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Theorem List for Metamath Proof Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremunissb 4501* Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)

Theoremuniss2 4502* A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. See iunss2 4597 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.)
(∀𝑥𝐴𝑦𝐵 𝑥𝑦 𝐴 𝐵)

Theoremunidif 4503* If the difference 𝐴𝐵 contains the largest members of 𝐴, then the union of the difference is the union of 𝐴. (Contributed by NM, 22-Mar-2004.)
(∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 (𝐴𝐵) = 𝐴)

Theoremssunieq 4504* Relationship implying union. (Contributed by NM, 10-Nov-1999.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → 𝐴 = 𝐵)

Theoremunimax 4505* Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
(𝐴𝐵 {𝑥𝐵𝑥𝐴} = 𝐴)

Theorempwuni 4506 A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)
𝐴 ⊆ 𝒫 𝐴

2.1.19  The intersection of a class

Syntaxcint 4507 Extend class notation to include the intersection of a class (read: 'intersect 𝐴').
class 𝐴

Definitiondf-int 4508* Define the intersection of a class. Definition 7.35 of [TakeutiZaring] p. 44. For example, {{1, 3}, {1, 8}} = {1}. Compare this with the intersection of two classes, df-in 3614. (Contributed by NM, 18-Aug-1993.)
𝐴 = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}

Theoremdfint2 4509* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}

Theoreminteq 4510 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
(𝐴 = 𝐵 𝐴 = 𝐵)

Theoreminteqi 4511 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
𝐴 = 𝐵        𝐴 = 𝐵

Theoreminteqd 4512 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
(𝜑𝐴 = 𝐵)       (𝜑 𝐴 = 𝐵)

Theoremelint 4513* Membership in class intersection. (Contributed by NM, 21-May-1994.)
𝐴 ∈ V       (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))

Theoremelint2 4514* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
𝐴 ∈ V       (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)

Theoremelintg 4515* Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.) (Proof shortened by JJ, 26-Jul-2021.)
(𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))

TheoremelintgOLD 4516* Obsolete proof of elintg 4515 as of 26-Jul-2021. (Contributed by NM, 20-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))

Theoremelinti 4517 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴 𝐵 → (𝐶𝐵𝐴𝐶))

Theoremnfint 4518 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
𝑥𝐴       𝑥 𝐴

Theoremelintab 4519* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))

Theoremelintrab 4520* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
𝐴 ∈ V       (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))

Theoremelintrabg 4521* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
(𝐴𝑉 → (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))

Theoremint0 4522 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)
∅ = V

Theoremint0OLD 4523 Obsolete proof of int0 4522 as of 26-Jul-2021. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
∅ = V

Theoremintss1 4524 An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)
(𝐴𝐵 𝐵𝐴)

Theoremssint 4525* Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
(𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)

Theoremssintab 4526* Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))

Theoremssintub 4527* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
𝐴 {𝑥𝐵𝐴𝑥}

Theoremssmin 4528* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
𝐴 {𝑥 ∣ (𝐴𝑥𝜑)}

Theoremintmin 4529* Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴𝐵 {𝑥𝐵𝐴𝑥} = 𝐴)

Theoremintss 4530 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.) (Proof shortened by OpenAI, 25-Mar-2020.)
(𝐴𝐵 𝐵 𝐴)

Theoremintssuni 4531 The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
(𝐴 ≠ ∅ → 𝐴 𝐴)

Theoremssintrab 4532* Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
(𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))

Theoremunissint 4533 If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4546). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
( 𝐴 𝐴 ↔ (𝐴 = ∅ ∨ 𝐴 = 𝐴))

Theoremintssuni2 4534 Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)
((𝐴𝐵𝐴 ≠ ∅) → 𝐴 𝐵)

Theoremintminss 4535* 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 → (𝜑𝜓))       ((𝐴𝐵𝜓) → {𝑥𝐵𝜑} ⊆ 𝐴)

Theoremintmin2 4536* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V        {𝑥𝐴𝑥} = 𝐴

Theoremintmin3 4537* 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 → (𝜑𝜓))    &   𝜓       (𝐴𝑉 {𝑥𝜑} ⊆ 𝐴)

Theoremintmin4 4538* Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
(𝐴 {𝑥𝜑} → {𝑥 ∣ (𝐴𝑥𝜑)} = {𝑥𝜑})

Theoremintab 4539* 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 7190 or abexssex 7191 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(𝜑𝑥 = 𝐴)} ∈ V        {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} = {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)}

Theoremint0el 4540 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
(∅ ∈ 𝐴 𝐴 = ∅)

Theoremintun 4541 The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)
(𝐴𝐵) = ( 𝐴 𝐵)

Theoremintpr 4542 The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)
𝐴 ∈ V    &   𝐵 ∈ V        {𝐴, 𝐵} = (𝐴𝐵)

Theoremintprg 4543 The intersection of a pair is the intersection of its members. Closed form of intpr 4542. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))

Theoremintsng 4544 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴𝑉 {𝐴} = 𝐴)

Theoremintsn 4545 The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)
𝐴 ∈ V        {𝐴} = 𝐴

Theoremuniintsn 4546* Two ways to express "𝐴 is a singleton." See also en1 8064, en1b 8065, card1 8832, and eusn 4297. (Contributed by NM, 2-Aug-2010.)
( 𝐴 = 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})

Theoremuniintab 4547 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 {𝑥𝜑} = {𝑥𝜑})

Theoremintunsn 4548 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
𝐵 ∈ V        (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)

Theoremrint0 4549 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (𝐴 𝑋) = 𝐴)

Theoremelrint 4550* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))

Theoremelrint2 4551* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋𝐴 → (𝑋 ∈ (𝐴 𝐵) ↔ ∀𝑦𝐵 𝑋𝑦))

2.1.20  Indexed union and intersection

Syntaxciun 4552 Extend class notation to include indexed union. Note: Historically (prior to 21-Oct-2005), set.mm used the notation 𝑥𝐴𝐵, with the same union symbol as cuni 4468. 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 𝑥𝐴 𝐵

Syntaxciin 4553 Extend class notation to include indexed intersection. Note: Historically (prior to 21-Oct-2005), set.mm used the notation 𝑥𝐴𝐵, with the same intersection symbol as cint 4507. 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 𝑥𝐴 𝐵

Definitiondf-iun 4554* 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, 𝑥𝐴 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 4586. Theorem uniiun 4605 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6545 and funiunfv 6546 are useful when 𝐵 is a function. (Contributed by NM, 27-Jun-1998.)
𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}

Definitiondf-iin 4555* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4554. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4587. Theorem intiin 4606 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)
𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}

Theoremeliun 4556* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
(𝐴 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝐴𝐶)

Theoremeliin 4557* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
(𝐴𝑉 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))

Theoremeliuni 4558* Membership in an indexed union, one way. (Contributed by JJ, 27-Jul-2021.)
(𝑥 = 𝐴𝐵 = 𝐶)       ((𝐴𝐷𝐸𝐶) → 𝐸 𝑥𝐷 𝐵)

Theoremiuncom 4559* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 𝑥𝐴 𝐶

Theoremiuncom4 4560 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)
𝑥𝐴 𝐵 = 𝑥𝐴 𝐵

Theoremiunconst 4561* 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 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)

Theoremiinconst 4562* Indexed intersection of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Mario Carneiro, 6-Feb-2015.)
(𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)

Theoremiuniin 4563* 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.)
𝑥𝐴 𝑦𝐵 𝐶 𝑦𝐵 𝑥𝐴 𝐶

Theoremiunss1 4564* Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)

Theoremiinss1 4565* Subclass theorem for indexed intersection. (Contributed by NM, 24-Jan-2012.)
(𝐴𝐵 𝑥𝐵 𝐶 𝑥𝐴 𝐶)

Theoremiuneq1 4566* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
(𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremiineq1 4567* Equality theorem for indexed intersection. (Contributed by NM, 27-Jun-1998.)
(𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremss2iun 4568 Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 𝑥𝐴 𝐶)

Theoremiuneq2 4569 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
(∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiineq2 4570 Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiuneq2i 4571 Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)
(𝑥𝐴𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑥𝐴 𝐶

Theoremiineq2i 4572 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
(𝑥𝐴𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑥𝐴 𝐶

Theoremiineq2d 4573 Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiuneq2dv 4574* Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiineq2dv 4575* Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiuneq12df 4576 Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremiuneq1d 4577* Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
(𝜑𝐴 = 𝐵)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremiuneq12d 4578* Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremiuneq2d 4579* Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
(𝜑𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremnfiun 4580 Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
𝑦𝐴    &   𝑦𝐵       𝑦 𝑥𝐴 𝐵

Theoremnfiin 4581 Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
𝑦𝐴    &   𝑦𝐵       𝑦 𝑥𝐴 𝐵

Theoremnfiu1 4582 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
𝑥 𝑥𝐴 𝐵

Theoremnfii1 4583 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
𝑥 𝑥𝐴 𝐵

Theoremdfiun2g 4584* 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.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})

Theoremdfiin2g 4585* Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})

Theoremdfiun2 4586* 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        𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}

Theoremdfiin2 4587* 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        𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}

Theoremdfiunv2 4588* Define double indexed union. (Contributed by FL, 6-Nov-2013.)
𝑥𝐴 𝑦𝐵 𝐶 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}

Theoremcbviun 4589* 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.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviin 4590* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviunv 4591* 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.)
(𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviinv 4592* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)
(𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremiunss 4593* Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)

Theoremssiun 4594* Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∃𝑥𝐴 𝐶𝐵𝐶 𝑥𝐴 𝐵)

Theoremssiun2 4595 Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝑥𝐴𝐵 𝑥𝐴 𝐵)

Theoremssiun2s 4596* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
(𝑥 = 𝐶𝐵 = 𝐷)       (𝐶𝐴𝐷 𝑥𝐴 𝐵)

Theoremiunss2 4597* 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 4502. (Contributed by NM, 9-Dec-2004.)
(∀𝑥𝐴𝑦𝐵 𝐶𝐷 𝑥𝐴 𝐶 𝑦𝐵 𝐷)

Theoremiunab 4598* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑}

Theoremiunrab 4599* The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑}

Theoremiunxdif2 4600* Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
(𝑥 = 𝑦𝐶 = 𝐷)       (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶)

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