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Theorem List for Metamath Proof Explorer - 41001-41100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfunbrafv 41001 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6221. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))

Theoremfunbrafv2b 41002 Function value in terms of a binary relation, analogous to funbrfv2b 6227. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵)))

Theoremdfafn5a 41003* Representation of a function in terms of its values, analogous to dffn5 6228 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))

Theoremdfafn5b 41004* Representation of a function in terms of its values, analogous to dffn5 6228 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)
(∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥))))

Theoremfnrnafv 41005* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6229. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})

Theoremafvelrnb 41006* A member of a function's range is a value of the function, analogous to fvelrnb 6230 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))

Theoremafvelrnb0 41007* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6230. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
(𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))

Theoremdfaimafn 41008* Alternate definition of the image of a function, analogous to dfimafn 6232. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})

Theoremdfaimafn2 41009* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6233. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})

Theoremafvelima 41010* Function value in an image, analogous to fvelima 6235. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → ∃𝑥𝐵 (𝐹'''𝑥) = 𝐴)

Theoremafvelrn 41011 A function's value belongs to its range, analogous to fvelrn 6338. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹)

Theoremfnafvelrn 41012 A function's value belongs to its range, analogous to fnfvelrn 6342. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)

Theoremfafvelrn 41013 A function's value belongs to its codomain, analogous to ffvelrn 6343. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ 𝐵)

Theoremffnafv 41014* A function maps to a class to which all values belong, analogous to ffnfv 6374. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))

Theoremafvres 41015 The value of a restricted function, analogous to fvres 6194. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴𝐵 → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))

Theoremtz6.12-afv 41016* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6198. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)

Theoremtz6.12-1-afv 41017* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6197. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦)

Theoremdmfcoafv 41018 Domains of a function composition, analogous to dmfco 6259. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))

Theoremafvco2 41019 Value of a function composition, analogous to fvco2 6260. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))

Theoremrlimdmafv 41020 Two ways to express that a function has a limit, analogous to rlimdm 14263. (Contributed by Alexander van der Vekens, 27-Nov-2017.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)       (𝜑 → (𝐹 ∈ dom ⇝𝑟𝐹𝑟 ( ⇝𝑟 '''𝐹)))

20.34.2.8  Alternative definition of the value of an operation

Theoremaoveq123d 41021 Equality deduction for operation value, analogous to oveq123d 6656. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

Theoremnfaov 41022 Bound-variable hypothesis builder for operation value, analogous to nfov 6661. To prove a deduction version of this analogous to nfovd 6660 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 40979). (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐵       𝑥 ((𝐴𝐹𝐵))

Theoremcsbaovg 41023 Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐴𝐷𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )

Theoremaovfundmoveq 41024 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))

Theoremaovnfundmuv 41025 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)

Theoremndmaov 41026 The value of an operation outside its domain, analogous to ndmafv 40983. (Contributed by Alexander van der Vekens, 26-May-2017.)
(¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)

Theoremndmaovg 41027 The value of an operation outside its domain, analogous to ndmovg 6802. (Contributed by Alexander van der Vekens, 26-May-2017.)
((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → ((𝐴𝐹𝐵)) = V)

Theoremaovvdm 41028 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)

Theoremnfunsnaov 41029 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)

Theoremaovvfunressn 41030 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⟩}))

Theoremaovprc 41031 The value of an operation when the one of the arguments is a proper class, analogous to ovprc 6668. (Contributed by Alexander van der Vekens, 26-May-2017.)
Rel dom 𝐹       (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)

Theoremaovrcl 41032 Reverse closure for an operation value, analogous to afvvv 40988. In contrast to ovrcl 6671, 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))

Theoremaovpcov0 41033 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) = ∅)

Theoremaovnuoveq 41034 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))

Theoremaovvoveq 41035 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))

Theoremaov0ov0 41036 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) = ∅)

Theoremaovovn0oveq 41037 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))

Theoremaov0nbovbi 41038 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))

Theoremaovov0bi 41039 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))

Theoremrspceaov 41040* A frequently used special case of rspc2ev 3319 for operation values, analogous to rspceov 6677. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐶𝐴𝐷𝐵𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥𝐴𝑦𝐵 𝑆 = ((𝑥𝐹𝑦)) )

Theoremfnotaovb 41041 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6224. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ( ((𝐶𝐹𝐷)) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))

Theoremffnaov 41042* An operation maps to a class to which all values belong, analogous to ffnov 6749. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶))

Theoremfaovcl 41043 Closure law for an operation, analogous to fovcl 6750. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹:(𝑅 × 𝑆)⟶𝐶       ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)

Theoremaovmpt4g 41044* Value of a function given by the "maps to" notation, analogous to ovmpt4g 6768. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = 𝐶)

Theoremaoprssdm 41045* Domain of closure of an operation. In contrast to oprssdm 6800, no additional property for S (¬ ∅ ∈ 𝑆) is required! (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆)       (𝑆 × 𝑆) ⊆ dom 𝐹

Theoremndmaovcl 41046 The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6804 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)    &   ((𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)    &    ((𝐴𝐹𝐵)) ∈ V        ((𝐴𝐹𝐵)) ∈ 𝑆

Theoremndmaovrcl 41047 Reverse closure law, in contrast to ndmovrcl 6805 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 → (𝐴𝑆𝐵𝑆))

Theoremndmaovcom 41048 Any operation is commutative outside its domain, analogous to ndmovcom 6806. (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )

Theoremndmaovass 41049 Any operation is associative outside its domain. In contrast to ndmovass 6807 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)       (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) )

Theoremndmaovdistr 41050 Any operation is distributive outside its domain. In contrast to ndmovdistr 6808 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)       (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) )

20.34.3  General auxiliary theorems

20.34.3.1  Negated membership (alternative)

Syntaxcnelbr 41051 Extend wff notation to include the 'not elemet of' relation.
class _∉

Definitiondf-nelbr 41052* Define negated membership as binary relation. Analogous to df-eprel 5019 (the epsilon relation). (Contributed by AV, 26-Dec-2021.)
_∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}

Theoremdfnelbr2 41053 Alternate definition of the negated membership as binary relation. (Proposed by BJ, 27-Dec-2021.) (Contributed by AV, 27-Dec-2021.)
_∉ = ((V × V) ∖ E )

Theoremnelbr 41054 The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))

Theoremnelbrim 41055 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 ¬ 𝐴𝐵 would be true, but not 𝐴 _∉ 𝐵. (Contributed by AV, 26-Dec-2021.)
(𝐴 _∉ 𝐵 → ¬ 𝐴𝐵)

Theoremnelbrnel 41056 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.)
((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵𝐴𝐵))

Theoremnelbrnelim 41057 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 _∉ 𝐵𝐴𝐵)

20.34.3.2  The empty set - extension

Theoremralralimp 41058* 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.)
((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 ((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))

20.34.3.3  Unordered and ordered pairs - extension

Theoremelprneb 41059 An element of a proper unordered pair is the first element iff it is not the second element. (Contributed by AV, 18-Jun-2020.)
((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐵𝐶) → (𝐴 = 𝐵𝐴𝐶))

Theoremopidg 41060 The ordered pair 𝐴, 𝐴 in Kuratowski's representation. Closed form of opid 4412. (Contributed by AV, 18-Sep-2020.) (Revised by AV, 18-Sep-2021.)
(𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

20.34.3.4  Indexed union and intersection - extension

TheoremotiunsndisjX 41061* 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.)
(𝐵𝑋Disj 𝑎𝑉 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩})

20.34.3.5  Functions - extension

Theoremfvifeq 41062 Equality of function values with conditional arguments, see also fvif 6191. (Contributed by Alexander van der Vekens, 21-May-2018.)
(𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = if(𝜑, (𝐹𝐵), (𝐹𝐶)))

Theoremrnfdmpr 41063 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.)
((𝑋𝑉𝑌𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))

Theoremimarnf1pr 41064 The image of the range of a function 𝐹 under a function 𝐸 if 𝐹 is a function of a pair into the domain of 𝐸. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
((𝑋𝑉𝑌𝑊) → (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵}))

Theoremfunop1 41065* A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
(∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))

Theoremfun2dmnopgexmpl 41066 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.)
(𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))

Theoremopabresex0d 41067* 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.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝜃)    &   ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)

Theoremopabbrfex0d 41068* A collection of ordered pairs, the class of all possible second components being a set, is a set. (Contributed by AV, 15-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝜃)    &   ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)

Theoremopabresexd 41069* 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.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝑦:𝐴𝐵)    &   ((𝜑𝑥𝐶) → 𝐴𝑈)    &   ((𝜑𝑥𝐶) → 𝐵𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)

Theoremopabbrfexd 41070* A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝑦:𝐴𝐵)    &   ((𝜑𝑥𝐶) → 𝐴𝑈)    &   ((𝜑𝑥𝐶) → 𝐵𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)

20.34.3.6  Ordering on reals - extension

Theoremleltletr 41071 Transitive law, weaker form of lelttr 10113. (Contributed by AV, 14-Oct-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵𝐵 < 𝐶) → 𝐴𝐶))

20.34.3.7  Subtraction - extension

Theoremcnambpcma 41072 ((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) → (((𝐴𝐵) + 𝐶) − 𝐴) = (𝐶𝐵))

Theoremcnapbmcpd 41073 ((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))

20.34.3.8  Ordering on reals (cont.) - extension

Theoremleaddsuble 41074 Addition and subtraction on one side of 'less or equal'. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵𝐶 ↔ ((𝐴 + 𝐵) − 𝐶) ≤ 𝐴))

Theorem2leaddle2 41075 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 < 𝐶𝐵 < 𝐶) → (𝐴 + 𝐵) < (2 · 𝐶)))

Theoremltnltne 41076 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)))

Theoremp1lep2 41077 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))

Theoremltsubsubaddltsub 41078 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 < ((𝐿𝑀) − 𝑁) ↔ (𝐽 + 𝑀) < (𝐿𝑁)))

Theoremzm1nn 41079 An integer minus 1 is positive under certain circumstances. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
((𝑁 ∈ ℕ0𝐿 ∈ ℤ) → ((𝐽 ∈ ℝ ∧ 0 ≤ 𝐽𝐽 < ((𝐿𝑁) − 1)) → (𝐿 − 1) ∈ ℕ))

20.34.3.9  Nonnegative integers (as a subset of complex numbers) - extension

Theoremnn0resubcl 41080 Closure law for subtraction of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴𝐵) ∈ ℝ)

20.34.3.10  Integers (as a subset of complex numbers) - extension

Theoremzgeltp1eq 41081 If an integer is between another integer and its successor, the integer is equal to the other integer. (Contributed by AV, 30-May-2020.)
((𝐼 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐴𝐼𝐼 < (𝐴 + 1)) → 𝐼 = 𝐴))

20.34.3.11  Decimal arithmetic - extension

Theorem1t10e1p1e11 41082 11 is 1 times 10 to the power of 1, plus 1. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
11 = ((1 · (10↑1)) + 1)

Theorem1t10e1p1e11OLD 41083 Obsolete version of 1t10e1p1e11 41082 as of 9-Sep-2021. (Contributed by AV, 4-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
11 = ((1 · (10↑1)) + 1)

Theoremdeccarry 41084 Add 1 to a 2 digit number with carry. This is a special case of decsucc 11535, but in closed form. As observed by ML, this theorem allows for carrying the 1 down multiple decimal constructors, so we can carry the 1 multiple times down a multi-digit number, e.g. by applying this theorem three times we get (999 + 1) = 1000. (Contributed by AV, 4-Aug-2020.) (Revised by ML, 8-Aug-2020.) (Proof shortened by AV, 10-Sep-2021.)
(𝐴 ∈ ℕ → (𝐴9 + 1) = (𝐴 + 1)0)

20.34.3.12  Upper sets of integers - extension

Theoremeluzge0nn0 41085 If an integer is greater than or equal to a nonnegative integer, then it is a nonnegative integer. (Contributed by Alexander van der Vekens, 27-Aug-2018.)
(𝑁 ∈ (ℤ𝑀) → (0 ≤ 𝑀𝑁 ∈ ℕ0))

20.34.3.13  Infinity and the extended real number system (cont.) - extension

Theoremnltle2tri 41086 Negated extended trichotomy law for 'less than' and 'less than or equal to'. (Contributed by AV, 18-Jul-2020.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ¬ (𝐴 < 𝐵𝐵𝐶𝐶𝐴))

20.34.3.14  Finite intervals of integers - extension

Theoremssfz12 41087 Subset relationship for finite sets of sequential integers. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾𝐿) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝑀𝐾𝐿𝑁)))

Theoremelfz2z 41088 Membership of an integer in a finite set of sequential integers starting at 0. (Contributed by Alexander van der Vekens, 25-May-2018.)
((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (0...𝑁) ↔ (0 ≤ 𝐾𝐾𝑁)))

Theorem2elfz3nn0 41089 If there are two elements in a finite set of sequential integers starting at 0, these two elements as well as the upper bound are nonnegative integers. (Contributed by Alexander van der Vekens, 7-Apr-2018.)
((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0))

Theoremfz0addcom 41090 The addition of two members of a finite set of sequential integers starting at 0 is commutative. (Contributed by Alexander van der Vekens, 22-May-2018.) (Revised by Alexander van der Vekens, 9-Jun-2018.)
((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theorem2elfz2melfz 41091 If the sum of two integers of a 0 based finite set of sequential integers is greater than the upper bound, the difference between one of the integers and the difference between the upper bound and the other integer is in the 0 based finite set of sequential integers with the first integer as upper bound. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by Alexander van der Vekens, 31-May-2018.)
((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝑁 < (𝐴 + 𝐵) → (𝐵 − (𝑁𝐴)) ∈ (0...𝐴)))

Theoremfz0addge0 41092 The sum of two integers in 0 based finite sets of sequential integers is greater than or equal to zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
((𝐴 ∈ (0...𝑀) ∧ 𝐵 ∈ (0...𝑁)) → 0 ≤ (𝐴 + 𝐵))

Theoremelfzlble 41093 Membership of an integer in a finite set of sequential integers with the integer as upper bound and a lower bound less than or equal to the integer. (Contributed by AV, 21-Oct-2018.)
((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → 𝑁 ∈ ((𝑁𝑀)...𝑁))

Theoremelfzelfzlble 41094 Membership of an element of a finite set of sequential integers in a finite set of sequential integers with the same upper bound and a lower bound less than the upper bound. (Contributed by AV, 21-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑁 < (𝑀 + 𝐾)) → 𝐾 ∈ ((𝑁𝑀)...𝑁))

20.34.3.15  Half-open integer ranges - extension

Theoremfzopred 41095 Join a predecessor to the beginning of an open integer interval. Generalization of fzo0sn0fzo1 12541. (Contributed by AV, 14-Jul-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → (𝑀..^𝑁) = ({𝑀} ∪ ((𝑀 + 1)..^𝑁)))

Theoremfzopredsuc 41096 Join a predecessor and a successor to the beginning and the end of an open integer interval. This theorem holds even if 𝑁 = 𝑀 (then (𝑀...𝑁) = {𝑀} = ({𝑀} ∪ ∅) ∪ {𝑀}). (Contributed by AV, 14-Jul-2020.)
(𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = (({𝑀} ∪ ((𝑀 + 1)..^𝑁)) ∪ {𝑁}))

Theorem1fzopredsuc 41097 Join 0 and a successor to the beginning and the end of an open integer interval starting at 1. (Contributed by AV, 14-Jul-2020.)
(𝑁 ∈ ℕ0 → (0...𝑁) = (({0} ∪ (1..^𝑁)) ∪ {𝑁}))

Theoremel1fzopredsuc 41098 An element of an open integer interval starting at 1 joined by 0 and a successor at the beginning and the end is either 0 or an element of the open integer interval or the successor. (Contributed by AV, 14-Jul-2020.)
(𝑁 ∈ ℕ0 → (𝐼 ∈ (0...𝑁) ↔ (𝐼 = 0 ∨ 𝐼 ∈ (1..^𝑁) ∨ 𝐼 = 𝑁)))

Theoremsubsubelfzo0 41099 Subtracting a difference from a number which is not less than the difference results in a bounded nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
((𝐴 ∈ (0..^𝑁) ∧ 𝐼 ∈ (0..^𝑁) ∧ ¬ 𝐼 < (𝑁𝐴)) → (𝐼 − (𝑁𝐴)) ∈ (0..^𝐴))

Theoremfzoopth 41100 A half-open integer range can represent an ordered pair, analogous to fzopth 12363. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → ((𝑀..^𝑁) = (𝐽..^𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))

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