P. 474 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47301-47400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	aoprssdm 47301*	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 47302	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 47303	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 47304	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 47305	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 47306	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 47219. 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 47218. 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 47308) 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 47310). 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 47323, 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 47311). 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 47359), but (𝐹‘𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 47364. The alternate definition, however, corresponds to the current definition ((𝐹‘𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 47360). With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 47326. 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 47315, fveq2 6822-> afv2eq2 47316, nffv 6832-> nfafv2 47317, csbfv12 6867-> csbafv212g , rlimdm 15458-> rlimdmafv2 47357, tz6.12-1 6845-> tz6.12-1-afv2 47340, fveu 6811-> afv2eu 47337. Six theorems proved by directly using df-fv 6489 are within a mathbox (fvsb 44543, uncov 37640) or not used (rlimdmafv 47276, avril1 30443) or experimental (dfafv2 47231, dfafv22 47358). 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 47312 resp. afv2ex 47313). 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 47338). 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 47338 can be used instead of fvres 6841. * tz6.12-2 6809 (-> tz6.12-2-afv2 47336): root theorem of many theorems which have not a strict analogue, and which are used many times: ** fvprc 6814 (-> afv2prc 47325), used in 193 proofs, ** tz6.12i 6848 (-> tz6.12i-afv2 47342), 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 10338 (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 35989, setrec2lem1 49793 (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 44990), by shftval 14981 (itself used in 11 proofs), by dffv5 35966 (mathbox) and by fvco2 6919 (-> afv2co2 47356). * fvopab5 6962: used only by ajval 30841 (not used) and by adjval 31870, which is used in adjval2 31871 (not used) and in adjbdln 32063 (used in 7 proofs). * zsum 15625: used (via isum 15626, sum0 15628, sumss 15631 and fsumsers 15635) in 76 proofs. * isumshft 15746: used in pserdv2 26367 (used in logtayl 26596, binomcxplemdvsum 44447) , eftlub 16018 (used in 4 proofs), binomcxplemnotnn0 44448 (used in binomcxp 44449 only) and logtayl 26596 (used in 4 proofs). * ovtpos 8171: used in 16 proofs. * zprod 15844: used in 3 proofs: iprod 15845, zprodn0 15846 and prodss 15854 * iprodclim3 15907: 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 15625, isumshft 15746, ovtpos 8171 and zprod 15844 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 47307	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 47216.
class (𝐹''''𝐴)
Definition	df-afv2 47308*	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 47309*	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 47310	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 47311	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 47312	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 47313	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 47314	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 47315	Equality theorem for function value, analogous to fveq1 6821. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴))
Theorem	afv2eq2 47316	Equality theorem for function value, analogous to fveq2 6822. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵))
Theorem	nfafv2 47317	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 47226 cannot be shown easily. (Contributed by AV, 4-Sep-2022.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(𝐹''''𝐴)
Theorem	csbafv212g 47318	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 47319	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 47320	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 47321	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 47322	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 47323	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 47324	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 47325	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 47326	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 47327	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 47328	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 47329	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 47330	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 47331	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 47332	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 47333	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 47334	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 47335	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 47336*	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 47337*	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 47338	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 47339*	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 47340*	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 47341*	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 47342	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 47343	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 47344	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 47312). (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	dfatopafv2b 47345	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 47346	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 47347	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 47348	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 47349	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 47350	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 47351	Singleton of function value, analogous to fnsnfv 6901. (Contributed by AV, 7-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))
Theorem	dfafv23 47352*	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 47353	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 47354*	Lemma for dfatco 47355. (Contributed by AV, 8-Sep-2022.)
⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ∃!𝑦 𝑋(𝐹 ∘ 𝐺)𝑦)
Theorem	dfatco 47355	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 47356	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 47357	Two ways to express that a function has a limit, analogous to rlimdm 15458. (Contributed by AV, 5-Sep-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝𝑟 ↔ 𝐹 ⇝𝑟 ( ⇝𝑟 ''''𝐹)))
Theorem	dfafv22 47358	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 47359	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 47360	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 47361	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 47362	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 47363	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 47364	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 47365	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 47366	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 47367	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 47368	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 47369	Rearrangement of 4 conjuncts with a triple conjunction. (Contributed by AV, 4-Mar-2022.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒 ∧ 𝜃)))
21.48.6.3  Negated membership (alternative)
Syntax	cnelbr 47370	Extend wff notation to include the 'not element of' relation.
class _∉
Definition	df-nelbr 47371*	Define negated membership as binary relation. Analogous to df-eprel 5514 (the membership relation). (Contributed by AV, 26-Dec-2021.)
⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
Theorem	dfnelbr2 47372	Alternate definition of the negated membership as binary relation. (Proposed by BJ, 27-Dec-2021.) (Contributed by AV, 27-Dec-2021.)
⊢ _∉ = ((V × V) ∖ E )
Theorem	nelbr 47373	The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵))
Theorem	nelbrim 47374	If a set is related to another set by the negated membership relation, then it is not a member of the other set. The other direction of the implication is not generally true, because if 𝐴 is a proper class, then ¬ 𝐴 ∈ 𝐵 would be true, but not 𝐴 _∉ 𝐵. (Contributed by AV, 26-Dec-2021.)
⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵)
Theorem	nelbrnel 47375	A set is related to another set by the negated membership relation iff it is not a member of the other set. (Contributed by AV, 26-Dec-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 _∉ 𝐵 ↔ 𝐴 ∉ 𝐵))
Theorem	nelbrnelim 47376	If a set is related to another set by the negated membership relation, then it is not a member of the other set. (Contributed by AV, 26-Dec-2021.)
⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵)
21.48.6.4  The empty set - extension
Theorem	ralralimp 47377*	Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.)
⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (∀𝑥 ∈ 𝐴 ((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏))
21.48.6.5  Indexed union and intersection - extension
Theorem	otiunsndisjX 47378*	The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩})
21.48.6.6  Functions - extension
Theorem	fvifeq 47379	Equality of function values with conditional arguments, see also fvif 6838. (Contributed by Alexander van der Vekens, 21-May-2018.)
⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶)))
Theorem	rnfdmpr 47380	The range of a one-to-one function 𝐹 of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)}))
Theorem	imarnf1pr 47381	The image of the range of a function 𝐹 under a function 𝐸 if 𝐹 is a function from a pair into the domain of 𝐸. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵)) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵}))
Theorem	funop1 47382*	A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Avoid depending on this detail.)
⊢ (∃𝑥∃𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
Theorem	fun2dmnopgexmpl 47383	A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Avoid depending on this detail.)
⊢ (𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))
Theorem	opabresex0d 47384*	A collection of ordered pairs, the class of all possible second components being a set, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 1-Jan-2021.)
⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)
Theorem	opabbrfex0d 47385*	A collection of ordered pairs, the class of all possible second components being a set, is a set. (Contributed by AV, 15-Jan-2021.)
⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)
Theorem	opabresexd 47386*	A collection of ordered pairs, the second component being a function, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.)
⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝑈)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)
Theorem	opabbrfexd 47387*	A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.)
⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝑈)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)
Theorem	f1oresf1orab 47388*	Build a bijection by restricting the domain of a bijection. (Contributed by AV, 1-Aug-2022.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    &   ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝜒 ↔ 𝑥 ∈ 𝐷))    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
Theorem	f1oresf1o 47389*	Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)    &   ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝜒)))    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
Theorem	f1oresf1o2 47390*	Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
21.48.6.7  Maps-to notation - extension
Theorem	fvmptrab 47391*	Value of a function mapping a set to a class abstraction restricting a class depending on the argument of the function. More general version of fvmptrabfv 6961, but relying on the fact that out-of-domain arguments evaluate to the empty set, which relies on set.mm's particular encoding. (Contributed by AV, 14-Feb-2022.)
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑})    &   ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑋 → 𝑀 = 𝑁)    &   ⊢ (𝑋 ∈ 𝑉 → 𝑁 ∈ V)    &   ⊢ (𝑋 ∉ 𝑉 → 𝑁 = ∅)    ⇒   ⊢ (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓}
Theorem	fvmptrabdm 47392*	Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 6961. (Suggested by BJ, 18-Feb-2022.) (Contributed by AV, 18-Feb-2022.)
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑})    &   ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑌 ∈ dom 𝐺 → 𝑋 ∈ dom 𝐹)    ⇒   ⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓}
21.48.6.8  Subtraction - extension
Theorem	cnambpcma 47393	((a-b)+c)-a = c-a holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 − 𝐵) + 𝐶) − 𝐴) = (𝐶 − 𝐵))
Theorem	cnapbmcpd 47394	((a+b)-c)+d = ((a+d)+b)-c holds for complex numbers a,b,c,d. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴 + 𝐵) − 𝐶) + 𝐷) = (((𝐴 + 𝐷) + 𝐵) − 𝐶))
Theorem	addsubeq0 47395	The sum of two complex numbers is equal to the difference of these two complex numbers iff the subtrahend is 0. (Contributed by AV, 8-May-2023.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 − 𝐵) ↔ 𝐵 = 0))
21.48.6.9  Ordering on reals (cont.) - extension
Theorem	leaddsuble 47396	Addition and subtraction on one side of "less than or equal to". (Contributed by Alexander van der Vekens, 18-Mar-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ≤ 𝐶 ↔ ((𝐴 + 𝐵) − 𝐶) ≤ 𝐴))
Theorem	2leaddle2 47397	If two real numbers are less than a third real number, the sum of the real numbers is less than twice the third real number. (Contributed by Alexander van der Vekens, 21-May-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐶 ∧ 𝐵 < 𝐶) → (𝐴 + 𝐵) < (2 · 𝐶)))
Theorem	ltnltne 47398	Variant of trichotomy law for 'less than'. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐵 = 𝐴)))
Theorem	p1lep2 47399	A real number increasd by 1 is less than or equal to the number increased by 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ≤ (𝑁 + 2))
Theorem	ltsubsubaddltsub 47400	If the result of subtracting two numbers is greater than a number, the result of adding one of these subtracted numbers to the number is less than the result of subtracting the other subtracted number only. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
⊢ ((𝐽 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ)) → (𝐽 < ((𝐿 − 𝑀) − 𝑁) ↔ (𝐽 + 𝑀) < (𝐿 − 𝑁)))