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Theorem List for Metamath Proof Explorer - 41701-41800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfnchoice 41701* For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝐴 ∈ Fin → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)))

Theoremrefsumcn 41702* A finite sum of continuous real functions, from a common topological space, is continuous. The class expression for B normally contains free variables k and x to index it. See fsumcn 23485 for the analogous theorem on continuous complex functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ Σ𝑘𝐴 𝐵) ∈ (𝐽 Cn 𝐾))

Theoremrfcnpre2 41703 If 𝐹 is a continuous function with respect to the standard topology, then the preimage A of the values smaller than a given extended real 𝐵, is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐵    &   𝑥𝐹    &   𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑋 = 𝐽    &   𝐴 = {𝑥𝑋 ∣ (𝐹𝑥) < 𝐵}    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴𝐽)

Theoremcncmpmax 41704* When the hypothesis for the extreme value theorem hold, then the sup of the range of the function belongs to the range, it is real and it an upper bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑇 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑇 ≠ ∅)       (𝜑 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ ∀𝑡𝑇 (𝐹𝑡) ≤ sup(ran 𝐹, ℝ, < )))

Theoremrfcnpre3 41705* If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than or equal to a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐴 = {𝑡𝑇𝐵 ≤ (𝐹𝑡)}    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴 ∈ (Clsd‘𝐽))

Theoremrfcnpre4 41706* If F is a continuous function with respect to the standard topology, then the preimage A of the values less than or equal to a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐴 = {𝑡𝑇 ∣ (𝐹𝑡) ≤ 𝐵}    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴 ∈ (Clsd‘𝐽))

Theoremsumpair 41707* Sum of two distinct complex values. The class expression for 𝐴 and 𝐵 normally contain free variable 𝑘 to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑘𝐷)    &   (𝜑𝑘𝐸)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐸 ∈ ℂ)    &   (𝜑𝐴𝐵)    &   ((𝜑𝑘 = 𝐴) → 𝐶 = 𝐷)    &   ((𝜑𝑘 = 𝐵) → 𝐶 = 𝐸)       (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))

Theoremrfcnnnub 41708* Given a real continuous function 𝐹 defined on a compact topological space, there is always a positive integer that is a strict upper bound of its range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   𝑇 = 𝐽    &   (𝜑𝑇 ≠ ∅)    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐹𝐶)       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < 𝑛)

Theoremrefsum2cnlem1 41709* This is the core Lemma for refsum2cn 41710: the sum of two continuous real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐺    &   𝑥𝜑    &   𝐴 = (𝑘 ∈ {1, 2} ↦ if(𝑘 = 1, 𝐹, 𝐺))    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ ((𝐹𝑥) + (𝐺𝑥))) ∈ (𝐽 Cn 𝐾))

Theoremrefsum2cn 41710* The sum of two continuus real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐹    &   𝑥𝐺    &   𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ ((𝐹𝑥) + (𝐺𝑥))) ∈ (𝐽 Cn 𝐾))

Theoremelunnel2 41711 A member of a union that is not a member of the second class, is a member of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ (𝐵𝐶) ∧ ¬ 𝐴𝐶) → 𝐴𝐵)

Theoremadantlllr 41712 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (((((𝜑𝜂) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem3adantlr3 41713 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)       (((𝜑 ∧ (𝜓𝜒𝜂)) ∧ 𝜃) → 𝜏)

Theoremnnxrd 41714 A natural number is an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℝ*)

Theorem3adantll2 41715 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       ((((𝜑𝜂𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem3adantll3 41716 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       ((((𝜑𝜓𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theoremssnel 41717 If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴𝐵 ∧ ¬ 𝐶𝐵) → ¬ 𝐶𝐴)

Theoremelabrexg 41718* Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝑥𝐴𝐵𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})

Theoremsncldre 41719 A singleton is closed w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ → {𝐴} ∈ (Clsd‘(topGen‘ran (,))))

Theoremn0p 41720 A polynomial with a nonzero coefficient is not the zero polynomial. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
((𝑃 ∈ (Poly‘ℤ) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘𝑃)‘𝑁) ≠ 0) → 𝑃 ≠ 0𝑝)

Theorempm2.65ni 41721 Inference rule for proof by contradiction. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝜑𝜓)    &   𝜑 → ¬ 𝜓)       𝜑

Theorempwssfi 41722 Every element of the power set of 𝐴 is finite if and only if 𝐴 is finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴𝑉 → (𝐴 ∈ Fin ↔ 𝒫 𝐴 ⊆ Fin))

Theoremiuneq2df 41723 Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremnnfoctb 41724* There exists a mapping from onto any (nonempty) countable set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto𝐴)

Theoremssinss1d 41725 Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝐶)       (𝜑 → (𝐴𝐵) ⊆ 𝐶)

Theoremelpwinss 41726 An element of the powerset of 𝐵 intersected with anything, is a subset of 𝐵. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴 ∈ (𝒫 𝐵𝐶) → 𝐴𝐵)

Theoremunidmex 41727 If 𝐹 is a set, then dom 𝐹 is a set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹𝑉)    &   𝑋 = dom 𝐹       (𝜑𝑋 ∈ V)

Theoremndisj2 41728* A non-disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑥 = 𝑦𝐵 = 𝐶)       Disj 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))

Theoremzenom 41729 The set of integer numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ℤ ≈ ω

Theoremuzwo4 41730* Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑗𝜓    &   (𝑗 = 𝑘 → (𝜑𝜓))       ((𝑆 ⊆ (ℤ𝑀) ∧ ∃𝑗𝑆 𝜑) → ∃𝑗𝑆 (𝜑 ∧ ∀𝑘𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))

Theoremunisn0 41731 The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
{∅} = ∅

Theoremssin0 41732 If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) = ∅)

Theoreminabs3 41733 Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐶𝐵 → ((𝐴𝐵) ∩ 𝐶) = (𝐴𝐶))

Theorempwpwuni 41734 Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵))

Theoremdisjiun2 41735* In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑Disj 𝑥𝐴 𝐵)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷 ∈ (𝐴𝐶))    &   (𝑥 = 𝐷𝐵 = 𝐸)       (𝜑 → ( 𝑥𝐶 𝐵𝐸) = ∅)

Theorem0pwfi 41736 The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
∅ ∈ (𝒫 𝐴 ∩ Fin)

Theoremssinss2d 41737 Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐵𝐶)       (𝜑 → (𝐴𝐵) ⊆ 𝐶)

Theoremzct 41738 The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ℤ ≼ ω

Theorempwfin0 41739 A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝒫 𝐴 ∩ Fin) ≠ ∅

Theoremuzct 41740 An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑍 = (ℤ𝑁)       𝑍 ≼ ω

Theoremiunxsnf 41741* A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐶    &   𝐴 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)        𝑥 ∈ {𝐴}𝐵 = 𝐶

Theoremfiiuncl 41742* If a set is closed under the union of two sets, then it is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐷)    &   ((𝜑𝑦𝐷𝑧𝐷) → (𝑦𝑧) ∈ 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)       (𝜑 𝑥𝐴 𝐵𝐷)

Theoremiunp1 41743* The addition of the next set to a union indexed by a finite set of sequential integers. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝐵    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)       (𝜑 𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = ( 𝑘 ∈ (𝑀...𝑁)𝐴𝐵))

Theoremfiunicl 41744* If a set is closed under the union of two sets, then it is closed under finite union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝜑𝑥𝐴𝑦𝐴) → (𝑥𝑦) ∈ 𝐴)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)       (𝜑 𝐴𝐴)

Theoremixpeq2d 41745 Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)

Theoremdisjxp1 41746* The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑Disj 𝑥𝐴 𝐵)       (𝜑Disj 𝑥𝐴 (𝐵 × 𝐶))

Theoremdisjsnxp 41747* The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Disj 𝑗𝐴 ({𝑗} × 𝐵)

Theoremeliind 41748* Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 𝑥𝐵 𝐶)    &   (𝜑𝐾𝐵)    &   (𝑥 = 𝐾 → (𝐴𝐶𝐴𝐷))       (𝜑𝐴𝐷)

Theoremrspcef 41749 Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑥𝜓    &   𝑥𝐴    &   𝑥𝐵    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)

Theoreminn0f 41750 A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑥𝐴    &   𝑥𝐵       ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)

Theoremixpssmapc 41751* An infinite Cartesian product is a subset of set exponentiation. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑥𝜑    &   (𝜑𝐶𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑X𝑥𝐴 𝐵 ⊆ (𝐶m 𝐴))

Theoreminn0 41752* A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)

Theoremelintd 41753* Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝑥)       (𝜑𝐴 𝐵)

Theoremssdf 41754* A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑𝐴𝐵)

Theorembrneqtrd 41755 Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑 → ¬ 𝐴𝑅𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑 → ¬ 𝐴𝑅𝐶)

Theoremssnct 41756 A set containing an uncountable set is itself uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑 → ¬ 𝐴 ≼ ω)    &   (𝜑𝐴𝐵)       (𝜑 → ¬ 𝐵 ≼ ω)

Theoremssuniint 41757* Sufficient condition for being a subclass of the union of an intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝑥)       (𝜑𝐴 𝐵)

Theoremelintdv 41758* Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝑥)       (𝜑𝐴 𝐵)

Theoremssd 41759* A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑𝐴𝐵)

Theoremralimralim 41760 Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝜓𝜑))

Theoremsnelmap 41761 Membership of the element in the range of a constant map. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → (𝐴 × {𝑥}) ∈ (𝐵m 𝐴))       (𝜑𝑥𝐵)

Theoremxrnmnfpnf 41762 An extended real that is neither real nor minus infinity, is plus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑 → ¬ 𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ -∞)       (𝜑𝐴 = +∞)

Theoremnelrnmpt 41763* Non-membership in the range of a function in maps-to notaion. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐶𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)       (𝜑 → ¬ 𝐶 ∈ ran 𝐹)

Theoremiuneq1i 41764* Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐴 = 𝐵        𝑥𝐴 𝐶 = 𝑥𝐵 𝐶

Theoremnssrex 41765* Negation of subclass relationship. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐴𝐵 ↔ ∃𝑥𝐴 ¬ 𝑥𝐵)

Theoremssinc 41766* Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑚 ∈ (𝑀..^𝑁)) → (𝐹𝑚) ⊆ (𝐹‘(𝑚 + 1)))       (𝜑 → (𝐹𝑀) ⊆ (𝐹𝑁))

Theoremssdec 41767* Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑚 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑚 + 1)) ⊆ (𝐹𝑚))       (𝜑 → (𝐹𝑁) ⊆ (𝐹𝑀))

Theoremelixpconstg 41768* Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝐹𝑉 → (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵))

Theoremiineq1d 41769* Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremmetpsmet 41770 A metric is a pseudometric. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))

Theoremixpssixp 41771 Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑X𝑥𝐴 𝐵X𝑥𝐴 𝐶)

Theoremballss3 41772* A sufficient condition for a ball being a subset. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   (𝜑𝐷 ∈ (PsMet‘𝑋))    &   (𝜑𝑃𝑋)    &   (𝜑𝑅 ∈ ℝ*)    &   ((𝜑𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥𝐴)       (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)

Theoremiunincfi 41773* Given a sequence of increasing sets, the union of a finite subsequence, is its last element. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))       (𝜑 𝑛 ∈ (𝑀...𝑁)(𝐹𝑛) = (𝐹𝑁))

Theoremnsstr 41774 If it's not a subclass, it's not a subclass of a smaller one. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
((¬ 𝐴𝐵𝐶𝐵) → ¬ 𝐴𝐶)

Theoremrexanuz3 41775* Combine two different upper integer properties into one, for a single integer. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑗𝜑    &   𝑍 = (ℤ𝑀)    &   (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜒)    &   (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓)    &   (𝑘 = 𝑗 → (𝜒𝜃))    &   (𝑘 = 𝑗 → (𝜓𝜏))       (𝜑 → ∃𝑗𝑍 (𝜃𝜏))

Theoremcbvmpo2 41776* Rule to change the second bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑦𝐴    &   𝑤𝐴    &   𝑤𝐶    &   𝑦𝐸    &   (𝑦 = 𝑤𝐶 = 𝐸)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑤𝐵𝐸)

Theoremcbvmpo1 41777* Rule to change the first bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵    &   𝑧𝐵    &   𝑧𝐶    &   𝑥𝐸    &   (𝑥 = 𝑧𝐶 = 𝐸)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑦𝐵𝐸)

Theoremeliuniin 41778* Indexed union of indexed intersections. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = 𝑥𝐵 𝑦𝐶 𝐷       (𝑍𝑉 → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))

Theoremssabf 41779 Subclass of a class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴       (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Theorempssnssi 41780 A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴𝐵        ¬ 𝐵𝐴

Theoremrabidim2 41781 Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝑥 ∈ {𝑥𝐴𝜑} → 𝜑)

Theoremeluni2f 41782* Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴    &   𝑥𝐵       (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)

Theoremeliin2f 41783* Membership in indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵       (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))

Theoremnssd 41784 Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋𝐴)    &   (𝜑 → ¬ 𝑋𝐵)       (𝜑 → ¬ 𝐴𝐵)

Theoremiineq12dv 41785* Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐵) → 𝐶 = 𝐷)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremsupxrcld 41786 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ*)       (𝜑 → sup(𝐴, ℝ*, < ) ∈ ℝ*)

Theoremelrestd 41787 A sufficient condition for being an open set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐽𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝑋𝐽)    &   𝐴 = (𝑋𝐵)       (𝜑𝐴 ∈ (𝐽t 𝐵))

Theoremeliuniincex 41788* Counterexample to show that the additional conditions in eliuniin 41778 and eliuniin2 41798 are actually needed. Notice that the definition of 𝐴 is not even needed (it can be any class). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐵 = {∅}    &   𝐶 = ∅    &   𝐷 = ∅    &   𝑍 = V        ¬ (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷)

Theoremeliincex 41789* Counterexample to show that the additional conditions in eliin 4887 and eliin2 41794 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = V    &   𝐵 = ∅        ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶)

Theoremeliinid 41790* Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴𝐶)

Theoremabssf 41791 Class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴       ({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Theoremsupxrubd 41792 A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵𝐴)    &   𝑆 = sup(𝐴, ℝ*, < )       (𝜑𝐵𝑆)

Theoremssrabf 41793 Subclass of a restricted class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵    &   𝑥𝐴       (𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))

Theoremeliin2 41794* Membership in indexed intersection. See eliincex 41789 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4887 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 41783). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))

Theoremssrab2f 41795 Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴       {𝑥𝐴𝜑} ⊆ 𝐴

Theoremrestuni3 41796 The underlying set of a subspace induced by the subspace operator t. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))

Theoremrabssf 41797 Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵       ({𝑥𝐴𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜑𝑥𝐵))

Theoremeliuniin2 41798* Indexed union of indexed intersections. See eliincex 41789 for a counterexample showing that the precondition 𝐶 ≠ ∅ cannot be simply dropped. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐶    &   𝐴 = 𝑥𝐵 𝑦𝐶 𝐷       (𝐶 ≠ ∅ → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))

Theoremrestuni4 41799 The underlying set of a subspace induced by the t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 𝐴)       (𝜑 (𝐴t 𝐵) = 𝐵)

Theoremrestuni6 41800 The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))

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