P. 472 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47101-47200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nfunsnaov 47101	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 47102	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 47103	The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7486. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Rel dom 𝐹    ⇒   ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)
Theorem	aovrcl 47104	Reverse closure for an operation value, analogous to afvvv 47060. In contrast to ovrcl 7489, 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 47105	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 47106	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 47107	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 47108	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 47109	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 47110	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 47111	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 47112*	A frequently used special case of rspc2ev 3648 for operation values, analogous to rspceov 7497. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) )
Theorem	fnotaovb 47113	Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6974. (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 47114*	An operation maps to a class to which all values belong, analogous to ffnov 7576. (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 47115	Closure law for an operation, analogous to fovcl 7578. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)
Theorem	aovmpt4g 47116*	Value of a function given by the maps-to notation, analogous to ovmpt4g 7597. (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 47117*	Domain of closure of an operation. In contrast to oprssdm 7631, 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 47118	The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7635 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 47119	Reverse closure law, in contrast to ndmovrcl 7636 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 47120	Any operation is commutative outside its domain, analogous to ndmovcom 7637. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    ⇒   ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )
Theorem	ndmaovass 47121	Any operation is associative outside its domain. In contrast to ndmovass 7638 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 47122	Any operation is distributive outside its domain. In contrast to ndmovdistr 7639 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 47035. The current definition of the value (𝐹‘𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6581) assures that this value is always a set, see fex 7263. 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 6955 and fvprc 6912). "(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 47034. 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 6956). To avoid such an ambiguity, an alternative definition (𝐹''''𝐴) (see df-afv2 47124) 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 47126). 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 47139, 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 47127). 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 47175), but (𝐹‘𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 47180. The alternate definition, however, corresponds to the current definition ((𝐹‘𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 47176). With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 47142. 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 6581 of (𝐹‘𝐴), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6919-> afv2eq1 47131, fveq2 6920-> afv2eq2 47132, nffv 6930-> nfafv2 47133, csbfv12 6968-> csbafv212g , rlimdm 15597-> rlimdmafv2 47173, tz6.12-1 6943-> tz6.12-1-afv2 47156, fveu 6909-> afv2eu 47153. Six theorems proved by directly using df-fv 6581 are within a mathbox (fvsb 44421, uncov 37561) or not used (rlimdmafv 47092, avril1 30495) or experimental (dfafv2 47047, dfafv22 47174). However, the remaining 11 theorems proved by directly using df-fv 6581 are used more or less often: * fvex 6933: 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 47128 resp. afv2ex 47129). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6933. * fvres 6939: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 47154). 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 47154 can be used instead of fvres 6939. * tz6.12-2 6908 (-> tz6.12-2-afv2 47152): root theorem of many theorems which have not a strict analogue, and which are used many times: ** fvprc 6912 (-> afv2prc 47141), used in 193 proofs, ** tz6.12i 6948 (-> tz6.12i-afv2 47158), used - indirectly via fvbr0 6949 and fvrn0 6950 - in 19 proofs, and in fvclss 7278 used in fvclex 7999 used in fvresex 8000 (which is not used!) and in dcomex 10516 (used in 4 proofs), ** ndmfv 6955 (-> ndmafv2nrn ), used in 124 proofs ** nfunsn 6962 (-> nfunsnafv2 ), used by fvfundmfvn0 6963 (used in 3 proofs), and dffv2 7017 (not used) ** funpartfv 35909, setrec2lem1 48785 (mathboxes) * fv2 6915: only used by elfv 6918, which is only used by fv3 6938, which is not used. * dffv3 6916 (-> dfafv23 ): used by dffv4 6917 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 44870), by shftval 15123 (itself used in 11 proofs), by dffv5 35888 (mathbox) and by fvco2 7019 (-> afv2co2 47172). * fvopab5 7062: used only by ajval 30893 (not used) and by adjval 31922, which is used in adjval2 31923 (not used) and in adjbdln 32115 (used in 7 proofs). * zsum 15766: used (via isum 15767, sum0 15769, sumss 15772 and fsumsers 15776) in 76 proofs. * isumshft 15887: used in pserdv2 26492 (used in logtayl 26720, binomcxplemdvsum 44324) , eftlub 16157 (used in 4 proofs), binomcxplemnotnn0 44325 (used in binomcxp 44326 only) and logtayl 26720 (used in 4 proofs). * ovtpos 8282: used in 16 proofs. * zprod 15985: used in 3 proofs: iprod 15986, zprodn0 15987 and prodss 15995 * iprodclim3 16048: 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 6915, dffv3 6916, fvopab5 7062, zsum 15766, isumshft 15887, ovtpos 8282 and zprod 15985 are not critical or are, hopefully, also valid for the alternative definition, fvex 6933, fvres 6939 and tz6.12-2 6908 (and the theorems based on them) are essential for the current definition of function values.
Syntax	cafv2 47123	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 47032.
class (𝐹''''𝐴)
Definition	df-afv2 47124*	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 6581, and especially ndmfv 6955), (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 47125*	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 47126	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 47127	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 47128	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 47129	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 47130	Equality deduction for function value, analogous to fveq12d 6927. (Contributed by AV, 4-Sep-2022.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))
Theorem	afv2eq1 47131	Equality theorem for function value, analogous to fveq1 6919. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴))
Theorem	afv2eq2 47132	Equality theorem for function value, analogous to fveq2 6920. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵))
Theorem	nfafv2 47133	Bound-variable hypothesis builder for function value, analogous to nffv 6930. To prove a deduction version of this analogous to nffvd 6932 is not easily possible because a deduction version of nfdfat 47042 cannot be shown easily. (Contributed by AV, 4-Sep-2022.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(𝐹''''𝐴)
Theorem	csbafv212g 47134	Move class substitution in and out of a function value, analogous to csbfv12 6968, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7492. (Contributed by AV, 4-Sep-2022.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵))
Theorem	fexafv2ex 47135	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 47136	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 47137	The value of a class outside its domain is not in the range, compare with ndmfv 6955. (Contributed by AV, 2-Sep-2022.)
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	funressndmafv2rn 47138	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 47139	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 47140	If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6962. (Contributed by AV, 2-Sep-2022.)
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	afv2prc 47141	A function's value at a proper class is not defined, compare with fvprc 6912. (Contributed by AV, 5-Sep-2022.)
⊢ (¬ 𝐴 ∈ V → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	dfatafv2rnb 47142	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 47143	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 47144	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 47145	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 47146	An alternate function value belongs to the range of the function, analogous to fvelrn 7110. (Contributed by AV, 3-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹)
Theorem	afv20defat 47147	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 47148	An alternate function value belongs to the range of the function, analogous to fnfvelrn 7114. (Contributed by AV, 2-Sep-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ ran 𝐹)
Theorem	fafv2elcdm 47149	An alternate function value belongs to the codomain of the function, analogous to ffvelcdm 7115. (Contributed by AV, 2-Sep-2022.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ 𝐵)
Theorem	fafv2elrnb 47150	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 47151	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 47152*	Function value when 𝐹 is (locally) not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27, analogous to tz6.12-2 6908. (Contributed by AV, 5-Sep-2022.)
⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) ∉ ran 𝐹)
Theorem	afv2eu 47153*	The value of a function at a unique point, analogous to fveu 6909. (Contributed by AV, 5-Sep-2022.)
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
Theorem	afv2res 47154	The value of a restricted function for an argument at which the function is defined. Analog to fvres 6939. (Contributed by AV, 5-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (𝐹''''𝐴))
Theorem	tz6.12-afv2 47155*	Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6945. (Contributed by AV, 5-Sep-2022.)
⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦)
Theorem	tz6.12-1-afv2 47156*	Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12-1 6943. (Contributed by AV, 5-Sep-2022.)
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
Theorem	tz6.12c-afv2 47157*	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6942. (Contributed by AV, 5-Sep-2022.)
⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
Theorem	tz6.12i-afv2 47158	Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6948. (Contributed by AV, 5-Sep-2022.)
⊢ (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵))
Theorem	funressnbrafv2 47159	The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6971. (Contributed by AV, 7-Sep-2022.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Theorem	dfatbrafv2b 47160	Equivalence of function value and binary relation, analogous to fnbrfvb 6973 or funbrfvb 6975. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 47128). (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	dfatopafv2b 47161	Equivalence of function value and ordered pair membership, analogous to fnopfvb 6974 or funopfvb 6976. (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Theorem	funbrafv2 47162	The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6971. (Contributed by AV, 6-Sep-2022.)
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Theorem	fnbrafv2b 47163	Equivalence of function value and binary relation, analogous to fnbrfvb 6973. (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))
Theorem	fnopafv2b 47164	Equivalence of function value and ordered pair membership, analogous to fnopfvb 6974. (Contributed by AV, 6-Sep-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
Theorem	funbrafv22b 47165	Equivalence of function value and binary relation, analogous to funbrfvb 6975. (Contributed by AV, 6-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	funopafv2b 47166	Equivalence of function value and ordered pair membership, analogous to funopfvb 6976. (Contributed by AV, 6-Sep-2022.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Theorem	dfatsnafv2 47167	Singleton of function value, analogous to fnsnfv 7001. (Contributed by AV, 7-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))
Theorem	dfafv23 47168*	A definition of function value in terms of iota, analogous to dffv3 6916. (Contributed by AV, 6-Sep-2022.)
⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
Theorem	dfatdmfcoafv2 47169	Domain of a function composition, analogous to dmfco 7018. (Contributed by AV, 7-Sep-2022.)
⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺''''𝐴) ∈ dom 𝐹))
Theorem	dfatcolem 47170*	Lemma for dfatco 47171. (Contributed by AV, 8-Sep-2022.)
⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ∃!𝑦 𝑋(𝐹 ∘ 𝐺)𝑦)
Theorem	dfatco 47171	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 47172	Value of a function composition, analogous to fvco2 7019. (Contributed by AV, 8-Sep-2022.)
⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ((𝐹 ∘ 𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋)))
Theorem	rlimdmafv2 47173	Two ways to express that a function has a limit, analogous to rlimdm 15597. (Contributed by AV, 5-Sep-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝𝑟 ↔ 𝐹 ⇝𝑟 ( ⇝𝑟 ''''𝐹)))
Theorem	dfafv22 47174	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 47175	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 47176	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 47177	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 47178	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 47179	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 47180	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 47181	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 47182	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 47183	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 47184	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 47185	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 47186	Extend wff notation to include the 'not element of' relation.
class _∉
Definition	df-nelbr 47187*	Define negated membership as binary relation. Analogous to df-eprel 5599 (the membership relation). (Contributed by AV, 26-Dec-2021.)
⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
Theorem	dfnelbr2 47188	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 47189	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 47190	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 47191	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 47192	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 47193*	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 47194*	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 47195	Equality of function values with conditional arguments, see also fvif 6936. (Contributed by Alexander van der Vekens, 21-May-2018.)
⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶)))
Theorem	rnfdmpr 47196	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 47197	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 47198*	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 47199	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 47200*	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)