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Theorem List for Metamath Proof Explorer - 30401-30500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-2reu 30401 Define the double restricted existential uniqueness quantifier. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
 
Theorem2reucom 30402 Double restricted existential uniqueness commutes. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ ∃!𝑦𝐵 , 𝑥𝐴𝜑)
 
Theorem2reu2rex1 30403 Double restricted existential uniqueness implies double restricted existence. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)
 
Theorem2reureurex 30404 Double restricted existential uniqueness implies restricted existential uniqueness with restricted existence. (Contributed by AV, 5-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 → ∃!𝑥𝐴𝑦𝐵 𝜑)
 
Theorem2reu2reu2 30405* Double restricted existential uniqueness implies two nested restricted existential uniqueness. (Contributed by AV, 5-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
 
Theoremopreu2reu1 30406* Equivalent definition of the double restricted existential uniqueness quantifier, using uniqueness of ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(𝑝 = ⟨𝑥, 𝑦⟩ → (𝜒𝜑))       (∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒)
 
Theoremsq2reunnltb 30407* There exists a unique decomposition of a prime as a sum of squares of two different positive integers iff the prime is of the form 4𝑘 + 1. Double restricted existential uniqueness variant of 2sqreunnltb 26197. (Contributed by AV, 5-Jul-2023.)
(𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑎 ∈ ℕ , 𝑏 ∈ ℕ(𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
 
Theoremaddsqnot2reu 30408* For each complex number 𝐶, there does not uniquely exist two complex numbers 𝑎 and 𝑏, with 𝑏 squared and added to 𝑎 resulting in the given complex number 𝐶. Double restricted existential uniqueness variant of addsqn2reurex2 26181. (Contributed by AV, 5-Jul-2023.)
(𝐶 ∈ ℂ → ¬ ∃!𝑎 ∈ ℂ , 𝑏 ∈ ℂ(𝑎 + (𝑏↑2)) = 𝐶)
 
20.3.2.6  Substitution (without distinct variables) - misc additions
 
Theoremsbceqbidf 30409 Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))
 
Theoremsbcies 30410* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝐴 = (𝐸𝑊)    &   (𝑎 = 𝐴 → (𝜑𝜓))       (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎]𝜓𝜑))
 
20.3.2.7  Existential "at most one" - misc additions
 
Theoremmo5f 30411* Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)
𝑖𝜑    &   𝑗𝜑       (∃*𝑥𝜑 ↔ ∀𝑖𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
 
Theoremnmo 30412* Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)
𝑦𝜑       (¬ ∃*𝑥𝜑 ↔ ∀𝑦𝑥(𝜑𝑥𝑦))
 
20.3.2.8  Existential uniqueness - misc additions
 
Theoremreuxfrdf 30413* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfrd 3647 (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) (Revised by Thierry Arnoux, 30-Mar-2018.)
𝑦𝐵    &   ((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃*𝑦𝐶 𝑥 = 𝐴)       (𝜑 → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐶 𝜓))
 
Theoremrexunirn 30414* Restricted existential quantification over the union of the range of a function. Cf. rexrn 6863 and eluni2 4800. (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑥𝐴𝐵𝑉)       (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦 ran 𝐹𝜑)
 
20.3.2.9  Restricted "at most one" - misc additions
 
Theoremrmoxfrd 30415* Transfer "at most one" restricted quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐵 𝜓 ↔ ∃*𝑦𝐶 𝜒))
 
Theoremrmoun 30416 "At most one" restricted existential quantifier for a union implies the same quantifier on both sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
(∃*𝑥 ∈ (𝐴𝐵)𝜑 → (∃*𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑))
 
Theoremrmounid 30417* A case where an "at most one" restricted existential quantifier for a union is equivalent to such a quantifier for one of the sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
((𝜑𝑥𝐵) → ¬ 𝜓)       (𝜑 → (∃*𝑥 ∈ (𝐴𝐵)𝜓 ↔ ∃*𝑥𝐴 𝜓))
 
20.3.3  General Set Theory
 
20.3.3.1  Class abstractions (a.k.a. class builders)
 
Theoremdmrab 30418* Domain of a restricted class abstraction over a cartesian product. (Contributed by Thierry Arnoux, 3-Jul-2023.)
(𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥𝐴 ∣ ∃𝑦𝐵 𝜓}
 
Theoremdifrab2 30419 Difference of two restricted class abstractions. Compare with difrab 4197. (Contributed by Thierry Arnoux, 3-Jan-2022.)
({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
 
TheoremrabexgfGS 30420 Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.)
𝑥𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
 
Theoremrabsnel 30421* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.)
𝐵 ∈ V       ({𝑥𝐴𝜑} = {𝐵} → 𝐵𝐴)
 
Theoremrabeqsnd 30422* Conditions for a restricted class abstraction to be a singleton, in deduction form. (Contributed by Thierry Arnoux, 2-Dec-2021.)
(𝑥 = 𝐵 → (𝜓𝜒))    &   (𝜑𝐵𝐴)    &   (𝜑𝜒)    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝑥 = 𝐵)       (𝜑 → {𝑥𝐴𝜓} = {𝐵})
 
Theoremeqrrabd 30423* Deduce equality with a restricted abstraction. (Contributed by Thierry Arnoux, 11-Apr-2024.)
(𝜑𝐵𝐴)    &   ((𝜑𝑥𝐴) → (𝑥𝐵𝜓))       (𝜑𝐵 = {𝑥𝐴𝜓})
 
Theoremforesf1o 30424* From a surjective function, *choose* a subset of the domain, such that the restricted function is bijective. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝐴𝑉𝐹:𝐴onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
 
Theoremrabfodom 30425* Domination relation for restricted abstract class builders, based on a surjective function. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝜑𝑥𝐴𝑦 = (𝐹𝑥)) → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴onto𝐵)       (𝜑 → {𝑦𝐵𝜒} ≼ {𝑥𝐴𝜓})
 
20.3.3.2  Image Sets
 
Theoremabrexdomjm 30426* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑦𝐴 → ∃*𝑥𝜑)       (𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝜑} ≼ 𝐴)
 
Theoremabrexdom2jm 30427* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ≼ 𝐴)
 
Theoremabrexexd 30428* Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)
𝑥𝐴    &   (𝜑𝐴 ∈ V)       (𝜑 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
 
Theoremelabreximd 30429* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑥𝜑    &   𝑥𝜒    &   (𝐴 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐶) → 𝜓)       ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)
 
Theoremelabreximdv 30430* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
(𝐴 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐶) → 𝜓)       ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)
 
Theoremabrexss 30431* A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)
𝑥𝐶       (∀𝑥𝐴 𝐵𝐶 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶)
 
20.3.3.3  Set relations and operations - misc additions
 
Theoremelunsn 30432 Elementhood to a union with a singleton. (Contributed by Thierry Arnoux, 14-Dec-2023.)
(𝐴𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴𝐵𝐴 = 𝐶)))
 
Theoremnelun 30433 Negated membership for a union. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝐴 = (𝐵𝐶) → (¬ 𝑋𝐴 ↔ (¬ 𝑋𝐵 ∧ ¬ 𝑋𝐶)))
 
Theoremsnsssng 30434 If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.) (Revised by Thierry Arnoux, 11-Apr-2024.)
((𝐴𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵)
 
Theoremrabss3d 30435* Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})
 
Theoreminin 30436 Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
 
Theoreminindif 30437 See inundif 4368. (Contributed by Thierry Arnoux, 13-Sep-2017.)
((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
 
Theoremdifininv 30438 Condition for the intersections of two sets with a given set to be equal. (Contributed by Thierry Arnoux, 28-Dec-2021.)
((((𝐴𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶𝐴) ∩ 𝐵) = ∅) → (𝐴𝐵) = (𝐶𝐵))
 
Theoremdifeq 30439 Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
((𝐴𝐵) = 𝐶 ↔ ((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)))
 
Theoremeqdif 30440 If both set differences of two sets are empty, those sets are equal. (Contributed by Thierry Arnoux, 16-Nov-2023.)
(((𝐴𝐵) = ∅ ∧ (𝐵𝐴) = ∅) → 𝐴 = 𝐵)
 
Theoremundif5 30441 An equality involving class union and class difference. (Contributed by Thierry Arnoux, 26-Jun-2024.)
((𝐴𝐵) = ∅ → ((𝐴𝐵) ∖ 𝐵) = 𝐴)
 
Theoremindifbi 30442 Two ways to express equality relative to a class 𝐴. (Contributed by Thierry Arnoux, 23-Jun-2024.)
((𝐴𝐵) = (𝐴𝐶) ↔ (𝐴𝐵) = (𝐴𝐶))
 
Theoremdiffib 30443 Case where diffi 8827 is a biconditional. (Contributed by Thierry Arnoux, 27-Jun-2024.)
(𝐵 ∈ Fin → (𝐴 ∈ Fin ↔ (𝐴𝐵) ∈ Fin))
 
Theoremdifxp1ss 30444 Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
((𝐴𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)
 
Theoremdifxp2ss 30445 Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
(𝐴 × (𝐵𝐶)) ⊆ (𝐴 × 𝐵)
 
Theoremundifr 30446 Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023.)
(𝐴𝐵 ↔ ((𝐵𝐴) ∪ 𝐴) = 𝐵)
 
Theoremindifundif 30447 A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)
(((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵)) = (𝐴 ∖ (𝐵𝐶))
 
Theoremelpwincl1 30448 Closure of intersection with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐴 ∈ 𝒫 𝐶)       (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
 
Theoremelpwdifcl 30449 Closure of class difference with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐴 ∈ 𝒫 𝐶)       (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
 
Theoremelpwiuncl 30450* Closure of indexed union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 27-May-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ 𝒫 𝐶)       (𝜑 𝑘𝐴 𝐵 ∈ 𝒫 𝐶)
 
20.3.3.4  Unordered pairs
 
Theoremeqsnd 30451* Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.)
((𝜑𝑥𝐴) → 𝑥 = 𝐵)    &   (𝜑𝐵𝐴)       (𝜑𝐴 = {𝐵})
 
Theoremelpreq 30452 Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)
(𝜑𝑋 ∈ {𝐴, 𝐵})    &   (𝜑𝑌 ∈ {𝐴, 𝐵})    &   (𝜑 → (𝑋 = 𝐴𝑌 = 𝐴))       (𝜑𝑋 = 𝑌)
 
Theoremnelpr 30453 A set 𝐴 not in a pair is neither element of the pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝐴𝑉 → (¬ 𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴𝐵𝐴𝐶)))
 
Theoreminpr0 30454 Rewrite an empty intersection with a pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ (¬ 𝐵𝐴 ∧ ¬ 𝐶𝐴))
 
Theoremneldifpr1 30455 The first element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})
 
Theoremneldifpr2 30456 The second element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})
 
Theoremunidifsnel 30457 The other element of a pair is an element of the pair. (Contributed by Thierry Arnoux, 26-Aug-2017.)
((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ 𝑃)
 
Theoremunidifsnne 30458 The other element of a pair is not the known element. (Contributed by Thierry Arnoux, 26-Aug-2017.)
((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
 
20.3.3.5  Conditional operator - misc additions
 
Theoremifeqeqx 30459* An equality theorem tailored for ballotlemsf1o 32050. (Contributed by Thierry Arnoux, 14-Apr-2017.)
(𝑥 = 𝑋𝐴 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝑎)    &   (𝑥 = 𝑋 → (𝜒𝜃))    &   (𝑥 = 𝑌 → (𝜒𝜓))    &   (𝜑𝑎 = 𝐶)    &   ((𝜑𝜓) → 𝜃)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑊)       ((𝜑𝑥 = if(𝜓, 𝑋, 𝑌)) → 𝑎 = if(𝜒, 𝐴, 𝐵))
 
Theoremelimifd 30460 Elimination of a conditional operator contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒𝜃)))    &   (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒𝜏)))       (𝜑 → (𝜒 ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))
 
Theoremelim2if 30461 Elimination of two conditional operators contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒𝜃))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒𝜏))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒𝜂))       (𝜒 ↔ ((𝜑𝜃) ∨ (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))))
 
Theoremelim2ifim 30462 Elimination of two conditional operators for an implication. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒𝜃))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒𝜏))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒𝜂))    &   (𝜑𝜃)    &   ((¬ 𝜑𝜓) → 𝜏)    &   ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜂)       𝜒
 
Theoremifeq3da 30463 Given an expression 𝐶 containing if(𝜓, 𝐸, 𝐹), substitute (hypotheses .1 and .2) and evaluate (hypotheses .3 and .4) it for both cases at the same time. (Contributed by Thierry Arnoux, 13-Dec-2021.)
(if(𝜓, 𝐸, 𝐹) = 𝐸𝐶 = 𝐺)    &   (if(𝜓, 𝐸, 𝐹) = 𝐹𝐶 = 𝐻)    &   (𝜑𝐺 = 𝐴)    &   (𝜑𝐻 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)
 
20.3.3.6  Set union
 
Theoremuniinn0 30464* Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.)
(( 𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
 
Theoremuniin1 30465* Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
𝑥𝐴 (𝑥𝐵) = ( 𝐴𝐵)
 
Theoremuniin2 30466* Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
𝑥𝐵 (𝐴𝑥) = (𝐴 𝐵)
 
Theoremdifuncomp 30467 Express a class difference using unions and class complements. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝐴𝐶 → (𝐴𝐵) = (𝐶 ∖ ((𝐶𝐴) ∪ 𝐵)))
 
Theoremelpwunicl 30468 Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020.) (Proof shortened by BJ, 6-Apr-2024.)
(𝜑𝐴 ∈ 𝒫 𝒫 𝐵)       (𝜑 𝐴 ∈ 𝒫 𝐵)
 
20.3.3.7  Indexed union - misc additions
 
Theoremcbviunf 30469* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 
Theoremiuneq12daf 30470 Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → 𝐶 = 𝐷)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
 
Theoremiunin1f 30471 Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4944 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.)
𝑥𝐶        𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)
 
Theoremssiun3 30472* Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)
(∀𝑦𝐶𝑥𝐴 𝑦𝐵𝐶 𝑥𝐴 𝐵)
 
Theoremssiun2sf 30473 Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
𝑥𝐴    &   𝑥𝐶    &   𝑥𝐷    &   (𝑥 = 𝐶𝐵 = 𝐷)       (𝐶𝐴𝐷 𝑥𝐴 𝐵)
 
Theoremiuninc 30474* The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.)
(𝜑𝐹 Fn ℕ)    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))       ((𝜑𝑖 ∈ ℕ) → 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))
 
Theoremiundifdifd 30475* The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.)
(𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥))))
 
Theoremiundifdif 30476* The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 30475. (Contributed by Thierry Arnoux, 4-Sep-2016.)
𝑂 ∈ V    &   𝐴 ⊆ 𝒫 𝑂       (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))
 
Theoremiunrdx 30477* Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)
(𝜑𝐹:𝐴onto𝐶)    &   ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)       (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)
 
Theoremiunpreima 30478* Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)
(Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))
 
Theoremiunrnmptss 30479* A subset relation for an indexed union over the range of function expressed as a mapping. (Contributed by Thierry Arnoux, 27-Mar-2018.)
(𝑦 = 𝐵𝐶 = 𝐷)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 𝑦 ∈ ran (𝑥𝐴𝐵)𝐶 𝑥𝐴 𝐷)
 
Theoremiunxunsn 30480* Appending a set to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑥 = 𝑋𝐵 = 𝐶)       (𝑋𝑉 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = ( 𝑥𝐴 𝐵𝐶))
 
Theoremiunxunpr 30481* Appending two sets to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)       ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))
 
20.3.3.8  Indexed intersection - misc additions
 
Theoremiinabrex 30482* Rewriting an indexed intersection into an intersection of its image set. (Contributed by Thierry Arnoux, 15-Jun-2024.)
(∀𝑥𝐴 𝐵𝑉 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
 
20.3.3.9  Disjointness - misc additions
 
Theoremdisjnf 30483* In case 𝑥 is not free in 𝐵, disjointness is not so interesting since it reduces to cases where 𝐴 is a singleton. (Google Groups discussion with Peter Mazsa.) (Contributed by Thierry Arnoux, 26-Jul-2018.)
(Disj 𝑥𝐴 𝐵 ↔ (𝐵 = ∅ ∨ ∃*𝑥 𝑥𝐴))
 
Theoremcbvdisjf 30484* Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑥𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
 
Theoremdisjss1f 30485 A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
 
Theoremdisjeq1f 30486 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 
Theoremdisjxun0 30487* Simplify a disjoint union. (Contributed by Thierry Arnoux, 27-Nov-2023.)
((𝜑𝑥𝐵) → 𝐶 = ∅)       (𝜑 → (Disj 𝑥 ∈ (𝐴𝐵)𝐶Disj 𝑥𝐴 𝐶))
 
Theoremdisjdifprg 30488* A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝐴𝑉𝐵𝑊) → Disj 𝑥 ∈ {(𝐵𝐴), 𝐴}𝑥)
 
Theoremdisjdifprg2 30489* A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝐴𝑉Disj 𝑥 ∈ {(𝐴𝐵), (𝐴𝐵)}𝑥)
 
Theoremdisji2f 30490* Property of a disjoint collection: if 𝐵(𝑥) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑥𝑌, then 𝐵 and 𝐶 are disjoint. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑥𝐶    &   (𝑥 = 𝑌𝐵 = 𝐶)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ 𝑥𝑌) → (𝐵𝐶) = ∅)
 
Theoremdisjif 30491* Property of a disjoint collection: if 𝐵(𝑥) and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑥 = 𝑌. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑥𝐶    &   (𝑥 = 𝑌𝐵 = 𝐶)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)
 
Theoremdisjorf 30492* Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑖𝐴    &   𝑗𝐴    &   (𝑖 = 𝑗𝐵 = 𝐶)       (Disj 𝑖𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅))
 
Theoremdisjorsf 30493* Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝐴       (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
 
Theoremdisjif2 30494* Property of a disjoint collection: if 𝐵(𝑥) and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑥 = 𝑌. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝑌𝐵 = 𝐶)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)
 
Theoremdisjabrex 30495* Rewriting a disjoint collection into a partition of its image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
(Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
 
Theoremdisjabrexf 30496* Rewriting a disjoint collection into a partition of its image set. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Revised by Thierry Arnoux, 9-Mar-2017.)
𝑥𝐴       (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
 
Theoremdisjpreima 30497* A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
((Fun 𝐹Disj 𝑥𝐴 𝐵) → Disj 𝑥𝐴 (𝐹𝐵))
 
Theoremdisjrnmpt 30498* Rewriting a disjoint collection using the range of a mapping. (Contributed by Thierry Arnoux, 27-May-2020.)
(Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ ran (𝑥𝐴𝐵)𝑦)
 
Theoremdisjin 30499 If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐶𝐴))
 
Theoremdisjin2 30500 If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐴𝐶))
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