P. 404 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 40301-40400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	disjf1o 40301*	A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅}    &   ⊢ 𝐷 = (ran 𝐹 ∖ {∅})    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶–1-1-onto→𝐷)
Theorem	fompt 40302*	Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    ⇒   ⊢ (𝐹:𝐴–onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
Theorem	disjinfi 40303*	Only a finite number of disjoint sets can have a nonempty intersection with a finite set 𝐶 (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ Fin)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐵 ∩ 𝐶) ≠ ∅} ∈ Fin)
Theorem	fvovco 40304	Value of the composition of an operator, with a given function. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐹:𝑋⟶(𝑉 × 𝑊))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = ((1st ‘(𝐹‘𝑌))𝑂(2nd ‘(𝐹‘𝑌))))
Theorem	ssnnf1octb 40305*	There exists a bijection between a subset of ℕ and a given nonempty countable set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴))
Theorem	mapdm0OLD 40306	Obsolete version of mapdm0 8155 as of 3-Dec-2021. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = {∅})
Theorem	nnf1oxpnn 40307	There is a bijection between the set of positive integers and the Cartesian product of the set of positive integers with itself. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ ∃𝑓 𝑓:ℕ–1-1-onto→(ℕ × ℕ)
Theorem	rnmptssd 40308*	The range of an operation given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶)
Theorem	projf1o 40309*	A biijection from a set to a projection in a two dimensional space. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨𝐴, 𝑥⟩)    ⇒   ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→({𝐴} × 𝐵))
Theorem	fvmap 40310	Function value for a member of a set exponentiation. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐴 ↑𝑚 𝐵))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝐴)
Theorem	fvixp2 40311*	Projection of a factor of an indexed Cartesian product. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
Theorem	fidmfisupp 40312	A function with a finite domain is finitely supported. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐹:𝐷⟶𝑅)    &   ⊢ (𝜑 → 𝐷 ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
Theorem	choicefi 40313*	For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
Theorem	mpct 40314	The exponentiation of a countable set to a finite set is countable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    ⇒   ⊢ (𝜑 → (𝐴 ↑𝑚 𝐵) ≼ ω)
Theorem	cnmetcoval 40315	Value of the distance function of the metric space of complex numbers, composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ 𝐷 = (abs ∘ − )    &   ⊢ (𝜑 → 𝐹:𝐴⟶(ℂ × ℂ))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = (abs‘((1st ‘(𝐹‘𝐵)) − (2nd ‘(𝐹‘𝐵)))))
Theorem	fcomptss 40316*	Express composition of two functions as a maps-to applying both in sequence. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))))
Theorem	elmapsnd 40317	Membership in a set exponentiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹 Fn {𝐴})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑𝑚 {𝐴}))
Theorem	mapss2 40318	Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐶 ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)))
Theorem	fsneq 40319	Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ 𝐵 = {𝐴}    &   ⊢ (𝜑 → 𝐹 Fn 𝐵)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
Theorem	difmap 40320	Difference of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐶 ≠ ∅)    ⇒   ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ↑𝑚 𝐶) ⊆ ((𝐴 ↑𝑚 𝐶) ∖ (𝐵 ↑𝑚 𝐶)))
Theorem	unirnmap 40321	Given a subset of a set exponentiation, the base set can be restricted. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑𝑚 𝐴))    ⇒   ⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑𝑚 𝐴))
Theorem	inmap 40322	Intersection of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑍)    ⇒   ⊢ (𝜑 → ((𝐴 ↑𝑚 𝐶) ∩ (𝐵 ↑𝑚 𝐶)) = ((𝐴 ∩ 𝐵) ↑𝑚 𝐶))
Theorem	fcoss 40323	Composition of two mappings. Similar to fco 6308, but with a weaker condition on the domain of 𝐹. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐺:𝐷⟶𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐷⟶𝐵)
Theorem	fsneqrn 40324	Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ 𝐵 = {𝐴}    &   ⊢ (𝜑 → 𝐹 Fn 𝐵)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) ∈ ran 𝐺))
Theorem	difmapsn 40325	Difference of two sets exponentiatiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑍)    ⇒   ⊢ (𝜑 → ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶})) = ((𝐴 ∖ 𝐵) ↑𝑚 {𝐶}))
Theorem	mapssbi 40326	Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐶 ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)))
Theorem	unirnmapsn 40327	Equality theorem for a subset of a set exponentiation, where the exponent is a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐶 = {𝐴}    &   ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑𝑚 𝐶))    ⇒   ⊢ (𝜑 → 𝑋 = (ran ∪ 𝑋 ↑𝑚 𝐶))
Theorem	iunmapss 40328*	The indexed union of set exponentiations is a subset of the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑𝑚 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐶))
Theorem	ssmapsn 40329*	A subset 𝐶 of a set exponentiation to a singleton, is its projection 𝐷 exponentiated to the singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑓𝐷    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ⊆ (𝐵 ↑𝑚 {𝐴}))    &   ⊢ 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓    ⇒   ⊢ (𝜑 → 𝐶 = (𝐷 ↑𝑚 {𝐴}))
Theorem	iunmapsn 40330*	The indexed union of set exponentiations to a singleton is equal to the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑍)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑𝑚 {𝐶}) = (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶}))
Theorem	absfico 40331	Mapping domain and codomain of the absolute value function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ abs:ℂ⟶(0[,)+∞)
Theorem	icof 40332	The set of left-closed right-open intervals of extended reals maps to subsets of extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
Theorem	rnmpt0 40333*	The range of a function in maps-to notation is empty if and only if its domain is empty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
Theorem	rnmptn0 40334*	The range of a function in maps-to notation is nonempty if the domain is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → ran 𝐹 ≠ ∅)
Theorem	elpmrn 40335	The range of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → ran 𝐹 ⊆ 𝐴)
Theorem	imaexi 40336	The image of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ (𝐴 “ 𝐵) ∈ V
Theorem	axccdom 40337*	Relax the constraint on ax-cc to dominance instead of equinumerosity. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝑋 ≼ ω)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧))
Theorem	dmmptdf 40338*	The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → dom 𝐴 = 𝐵)
Theorem	elpmi2 40339	The domain of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → dom 𝐹 ⊆ 𝐵)
Theorem	dmrelrnrel 40340*	A relation preserving function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))
Theorem	fco3 40341	Functionality of a composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → Fun 𝐺)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)
Theorem	fvcod 40342	Value of a function composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝐺)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐺)    ⇒   ⊢ (𝜑 → (𝐻‘𝐴) = (𝐹‘(𝐺‘𝐴)))
Theorem	fcod 40343	Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
Theorem	freld 40344	A mapping is a relation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → Rel 𝐹)
Theorem	elrnmpt2id 40345*	Membership in the range of an operation class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) ∈ ran 𝐹)
Theorem	axccd 40346*	An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ≈ ω)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)
Theorem	axccd2 40347*	An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)
Theorem	funimassd 40348*	Sufficient condition for the image of a function being a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Fun 𝐹)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐵)
Theorem	fimassd 40349	The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐹 “ 𝑋) ⊆ 𝐵)
Theorem	feqresmptf 40350*	Express a restricted function as a mapping. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))
Theorem	fnmptd 40351*	The maps-to notation defines a function with domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝜑 → 𝐹 Fn 𝐴)
Theorem	elrnmpt1d 40352	Elementhood in an image set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 ∈ ran 𝐹)
Theorem	dmresss 40353	The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴
Theorem	mptima 40354*	Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
Theorem	dmmptssf 40355	The domain of a mapping is a subset of its base class. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐴    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ dom 𝐹 ⊆ 𝐴
Theorem	dmmptdf2 40356	The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → dom 𝐴 = 𝐵)
Theorem	dmuz 40357	Domain of the upper integers function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ dom ℤ≥ = ℤ
Theorem	fmptd2f 40358*	Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
Theorem	mpteq1df 40359	An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	mptexf 40360	If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg 6756. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐴    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
Theorem	fvmpt4 40361*	Value of a function given by the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
Theorem	fmptf 40362*	Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐵    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
Theorem	resimass 40363	The image of a restriction is a subset of the original image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ ((𝐴 ↾ 𝐵) “ 𝐶) ⊆ (𝐴 “ 𝐶)
Theorem	mptssid 40364	The mapping operation expressed with its actual domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐴    &   ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵)
Theorem	mptfnd 40365	The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
Theorem	mpteq12da 40366	An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
Theorem	rnmptlb 40367*	Boundness below of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
Theorem	rnmptbddlem 40368*	Boundness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
Theorem	rnmptbdd 40369*	Boundness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
Theorem	mptima2 40370*	Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ 𝐶 ↦ 𝐵))
Theorem	fvelimad 40371*	Function value in an image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐹 “ 𝐵))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑥) = 𝐶)
Theorem	funimaeq 40372*	Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)    &   ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴))
Theorem	rnmptssf 40373*	The range of an operation given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐶    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)
Theorem	rnmptbd2lem 40374*	Boundness below of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
Theorem	rnmptbd2 40375*	Boundness below of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
Theorem	infnsuprnmpt 40376*	The indexed infimum of real numbers is the negative of the indexed supremum of the negative values. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)    ⇒   ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
Theorem	suprclrnmpt 40377*	Closure of the indexed supremum of a nonempty bounded set of reals. Range of a function in maps-to notation can be used, to express an indexed supremum. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)    ⇒   ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) ∈ ℝ)
Theorem	suprubrnmpt2 40378*	A member of a nonempty indexed set of reals is less than or equal to the set's upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 ≤ sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ))
Theorem	suprubrnmpt 40379*	A member of a nonempty indexed set of reals is less than or equal to the set's upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ))
Theorem	rnmptssdf 40380*	The range of an operation given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶)
Theorem	rnmptbdlem 40381*	Boundness above of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
Theorem	rnmptbd 40382*	Boundness above of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
Theorem	rnmptss2 40383*	The range of an operation given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ran (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	elmptima 40384*	The image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 = 𝐵))
Theorem	ralrnmpt3 40385*	A restricted quantifier over an image set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
Theorem	fvelima2 40386*	Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ (𝐹 “ 𝐶)) → ∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵)
Theorem	funresd 40387	A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))
Theorem	rnmptssbi 40388*	The range of an operation given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (ran 𝐹 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶))
Theorem	fnfvima2 40389	Given an element of the preimage, its function value is in the image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶))
Theorem	fnfvelrnd 40390	A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹‘𝐵) ∈ ran 𝐹)
Theorem	imass2d 40391	Subset theorem for image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵))
Theorem	imassmpt 40392*	Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ 𝐷))
Theorem	fnssresd 40393	Restriction of a function to a subclass of its domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐵) Fn 𝐵)
Theorem	fpmd 40394	A total function is a partial function. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐶⟶𝐵)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑pm 𝐴))
Theorem	fconst7 40395*	An alternative way to express a constant function. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)    ⇒   ⊢ (𝜑 → 𝐹 = (𝐴 × {𝐵}))
20.33.3  Ordering on real numbers - Real and complex numbers basic operations
Theorem	sub2times 40396	Subtracting from a number, twice the number itself, gives negative the number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ ℂ → (𝐴 − (2 · 𝐴)) = -𝐴)
Theorem	abssubrp 40397	The distance of two distinct complex number is a strictly positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ≠ 𝐵) → (abs‘(𝐴 − 𝐵)) ∈ ℝ+)
Theorem	elfzfzo 40398	Relationship between membership in a half-open finite set of sequential integers and membership in a finite set of sequential intergers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ (𝑀..^𝑁) ↔ (𝐴 ∈ (𝑀...𝑁) ∧ 𝐴 < 𝑁))
Theorem	oddfl 40399	Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐾 ∈ ℤ ∧ (𝐾 mod 2) ≠ 0) → 𝐾 = ((2 · (⌊‘(𝐾 / 2))) + 1))
Theorem	abscosbd 40400	Bound for the absolute value of the cosine of a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ ℝ → (abs‘(cos‘𝐴)) ≤ 1)