovrcl 7532 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 47235	Any operation is commutative outside its domain, analogous to ndmovcom 7533. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    ⇒   ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )
Theorem	ndmaovass 47236	Any operation is associative outside its domain. In contrast to ndmovass 7534 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 47237	Any operation is distributive outside its domain. In contrast to ndmovdistr 7535 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)) )) )
21.48.5  Alternative definitions of function values (2) In the following, a second approach is followed to define function values alternately to df-afv 47150. The current definition of the value (𝐹‘𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6489) assures that this value is always a set, see fex 7160. 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 6854 and fvprc 6814). "(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 47149. 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 6855). To avoid such an ambiguity, an alternative definition (𝐹''''𝐴) (see df-afv2 47239) 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 47241). 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 47254, 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 47242). 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 47290), but (𝐹‘𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 47295. The alternate definition, however, corresponds to the current definition ((𝐹‘𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 47291). With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 47257. 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 6489 of (𝐹‘𝐴), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6821-> afv2eq1 47246, fveq2 6822-> afv2eq2 47247, nffv 6832-> nfafv2 47248, csbfv12 6867-> csbafv212g , rlimdm 15455-> rlimdmafv2 47288, tz6.12-1 6845-> tz6.12-1-afv2 47271, fveu 6811-> afv2eu 47268. Six theorems proved by directly using df-fv 6489 are within a mathbox (fvsb 44483, uncov 37640) or not used (rlimdmafv 47207, avril1 30438) or experimental (dfafv2 47162, dfafv22 47289). However, the remaining 11 theorems proved by directly using df-fv 6489 are used more or less often: * fvex 6835: 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 47243 resp. afv2ex 47244). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6835. * fvres 6841: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 47269). 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 47269 can be used instead of fvres 6841. * tz6.12-2 6809 (-> tz6.12-2-afv2 47267): root theorem of many theorems which have not a strict analogue, and which are used many times: ** fvprc 6814 (-> afv2prc 47256), used in 193 proofs, ** tz6.12i 6848 (-> tz6.12i-afv2 47273), used - indirectly via fvbr0 6849 and fvrn0 6850 - in 19 proofs, and in fvclss 7175 used in fvclex 7891 used in fvresex 7892 (which is not used!) and in dcomex 10335 (used in 4 proofs), ** ndmfv 6854 (-> ndmafv2nrn ), used in 124 proofs ** nfunsn 6861 (-> nfunsnafv2 ), used by fvfundmfvn0 6862 (used in 3 proofs), and dffv2 6917 (not used) ** funpartfv 35978, setrec2lem1 49724 (mathboxes) * fv2 6817: only used by elfv 6820, which is only used by fv3 6840, which is not used. * dffv3 6818 (-> dfafv23 ): used by dffv4 6819 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 44930), by shftval 14978 (itself used in 11 proofs), by dffv5 35957 (mathbox) and by fvco2 6919 (-> afv2co2 47287). * fvopab5 6962: used only by ajval 30836 (not used) and by adjval 31865, which is used in adjval2 31866 (not used) and in adjbdln 32058 (used in 7 proofs). * zsum 15622: used (via isum 15623, sum0 15625, sumss 15628 and fsumsers 15632) in 76 proofs. * isumshft 15743: used in pserdv2 26365 (used in logtayl 26594, binomcxplemdvsum 44387) , eftlub 16015 (used in 4 proofs), binomcxplemnotnn0 44388 (used in binomcxp 44389 only) and logtayl 26594 (used in 4 proofs). * ovtpos 8171: used in 16 proofs. * zprod 15841: used in 3 proofs: iprod 15842, zprodn0 15843 and prodss 15851 * iprodclim3 15904: 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 6817, dffv3 6818, fvopab5 6962, zsum 15622, isumshft 15743, ovtpos 8171 and zprod 15841 are not critical or are, hopefully, also valid for the alternative definition, fvex 6835, fvres 6841 and tz6.12-2 6809 (and the theorems based on them) are essential for the current definition of function values.
Syntax	cafv2 47238	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 47147.
class (𝐹''''𝐴)
Definition	df-afv2 47239*	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 6489, and especially ndmfv 6854), (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 47240*	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 47241	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 47242	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 47243	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 47244	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 47245	Equality deduction for function value, analogous to fveq12d 6829. (Contributed by AV, 4-Sep-2022.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))
Theorem	afv2eq1 47246	Equality theorem for function value, analogous to fveq1 6821. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴))
Theorem	afv2eq2 47247	Equality theorem for function value, analogous to fveq2 6822. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵))
Theorem	nfafv2 47248	Bound-variable hypothesis builder for function value, analogous to nffv 6832. To prove a deduction version of this analogous to nffvd 6834 is not easily possible because a deduction version of nfdfat 47157 cannot be shown easily. (Contributed by AV, 4-Sep-2022.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(𝐹''''𝐴)
Theorem	csbafv212g 47249	Move class substitution in and out of a function value, analogous to csbfv12 6867, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7390. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵))
Theorem	fexafv2ex 47250	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 47251	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 47252	The value of a class outside its domain is not in the range, compare with ndmfv 6854. (Contributed by AV, 2-Sep-2022.)
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	funressndmafv2rn 47253	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 47254	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 47255	If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6861. (Contributed by AV, 2-Sep-2022.)
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	afv2prc 47256	A function's value at a proper class is not defined, compare with fvprc 6814. (Contributed by AV, 5-Sep-2022.)
⊢ (¬ 𝐴 ∈ V → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	dfatafv2rnb 47257	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 47258	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 47259	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 47260	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 47261	An alternate function value belongs to the range of the function, analogous to fvelrn 7009. (Contributed by AV, 3-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹)
Theorem	afv20defat 47262	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 47263	An alternate function value belongs to the range of the function, analogous to fnfvelrn 7013. (Contributed by AV, 2-Sep-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ ran 𝐹)
Theorem	fafv2elcdm 47264	An alternate function value belongs to the codomain of the function, analogous to ffvelcdm 7014. (Contributed by AV, 2-Sep-2022.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ 𝐵)
Theorem	fafv2elrnb 47265	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 47266	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 47267*	Function value when 𝐹 is (locally) not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27, analogous to tz6.12-2 6809. (Contributed by AV, 5-Sep-2022.)
⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	afv2eu 47268*	The value of a function at a unique point, analogous to fveu 6811. (Contributed by AV, 5-Sep-2022.)
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
Theorem	afv2res 47269	The value of a restricted function for an argument at which the function is defined. Analog to fvres 6841. (Contributed by AV, 5-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (𝐹''''𝐴))
Theorem	tz6.12-afv2 47270*	Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6846. (Contributed by AV, 5-Sep-2022.)
⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦)
Theorem	tz6.12-1-afv2 47271*	Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12-1 6845. (Contributed by AV, 5-Sep-2022.)
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
Theorem	tz6.12c-afv2 47272*	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6844. (Contributed by AV, 5-Sep-2022.)
⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
Theorem	tz6.12i-afv2 47273	Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6848. (Contributed by AV, 5-Sep-2022.)
⊢ (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵))
Theorem	funressnbrafv2 47274	The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6870. (Contributed by AV, 7-Sep-2022.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Theorem	dfatbrafv2b 47275	Equivalence of function value and binary relation, analogous to fnbrfvb 6872 or funbrfvb 6875. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 47243). (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	dfatopafv2b 47276	Equivalence of function value and ordered pair membership, analogous to fnopfvb 6873 or funopfvb 6876. (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Theorem	funbrafv2 47277	The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6870. (Contributed by AV, 6-Sep-2022.)
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Theorem	fnbrafv2b 47278	Equivalence of function value and binary relation, analogous to fnbrfvb 6872. (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))
Theorem	fnopafv2b 47279	Equivalence of function value and ordered pair membership, analogous to fnopfvb 6873. (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
Theorem	funbrafv22b 47280	Equivalence of function value and binary relation, analogous to funbrfvb 6875. (Contributed by AV, 6-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	funopafv2b 47281	Equivalence of function value and ordered pair membership, analogous to funopfvb 6876. (Contributed by AV, 6-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Theorem	dfatsnafv2 47282	Singleton of function value, analogous to fnsnfv 6901. (Contributed by AV, 7-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))
Theorem	dfafv23 47283*	A definition of function value in terms of iota, analogous to dffv3 6818. (Contributed by AV, 6-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
Theorem	dfatdmfcoafv2 47284	Domain of a function composition, analogous to dmfco 6918. (Contributed by AV, 7-Sep-2022.)
⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺''''𝐴) ∈ dom 𝐹))
Theorem	dfatcolem 47285*	Lemma for dfatco 47286. (Contributed by AV, 8-Sep-2022.)
⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ∃!𝑦 𝑋(𝐹 ∘ 𝐺)𝑦)
Theorem	dfatco 47286	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 47287	Value of a function composition, analogous to fvco2 6919. (Contributed by AV, 8-Sep-2022.)
⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ((𝐹 ∘ 𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋)))
Theorem	rlimdmafv2 47288	Two ways to express that a function has a limit, analogous to rlimdm 15455. (Contributed by AV, 5-Sep-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝𝑟 ↔ 𝐹 ⇝𝑟 ( ⇝𝑟 ''''𝐹)))
Theorem	dfafv22 47289	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 47290	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 47291	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))
Theorem	afv2rnfveq 47292	If the alternate function value is defined, i.e., in the range of the function, the alternate function value equals the function's value. (Contributed by AV, 3-Sep-2022.)
⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → (𝐹''''𝐴) = (𝐹‘𝐴))
Theorem	afv20fv0 47293	If the alternate function value at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.)
⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅)
Theorem	afv2fvn0fveq 47294	If the function's value at an argument is not the empty set, it equals the alternate function value at this argument. (Contributed by AV, 3-Sep-2022.)
⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹''''𝐴) = (𝐹‘𝐴))
Theorem	afv2fv0 47295	If the function's value at an argument is the empty set, then the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022.)
⊢ ((𝐹‘𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
Theorem	afv2fv0b 47296	The function's value at an argument is the empty set if and only if the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022.)
⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
Theorem	afv2fv0xorb 47297	If a set is in the range of a function, the function's value at an argument is the empty set if and only if the alternate function value at this argument is either the empty set or undefined. (Contributed by AV, 11-Sep-2022.)
⊢ (∅ ∈ ran 𝐹 → ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))
21.48.6  General auxiliary theorems (2)
21.48.6.1  Logical conjunction - extension
Theorem	an4com24 47298	Rearrangement of 4 conjuncts: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜓)))
21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension
Theorem	3an4ancom24 47299	Commutative law for a conjunction with a triple conjunction: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜃 ∧ 𝜒) ∧ 𝜓))
Theorem	4an21 47300	Rearrangement of 4 conjuncts with a triple conjunction. (Contributed by AV, 4-Mar-2022.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒 ∧ 𝜃)))