P. 435 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43401-43500   *Has distinct variable group(s)

Type	Label	Description
Statement
20.41.5  Alternative definitions of function values (2) In the following, a second approach is followed to define function values alternately to df-afv 43313. The current definition of the value (𝐹‘𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6357) assures that this value is always a set, see fex 6983. 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 6694 and fvprc 6657). "(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 43312. 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 6695). To avoid such an ambiguity, an alternative definition (𝐹''''𝐴) (see df-afv2 43402) 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 43404). 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 43417, 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 43405). 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 43453), but (𝐹‘𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 43458. The alternate definition, however, corresponds to the current definition ((𝐹‘𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 43454). With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 43420. 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 6357 of (𝐹‘𝐴), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6663-> afv2eq1 43409, fveq2 6664-> afv2eq2 43410, nffv 6674-> nfafv2 43411, csbfv12 6707-> csbafv212g , rlimdm 14902-> rlimdmafv2 43451, tz6.12-1 6686-> tz6.12-1-afv2 43434, fveu 6655-> afv2eu 43431. Six theorems proved by directly using df-fv 6357 are within a mathbox (fvsb 40777, uncov 34867) or not used (rlimdmafv 43370, avril1 28236) or experimental (dfafv2 43325, dfafv22 43452). However, the remaining 11 theorems proved by directly using df-fv 6357 are used more or less often: * fvex 6677: 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 43406 resp. afv2ex 43407). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6677. * fvres 6683: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 43432). 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 43432 can be used instead of fvres 6683. * tz6.12-2 6654 (-> tz6.12-2-afv2 43430): root theorem of many theorems which have not a strict analogue, and which are used many times: ** fvprc 6657 (-> afv2prc 43419), used in 193 proofs, ** tz6.12i 6690 (-> tz6.12i-afv2 43436), used - indirectly via fvbr0 6691 and fvrn0 6692 - in 19 proofs, and in fvclss 6995 used in fvclex 7654 used in fvresex 7655 (which is not used!) and in dcomex 9863 (used in 4 proofs), ** ndmfv 6694 (-> ndmafv2nrn ), used in 124 proofs ** nfunsn 6701 (-> nfunsnafv2 ), used by fvfundmfvn0 6702 (used in 3 proofs), and dffv2 6750 (not used) ** funpartfv 33401, setrec2lem1 44790 (mathboxes) * fv2 6659: only used by elfv 6662, which is only used by fv3 6682, which is not used. * dffv3 6660 (-> dfafv23 ): used by dffv4 6661 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 41226), by shftval 14427 (itself used in 11 proofs), by dffv5 33380 (mathbox) and by fvco2 6752 (-> afv2co2 43450). * fvopab5 6794: used only by ajval 28632 (not used) and by adjval 29661, which is used in adjval2 29662 (not used) and in adjbdln 29854 (used in 7 proofs). * zsum 15069: used (via isum 15070, sum0 15072, sumss 15075 and fsumsers 15079) in 76 proofs. * isumshft 15188: used in pserdv2 25012 (used in logtayl 25237, binomcxplemdvsum 40680) , eftlub 15456 (used in 4 proofs), binomcxplemnotnn0 40681 (used in binomcxp 40682 only) and logtayl 25237 (used in 4 proofs). * ovtpos 7901: used in 16 proofs. * zprod 15285: used in 3 proofs: iprod 15286, zprodn0 15287 and prodss 15295 * iprodclim3 15348: 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 6659, dffv3 6660, fvopab5 6794, zsum 15069, isumshft 15188, ovtpos 7901 and zprod 15285 are not critical or are, hopefully, also valid for the alternative definition, fvex 6677, fvres 6683 and tz6.12-2 6654 (and the theorems based on them) are essential for the current definition of function values.
Syntax	cafv2 43401	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 43310.
class (𝐹''''𝐴)
Definition	df-afv2 43402*	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 6357, and especially ndmfv 6694), (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 43403*	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 43404	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 43405	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 43406	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 43407	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 43408	Equality deduction for function value, analogous to fveq12d 6671. (Contributed by AV, 4-Sep-2022.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))
Theorem	afv2eq1 43409	Equality theorem for function value, analogous to fveq1 6663. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴))
Theorem	afv2eq2 43410	Equality theorem for function value, analogous to fveq2 6664. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵))
Theorem	nfafv2 43411	Bound-variable hypothesis builder for function value, analogous to nffv 6674. To prove a deduction version of this analogous to nffvd 6676 is not easily possible because a deduction version of nfdfat 43320 cannot be shown easily. (Contributed by AV, 4-Sep-2022.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(𝐹''''𝐴)
Theorem	csbafv212g 43412	Move class substitution in and out of a function value, analogous to csbfv12 6707, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7192. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵))
Theorem	fexafv2ex 43413	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 43414	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 43415	The value of a class outside its domain is not in the range, compare with ndmfv 6694. (Contributed by AV, 2-Sep-2022.)
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	funressndmafv2rn 43416	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 43417	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 43418	If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6701. (Contributed by AV, 2-Sep-2022.)
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	afv2prc 43419	A function's value at a proper class is not defined, compare with fvprc 6657. (Contributed by AV, 5-Sep-2022.)
⊢ (¬ 𝐴 ∈ V → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	dfatafv2rnb 43420	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 43421	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 43422	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 43423	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 43424	An alternate function value belongs to the range of the function, analogous to fvelrn 6838. (Contributed by AV, 3-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹)
Theorem	afv20defat 43425	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 43426	An alternate function value belongs to the range of the function, analogous to fnfvelrn 6842. (Contributed by AV, 2-Sep-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ ran 𝐹)
Theorem	fafv2elrn 43427	An alternate function value belongs to the codomain of the function, analogous to ffvelrn 6843. (Contributed by AV, 2-Sep-2022.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ 𝐵)
Theorem	fafv2elrnb 43428	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	frnvafv2v 43429	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 43430*	Function value when 𝐹 is (locally) not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27, analogous to tz6.12-2 6654. (Contributed by AV, 5-Sep-2022.)
⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	afv2eu 43431*	The value of a function at a unique point, analogous to fveu 6655. (Contributed by AV, 5-Sep-2022.)
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
Theorem	afv2res 43432	The value of a restricted function for an argument at which the function is defined. Analog to fvres 6683. (Contributed by AV, 5-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (𝐹''''𝐴))
Theorem	tz6.12-afv2 43433*	Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6687. (Contributed by AV, 5-Sep-2022.)
⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦)
Theorem	tz6.12-1-afv2 43434*	Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12-1 6686. (Contributed by AV, 5-Sep-2022.)
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
Theorem	tz6.12c-afv2 43435*	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6689. (Contributed by AV, 5-Sep-2022.)
⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
Theorem	tz6.12i-afv2 43436	Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6690. (Contributed by AV, 5-Sep-2022.)
⊢ (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵))
Theorem	funressnbrafv2 43437	The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6710. (Contributed by AV, 7-Sep-2022.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Theorem	dfatbrafv2b 43438	Equivalence of function value and binary relation, analogous to fnbrfvb 6712 or funbrfvb 6714. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 43406). (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	dfatopafv2b 43439	Equivalence of function value and ordered pair membership, analogous to fnopfvb 6713 or funopfvb 6715. (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Theorem	funbrafv2 43440	The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6710. (Contributed by AV, 6-Sep-2022.)
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Theorem	fnbrafv2b 43441	Equivalence of function value and binary relation, analogous to fnbrfvb 6712. (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))
Theorem	fnopafv2b 43442	Equivalence of function value and ordered pair membership, analogous to fnopfvb 6713. (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
Theorem	funbrafv22b 43443	Equivalence of function value and binary relation, analogous to funbrfvb 6714. (Contributed by AV, 6-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	funopafv2b 43444	Equivalence of function value and ordered pair membership, analogous to funopfvb 6715. (Contributed by AV, 6-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Theorem	dfatsnafv2 43445	Singleton of function value, analogous to fnsnfv 6737. (Contributed by AV, 7-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))
Theorem	dfafv23 43446*	A definition of function value in terms of iota, analogous to dffv3 6660. (Contributed by AV, 6-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
Theorem	dfatdmfcoafv2 43447	Domain of a function composition, analogous to dmfco 6751. (Contributed by AV, 7-Sep-2022.)
⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺''''𝐴) ∈ dom 𝐹))
Theorem	dfatcolem 43448*	Lemma for dfatco 43449. (Contributed by AV, 8-Sep-2022.)
⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ∃!𝑦 𝑋(𝐹 ∘ 𝐺)𝑦)
Theorem	dfatco 43449	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 43450	Value of a function composition, analogous to fvco2 6752. (Contributed by AV, 8-Sep-2022.)
⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ((𝐹 ∘ 𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋)))
Theorem	rlimdmafv2 43451	Two ways to express that a function has a limit, analogous to rlimdm 14902. (Contributed by AV, 5-Sep-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝𝑟 ↔ 𝐹 ⇝𝑟 ( ⇝𝑟 ''''𝐹)))
Theorem	dfafv22 43452	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 43453	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 43454	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 43455	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 43456	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 43457	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 43458	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 43459	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 43460	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)))
20.41.6  General auxiliary theorems (2)
20.41.6.1  Logical conjunction - extension
Theorem	an4com24 43461	Rearrangement of 4 conjuncts: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜓)))
20.41.6.2  Abbreviated conjunction and disjunction of three wff's - extension
Theorem	3an4ancom24 43462	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 43463	Rearrangement of 4 conjuncts with a triple conjunction. (Contributed by AV, 4-Mar-2022.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒 ∧ 𝜃)))
20.41.6.3  Negated membership (alternative)
Syntax	cnelbr 43464	Extend wff notation to include the 'not elemet of' relation.
class _∉
Definition	df-nelbr 43465*	Define negated membership as binary relation. Analogous to df-eprel 5459 (the membership relation). (Contributed by AV, 26-Dec-2021.)
⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
Theorem	dfnelbr2 43466	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 43467	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 43468	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 43469	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 43470	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)
20.41.6.4  The empty set - extension
Theorem	ralralimp 43471*	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))
20.41.6.5  Indexed union and intersection - extension
Theorem	otiunsndisjX 43472*	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⟩})
20.41.6.6  Functions - extension
Theorem	fvifeq 43473	Equality of function values with conditional arguments, see also fvif 6680. (Contributed by Alexander van der Vekens, 21-May-2018.)
⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶)))
Theorem	rnfdmpr 43474	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 43475	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 43476*	A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
⊢ (∃𝑥∃𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
Theorem	fun2dmnopgexmpl 43477	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 43478*	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 43479*	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 43480*	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 43481*	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 43482*	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 43483*	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 43484*	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})
20.41.6.7  Maps-to notation - extension
Theorem	fvmptrab 43485*	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 6793, 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 43486*	Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 6793. (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}
20.41.6.8  Ordering on reals - extension
Theorem	leltletr 43487	Transitive law, weaker form of lelttr 10725. (Contributed by AV, 14-Oct-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 ≤ 𝐶))
20.41.6.9  Subtraction - extension
Theorem	cnambpcma 43488	((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 43489	((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 43490	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))
20.41.6.10  Ordering on reals (cont.) - extension
Theorem	leaddsuble 43491	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 43492	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 43493	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 43494	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 43495	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)))
Theorem	zm1nn 43496	An integer minus 1 is positive under certain circumstances. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℤ) → ((𝐽 ∈ ℝ ∧ 0 ≤ 𝐽 ∧ 𝐽 < ((𝐿 − 𝑁) − 1)) → (𝐿 − 1) ∈ ℕ))
20.41.6.11  Imaginary and complex number properties - extension
Theorem	readdcnnred 43497	The sum of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ ℝ))    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∉ ℝ)
Theorem	resubcnnred 43498	The difference of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ ℝ))    ⇒   ⊢ (𝜑 → (𝐴 − 𝐵) ∉ ℝ)
Theorem	recnmulnred 43499	The product of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ ℝ))    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∉ ℝ)
Theorem	cndivrenred 43500	The quotient of an imaginary number and a real number is not a real number. (Contributed by AV, 23-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ ℝ))    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐵 / 𝐴) ∉ ℝ)