P. 305 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30401-30500   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-2reu 30401	Define the double restricted existential uniqueness quantifier. (Contributed by Thierry Arnoux, 4-Jul-2023.)
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
Theorem	2reucom 30402	Double restricted existential uniqueness commutes. (Contributed by Thierry Arnoux, 4-Jul-2023.)
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑦 ∈ 𝐵 , 𝑥 ∈ 𝐴𝜑)
Theorem	2reu2rex1 30403	Double restricted existential uniqueness implies double restricted existence. (Contributed by Thierry Arnoux, 4-Jul-2023.)
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
Theorem	2reureurex 30404	Double restricted existential uniqueness implies restricted existential uniqueness with restricted existence. (Contributed by AV, 5-Jul-2023.)
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
Theorem	2reu2reu2 30405*	Double restricted existential uniqueness implies two nested restricted existential uniqueness. (Contributed by AV, 5-Jul-2023.)
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
Theorem	opreu2reu1 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⟩ → (𝜒 ↔ 𝜑))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒)
Theorem	sq2reunnltb 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)))
Theorem	addsqnot2reu 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
Theorem	sbceqbidf 30409	Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒))
Theorem	sbcies 30410*	A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
⊢ 𝐴 = (𝐸‘𝑊)    &   ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎]𝜓 ↔ 𝜑))
20.3.2.7  Existential "at most one" - misc additions
Theorem	mo5f 30411*	Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)
⊢ Ⅎ𝑖𝜑    &   ⊢ Ⅎ𝑗𝜑    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑖∀𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
Theorem	nmo 30412*	Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (¬ ∃*𝑥𝜑 ↔ ∀𝑦∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦))
20.3.2.8  Existential uniqueness - misc additions
Theorem	reuxfrdf 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.)
⊢ Ⅎ𝑦𝐵    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 𝜓))
Theorem	rexunirn 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 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ ∪ ran 𝐹𝜑)
20.3.2.9  Restricted "at most one" - misc additions
Theorem	rmoxfrd 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.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐵 𝜓 ↔ ∃*𝑦 ∈ 𝐶 𝜒))
Theorem	rmoun 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 ∈ (𝐴 ∪ 𝐵)𝜑 → (∃*𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑))
Theorem	rmounid 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.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ¬ 𝜓)    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜓))
20.3.3  General Set Theory
20.3.3.1  Class abstractions (a.k.a. class builders)
Theorem	dmrab 30418*	Domain of a restricted class abstraction over a cartesian product. (Contributed by Thierry Arnoux, 3-Jul-2023.)
⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ 𝐵 𝜓}
Theorem	difrab2 30419	Difference of two restricted class abstractions. Compare with difrab 4197. (Contributed by Thierry Arnoux, 3-Jan-2022.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐵 ∣ 𝜑}) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑}
Theorem	rabexgfGS 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.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	rabsnel 30421*	Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.)
⊢ 𝐵 ∈ V    ⇒   ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ 𝐴)
Theorem	rabeqsnd 30422*	Conditions for a restricted class abstraction to be a singleton, in deduction form. (Contributed by Thierry Arnoux, 2-Dec-2021.)
⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝑥 = 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝐵})
Theorem	eqrrabd 30423*	Deduce equality with a restricted abstraction. (Contributed by Thierry Arnoux, 11-Apr-2024.)
⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓))    ⇒   ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓})
Theorem	foresf1o 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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵)
Theorem	rabfodom 30425*	Domination relation for restricted abstract class builders, based on a surjective function. (Contributed by Thierry Arnoux, 27-Jan-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)    ⇒   ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
20.3.3.2  Image Sets
Theorem	abrexdomjm 30426*	An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴)
Theorem	abrexdom2jm 30427*	An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ≼ 𝐴)
Theorem	abrexexd 30428*	Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ (𝜑 → 𝐴 ∈ V)    ⇒   ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
Theorem	elabreximd 30429*	Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒)
Theorem	elabreximdv 30430*	Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒)
Theorem	abrexss 30431*	A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)
⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶)
20.3.3.3  Set relations and operations - misc additions
Theorem	elunsn 30432	Elementhood to a union with a singleton. (Contributed by Thierry Arnoux, 14-Dec-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐶)))
Theorem	nelun 30433	Negated membership for a union. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ (𝐴 = (𝐵 ∪ 𝐶) → (¬ 𝑋 ∈ 𝐴 ↔ (¬ 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐶)))
Theorem	snsssng 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.)
⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵)
Theorem	rabss3d 30435*	Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	inin 30436	Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵)
Theorem	inindif 30437	See inundif 4368. (Contributed by Thierry Arnoux, 13-Sep-2017.)
⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅
Theorem	difininv 30438	Condition for the intersections of two sets with a given set to be equal. (Contributed by Thierry Arnoux, 28-Dec-2021.)
⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = (𝐶 ∩ 𝐵))
Theorem	difeq 30439	Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
⊢ ((𝐴 ∖ 𝐵) = 𝐶 ↔ ((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)))
Theorem	eqdif 30440	If both set differences of two sets are empty, those sets are equal. (Contributed by Thierry Arnoux, 16-Nov-2023.)
⊢ (((𝐴 ∖ 𝐵) = ∅ ∧ (𝐵 ∖ 𝐴) = ∅) → 𝐴 = 𝐵)
Theorem	undif5 30441	An equality involving class union and class difference. (Contributed by Thierry Arnoux, 26-Jun-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴)
Theorem	indifbi 30442	Two ways to express equality relative to a class 𝐴. (Contributed by Thierry Arnoux, 23-Jun-2024.)
⊢ ((𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐶) ↔ (𝐴 ∖ 𝐵) = (𝐴 ∖ 𝐶))
Theorem	diffib 30443	Case where diffi 8827 is a biconditional. (Contributed by Thierry Arnoux, 27-Jun-2024.)
⊢ (𝐵 ∈ Fin → (𝐴 ∈ Fin ↔ (𝐴 ∖ 𝐵) ∈ Fin))
Theorem	difxp1ss 30444	Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
⊢ ((𝐴 ∖ 𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)
Theorem	difxp2ss 30445	Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
⊢ (𝐴 × (𝐵 ∖ 𝐶)) ⊆ (𝐴 × 𝐵)
Theorem	undifr 30446	Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023.)
⊢ (𝐴 ⊆ 𝐵 ↔ ((𝐵 ∖ 𝐴) ∪ 𝐴) = 𝐵)
Theorem	indifundif 30447	A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (((𝐴 ∩ 𝐵) ∖ 𝐶) ∪ (𝐴 ∖ 𝐵)) = (𝐴 ∖ (𝐵 ∩ 𝐶))
Theorem	elpwincl1 30448	Closure of intersection with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 𝐶)
Theorem	elpwdifcl 30449	Closure of class difference with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐶)
Theorem	elpwiuncl 30450*	Closure of indexed union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 27-May-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝒫 𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶)
20.3.3.4  Unordered pairs
Theorem	eqsnd 30451*	Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = {𝐵})
Theorem	elpreq 30452	Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)
⊢ (𝜑 → 𝑋 ∈ {𝐴, 𝐵})    &   ⊢ (𝜑 → 𝑌 ∈ {𝐴, 𝐵})    &   ⊢ (𝜑 → (𝑋 = 𝐴 ↔ 𝑌 = 𝐴))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	nelpr 30453	A set 𝐴 not in a pair is neither element of the pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)))
Theorem	inpr0 30454	Rewrite an empty intersection with a pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
⊢ ((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ (¬ 𝐵 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴))
Theorem	neldifpr1 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})
Theorem	neldifpr2 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})
Theorem	unidifsnel 30457	The other element of a pair is an element of the pair. (Contributed by Thierry Arnoux, 26-Aug-2017.)
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)
Theorem	unidifsnne 30458	The other element of a pair is not the known element. (Contributed by Thierry Arnoux, 26-Aug-2017.)
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)
20.3.3.5  Conditional operator - misc additions
Theorem	ifeqeqx 30459*	An equality theorem tailored for ballotlemsf1o 32050. (Contributed by Thierry Arnoux, 14-Apr-2017.)
⊢ (𝑥 = 𝑋 → 𝐴 = 𝐶)    &   ⊢ (𝑥 = 𝑌 → 𝐵 = 𝑎)    &   ⊢ (𝑥 = 𝑋 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑌 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → 𝑎 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑊)    ⇒   ⊢ ((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) → 𝑎 = if(𝜒, 𝐴, 𝐵))
Theorem	elimifd 30460	Elimination of a conditional operator contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒 ↔ 𝜃)))    &   ⊢ (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒 ↔ 𝜏)))    ⇒   ⊢ (𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜃) ∨ (¬ 𝜓 ∧ 𝜏))))
Theorem	elim2if 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 → (𝜒 ↔ 𝜃))    &   ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒 ↔ 𝜏))    &   ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒 ↔ 𝜂))    ⇒   ⊢ (𝜒 ↔ ((𝜑 ∧ 𝜃) ∨ (¬ 𝜑 ∧ ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂)))))
Theorem	elim2ifim 30462	Elimination of two conditional operators for an implication. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒 ↔ 𝜃))    &   ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒 ↔ 𝜏))    &   ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒 ↔ 𝜂))    &   ⊢ (𝜑 → 𝜃)    &   ⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜏)    &   ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜂)    ⇒   ⊢ 𝜒
Theorem	ifeq3da 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) = 𝐸 → 𝐶 = 𝐺)    &   ⊢ (if(𝜓, 𝐸, 𝐹) = 𝐹 → 𝐶 = 𝐻)    &   ⊢ (𝜑 → 𝐺 = 𝐴)    &   ⊢ (𝜑 → 𝐻 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)
20.3.3.6  Set union
Theorem	uniinn0 30464*	Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.)
⊢ ((∪ 𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) ≠ ∅)
Theorem	uniin1 30465*	Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
⊢ ∪ 𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = (∪ 𝐴 ∩ 𝐵)
Theorem	uniin2 30466*	Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝐵)
Theorem	difuncomp 30467	Express a class difference using unions and class complements. (Contributed by Thierry Arnoux, 21-Jun-2020.)
⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∖ 𝐵) = (𝐶 ∖ ((𝐶 ∖ 𝐴) ∪ 𝐵)))
Theorem	elpwunicl 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.)
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵)
20.3.3.7  Indexed union - misc additions
Theorem	cbviunf 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.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶
Theorem	iuneq12daf 30470	Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
Theorem	iunin1f 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.)
⊢ Ⅎ𝑥𝐶    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ 𝐶)
Theorem	ssiun3 30472*	Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)
⊢ (∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	ssiun2sf 30473	Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)    ⇒   ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	iuninc 30474*	The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.)
⊢ (𝜑 → 𝐹 Fn ℕ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))    ⇒   ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖))
Theorem	iundifdifd 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 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥))))
Theorem	iundifdif 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 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)))
Theorem	iunrdx 30477*	Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)
⊢ (𝜑 → 𝐹:𝐴–onto→𝐶)    &   ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷)
Theorem	iunpreima 30478*	Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)
⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))
Theorem	iunrnmptss 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 = 𝐵 → 𝐶 = 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∪ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐷)
Theorem	iunxunsn 30480*	Appending a set to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶))
Theorem	iunxunpr 30481*	Appending two sets to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)    &   ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷)))
20.3.3.8  Indexed intersection - misc additions
Theorem	iinabrex 30482*	Rewriting an indexed intersection into an intersection of its image set. (Contributed by Thierry Arnoux, 15-Jun-2024.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
20.3.3.9  Disjointness - misc additions
Theorem	disjnf 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 𝑥 ∈ 𝐴 𝐵 ↔ (𝐵 = ∅ ∨ ∃*𝑥 𝑥 ∈ 𝐴))
Theorem	cbvdisjf 30484*	Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶)
Theorem	disjss1f 30485	A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶))
Theorem	disjeq1f 30486	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))
Theorem	disjxun0 30487*	Simplify a disjoint union. (Contributed by Thierry Arnoux, 27-Nov-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = ∅)    ⇒   ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐶))
Theorem	disjdifprg 30488*	A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Disj 𝑥 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑥)
Theorem	disjdifprg2 30489*	A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥)
Theorem	disji2f 30490*	Property of a disjoint collection: if 𝐵(𝑥) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑥 ≠ 𝑌, then 𝐵 and 𝐶 are disjoint. (Contributed by Thierry Arnoux, 30-Dec-2016.)
⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐶)    ⇒   ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ 𝑥 ≠ 𝑌) → (𝐵 ∩ 𝐶) = ∅)
Theorem	disjif 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.)
⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐶)    ⇒   ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝑍 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶)) → 𝑥 = 𝑌)
Theorem	disjorf 30492*	Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑖𝐴    &   ⊢ Ⅎ𝑗𝐴    &   ⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)    ⇒   ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅))
Theorem	disjorsf 30493*	Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
Theorem	disjif2 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.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐶)    ⇒   ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝑍 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶)) → 𝑥 = 𝑌)
Theorem	disjabrex 30495*	Rewriting a disjoint collection into a partition of its image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑦)
Theorem	disjabrexf 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.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑦)
Theorem	disjpreima 30497*	A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
⊢ ((Fun 𝐹 ∧ Disj 𝑥 ∈ 𝐴 𝐵) → Disj 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))
Theorem	disjrnmpt 30498*	Rewriting a disjoint collection using the range of a mapping. (Contributed by Thierry Arnoux, 27-May-2020.)
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦)
Theorem	disjin 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 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴))
Theorem	disjin2 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 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶))