P. 50 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 4901-5000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	intss 4901	Intersection of subclasses. (Contributed by NM, 14-Oct-1999.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴)
Theorem	intssuni 4902	The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
Theorem	ssintrab 4903*	Subclass of the intersection of a restricted class abstraction. (Contributed by NM, 30-Jan-2015.)
⊢ (𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ⊆ 𝑥))
Theorem	unissint 4904	If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4919). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (∪ 𝐴 ⊆ ∩ 𝐴 ↔ (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴))
Theorem	intssuni2 4905	Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐵)
Theorem	intminss 4906*	Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴)
Theorem	intmin2 4907*	Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∩ {𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴
Theorem	intmin3 4908*	Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)
Theorem	intmin4 4909*	Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} = ∩ {𝑥 ∣ 𝜑})
Theorem	intab 4910*	The intersection of a special case of a class abstraction. 𝑦 may be free in 𝜑 and 𝐴, which can be thought of a 𝜑(𝑦) and 𝐴(𝑦). Typically, abrexex2 7821 or abexssex 7822 can be used to satisfy the second hypothesis. (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
⊢ 𝐴 ∈ V    &   ⊢ {𝑥 ∣ ∃𝑦(𝜑 ∧ 𝑥 = 𝐴)} ∈ V    ⇒   ⊢ ∩ {𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)} = {𝑥 ∣ ∃𝑦(𝜑 ∧ 𝑥 = 𝐴)}
Theorem	int0el 4911	The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅)
Theorem	intun 4912	The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)
⊢ ∩ (𝐴 ∪ 𝐵) = (∩ 𝐴 ∩ ∩ 𝐵)
Theorem	intprg 4913	The intersection of a pair is the intersection of its members. Closed form of intpr 4914. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.) (Proof shortened by BJ, 1-Sep-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))
Theorem	intpr 4914	The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.) Prove from intprg 4913. (Revised by BJ, 1-Sep-2024.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)
Theorem	intprOLD 4915	Obsolete version of intpr 4914 as of 1-Sep-2024. (Contributed by NM, 14-Oct-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)
Theorem	intprgOLD 4916	Obsolete version of intprg 4913 as of 1-Sep-2024. (Contributed by FL, 27-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))
Theorem	intsng 4917	Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)
Theorem	intsn 4918	The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∩ {𝐴} = 𝐴
Theorem	uniintsn 4919*	Two ways to express "𝐴 is a singleton". See also en1 8820, en1b 8822, card1 9735, and eusn 4667. (Contributed by NM, 2-Aug-2010.)
⊢ (∪ 𝐴 = ∩ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Theorem	uniintab 4920	The union and the intersection of a class abstraction are equal exactly when there is a unique satisfying value of 𝜑(𝑥). (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (∃!𝑥𝜑 ↔ ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
Theorem	intunsn 4921	Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
⊢ 𝐵 ∈ V    ⇒   ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵)
Theorem	rint0 4922	Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴)
Theorem	elrint 4923*	Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))
Theorem	elrint2 4924*	Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))
2.1.21  Indexed union and intersection
Syntax	ciun 4925	Extend class notation to include indexed union. Note: Historically (prior to 21-Oct-2005), set.mm used the notation ∪ 𝑥 ∈ 𝐴𝐵, with the same union symbol as cuni 4840. While that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses a distinguished symbol ∪ instead of ∪ and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
class ∪ 𝑥 ∈ 𝐴 𝐵
Syntax	ciin 4926	Extend class notation to include indexed intersection. Note: Historically (prior to 21-Oct-2005), set.mm used the notation ∩ 𝑥 ∈ 𝐴𝐵, with the same intersection symbol as cint 4880. Although that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses a distinguished symbol ∩ instead of ∩ and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
class ∩ 𝑥 ∈ 𝐴 𝐵
Definition	df-iun 4927*	Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, 𝐴 is independent of 𝑥 (although this is not required by the definition), and 𝐵 depends on 𝑥 i.e. can be read informally as 𝐵(𝑥). We call 𝑥 the index, 𝐴 the index set, and 𝐵 the indexed set. In most books, 𝑥 ∈ 𝐴 is written as a subscript or underneath a union symbol ∪. We use a special union symbol ∪ to make it easier to distinguish from plain class union. In many theorems, you will see that 𝑥 and 𝐴 are in the same distinct variable group (meaning 𝐴 cannot depend on 𝑥) and that 𝐵 and 𝑥 do not share a distinct variable group (meaning that can be thought of as 𝐵(𝑥) i.e. can be substituted with a class expression containing 𝑥). An alternate definition tying indexed union to ordinary union is dfiun2 4964. Theorem uniiun 4989 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 7129 and funiunfv 7130 are useful when 𝐵 is a function. (Contributed by NM, 27-Jun-1998.)
⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
Definition	df-iin 4928*	Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4927. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4965. Theorem intiin 4990 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)
⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
Theorem	eliun 4929*	Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
Theorem	eliin 4930*	Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
Theorem	eliuni 4931*	Membership in an indexed union, one way. (Contributed by JJ, 27-Jul-2021.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → 𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵)
Theorem	iuncom 4932*	Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶
Theorem	iuncom4 4933	Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 = ∪ ∪ 𝑥 ∈ 𝐴 𝐵
Theorem	iunconst 4934*	Indexed union of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝐴 ≠ ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵)
Theorem	iinconst 4935*	Indexed intersection of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Mario Carneiro, 6-Feb-2015.)
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐵)
Theorem	iuneqconst 4936*	Indexed union of identical classes. (Contributed by AV, 5-Mar-2024.)
⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
Theorem	iuniin 4937*	Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ⊆ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶
Theorem	iinssiun 4938*	An indexed intersection is a subset of the corresponding indexed union. (Contributed by Thierry Arnoux, 31-Dec-2021.)
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	iunss1 4939*	Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝐴 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶)
Theorem	iinss1 4940*	Subclass theorem for indexed intersection. (Contributed by NM, 24-Jan-2012.)
⊢ (𝐴 ⊆ 𝐵 → ∩ 𝑥 ∈ 𝐵 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶)
Theorem	iuneq1 4941*	Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
Theorem	iineq1 4942*	Equality theorem for indexed intersection. (Contributed by NM, 27-Jun-1998.)
⊢ (𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)
Theorem	ss2iun 4943	Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶)
Theorem	iuneq2 4944	Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
Theorem	iineq2 4945	Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)
Theorem	iuneq2i 4946	Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶
Theorem	iineq2i 4947	Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶
Theorem	iineq2d 4948	Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)
Theorem	iuneq2dv 4949*	Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
Theorem	iineq2dv 4950*	Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)
Theorem	iuneq12df 4951	Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
Theorem	iuneq1d 4952*	Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
Theorem	iuneq12d 4953*	Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
Theorem	iuneq2d 4954*	Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
Theorem	nfiun 4955*	Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2373. See nfiung 4957 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵
Theorem	nfiin 4956*	Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2373. See nfiing 4958 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵
Theorem	nfiung 4957	Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2373. See nfiun 4955 for a version with more disjoint variable conditions, but not requiring ax-13 2373. (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵
Theorem	nfiing 4958	Bound-variable hypothesis builder for indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2373. See nfiin 4956 for a version with more disjoint variable conditions, but not requiring ax-13 2373. (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵
Theorem	nfiu1 4959	Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
Theorem	nfii1 4960	Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵
Theorem	dfiun2g 4961*	Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Rohan Ridenour, 11-Aug-2023.) Avoid ax-10 2138, ax-12 2172. (Revised by SN, 11-Dec-2024.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
Theorem	dfiun2gOLD 4962*	Obsolete version of dfiun2g 4961 as of 11-Dec-2024. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Rohan Ridenour, 11-Aug-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
Theorem	dfiin2g 4963*	Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
Theorem	dfiun2 4964*	Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ 𝐵 ∈ V    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
Theorem	dfiin2 4965*	Alternate definition of indexed intersection when 𝐵 is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ 𝐵 ∈ V    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
Theorem	dfiunv2 4966*	Define double indexed union. (Contributed by FL, 6-Nov-2013.)
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶}
Theorem	cbviun 4967*	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.) Add disjoint variable condition to avoid ax-13 2373. See cbviung 4969 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶
Theorem	cbviin 4968*	Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2373. See cbviing 4970 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶
Theorem	cbviung 4969*	Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. Usage of this theorem is discouraged because it depends on ax-13 2373. See cbviun 4967 for a version with more disjoint variable conditions, but not requiring ax-13 2373. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶
Theorem	cbviing 4970*	Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2373. See cbviin 4968 for a version with more disjoint variable conditions, but not requiring ax-13 2373. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶
Theorem	cbviunv 4971*	Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.) Add disjoint variable condition to avoid ax-13 2373. See cbviunvg 4973 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶
Theorem	cbviinv 4972*	Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2373. See cbviinvg 4974 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶
Theorem	cbviunvg 4973*	Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. Usage of this theorem is discouraged because it depends on ax-13 2373. Usage of the weaker cbviunv 4971 is preferred. (Contributed by NM, 15-Sep-2003.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶
Theorem	cbviinvg 4974*	Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2373. Usage of the weaker cbviinv 4972 is preferred. (Contributed by Jeff Hankins, 26-Aug-2009.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶
Theorem	iunssf 4975	Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
Theorem	iunss 4976*	Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
Theorem	ssiun 4977*	Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	ssiun2 4978	Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	ssiun2s 4979*	Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)    ⇒   ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	iunss2 4980*	A subclass condition on the members of two indexed classes 𝐶(𝑥) and 𝐷(𝑦) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 4875. (Contributed by NM, 9-Dec-2004.)
⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)
Theorem	iunssd 4981*	Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
Theorem	iunab 4982*	The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
Theorem	iunrab 4983*	The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑}
Theorem	iunxdif2 4984*	Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶)
Theorem	ssiinf 4985	Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
Theorem	ssiin 4986*	Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
Theorem	iinss 4987*	Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
Theorem	iinss2 4988	An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)
Theorem	uniiun 4989*	Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
Theorem	intiin 4990*	Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
Theorem	iunid 4991*	An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
Theorem	iun0 4992	An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅
Theorem	0iun 4993	An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅
Theorem	0iin 4994	An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V
Theorem	viin 4995*	Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2196 and abid2 2883. When 𝐴 = 𝑥, this evaluates to ∅ by intiin 4990 and intv 5287. (Contributed by NM, 11-Sep-2008.)
⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
Theorem	iunsn 4996*	Indexed union of a singleton. Compare dfiun2 4964 and rnmpt 5867. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
Theorem	iunn0 4997*	There is a nonempty class in an indexed collection 𝐵(𝑥) iff the indexed union of them is nonempty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅)
Theorem	iinab 4998*	Indexed intersection of a class abstraction. (Contributed by NM, 6-Dec-2011.)
⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}
Theorem	iinrab 4999*	Indexed intersection of a restricted class abstraction. (Contributed by NM, 6-Dec-2011.)
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
Theorem	iinrab2 5000*	Indexed intersection of a restricted class abstraction. (Contributed by NM, 6-Dec-2011.)
⊢ (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑}