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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | intun 4901 | 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 | intpr 4902 | 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} = (𝐴 ∩ 𝐵) | ||
Theorem | intprg 4903 | The intersection of a pair is the intersection of its members. Closed form of intpr 4902. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)) | ||
Theorem | intsng 4904 | Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) | ||
Theorem | intsn 4905 | 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 4906* | Two ways to express "𝐴 is a singleton." See also en1 8565, en1b 8566, card1 9386, and eusn 4660. (Contributed by NM, 2-Aug-2010.) |
⊢ (∪ 𝐴 = ∩ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥}) | ||
Theorem | uniintab 4907 | 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 4908 | Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.) |
⊢ 𝐵 ∈ V ⇒ ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵) | ||
Theorem | rint0 4909 | Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) | ||
Theorem | elrint 4910* | Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦)) | ||
Theorem | elrint2 4911* | Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦)) | ||
Syntax | ciun 4912 | 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 4832. 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 4913 | 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 4869. 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 4914* | 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 4950. Theorem uniiun 4974 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6997 and funiunfv 6998 are useful when 𝐵 is a function. (Contributed by NM, 27-Jun-1998.) |
⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} | ||
Definition | df-iin 4915* | Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4914. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4951. Theorem intiin 4975 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 4916* | Membership in indexed union. (Contributed by NM, 3-Sep-2003.) |
⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) | ||
Theorem | eliin 4917* | Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) | ||
Theorem | eliuni 4918* | 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 4919* | Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.) |
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 | ||
Theorem | iuncom4 4920 | Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.) |
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 = ∪ ∪ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | iunconst 4921* | 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 4922* | 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 | iuniin 4923* | 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 4924* | 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 4925* | 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 4926* | Subclass theorem for indexed intersection. (Contributed by NM, 24-Jan-2012.) |
⊢ (𝐴 ⊆ 𝐵 → ∩ 𝑥 ∈ 𝐵 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iuneq1 4927* | Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.) |
⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) | ||
Theorem | iineq1 4928* | Equality theorem for indexed intersection. (Contributed by NM, 27-Jun-1998.) |
⊢ (𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶) | ||
Theorem | ss2iun 4929 | 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 4930 | Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iineq2 4931 | 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 4932 | Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.) |
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 | ||
Theorem | iineq2i 4933 | Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.) |
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶 | ||
Theorem | iineq2d 4934 | 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 4935* | 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 4936* | 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 4937 | 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 4938* | 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 4939* | 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 4940* | Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | nfiun 4941* | Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2383. See nfiung 4943 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfiin 4942* | Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2383. See nfiing 4944 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfiung 4943 | Bound-variable hypothesis builder for indexed union. See nfiun 4941 for a version with more disjoint variable conditions, but not requiring ax-13 2383. (Contributed by Mario Carneiro, 25-Jan-2014.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfiing 4944 | Bound-variable hypothesis builder for indexed intersection. See nfiin 4942 for a version with more disjoint variable conditions, but not requiring ax-13 2383. (Contributed by Mario Carneiro, 25-Jan-2014.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfiu1 4945 | Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.) |
⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfii1 4946 | Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.) |
⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | dfiun2g 4947* | 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.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | ||
Theorem | dfiun2gOLD 4948* | Obsolete proof of dfiun2g 4947 as of 11-Aug-2023. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | ||
Theorem | dfiin2g 4949* | 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 4950* | 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 4951* | 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 4952* | Define double indexed union. (Contributed by FL, 6-Nov-2013.) |
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶} | ||
Theorem | cbviun 4953* | 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 2383. See cbviung 4955 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 4954* | 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 2383. See cbviing 4956 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 4955* | Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. See cbviun 4953 for a version with more disjoint variable conditions, but not requiring ax-13 2383. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | cbviing 4956* | Change bound variables in an indexed intersection. See cbviin 4954 for a version with more disjoint variable conditions, but not requiring ax-13 2383. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | cbviunv 4957* | 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 2383. See cbviunvg 4959 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 4958* | Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2383. See cbviinvg 4960 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 4959* | 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 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | cbviinvg 4960* | Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | iunss 4961* | 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 4962* | 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 4963 | 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 4964* | Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.) |
⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) ⇒ ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iunss2 4965* | 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 4864. (Contributed by NM, 9-Dec-2004.) |
⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷) | ||
Theorem | iunssd 4966* | 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 4967* | The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.) |
⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} | ||
Theorem | iunrab 4968* | 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 4969* | 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 4970 | 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 4971* | Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.) |
⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) | ||
Theorem | iinss 4972* | 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 4973 | An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.) |
⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵) | ||
Theorem | uniiun 4974* | Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.) |
⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | ||
Theorem | intiin 4975* | Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.) |
⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | ||
Theorem | iunid 4976* | An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.) |
⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | ||
Theorem | iun0 4977 | 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 4978 | An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅ | ||
Theorem | 0iin 4979 | An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.) |
⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V | ||
Theorem | viin 4980* | Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2193 and abid2 2957. When 𝐴 = 𝑥, this evaluates to ∅ by intiin 4975 and intv 5256. (Contributed by NM, 11-Sep-2008.) |
⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} | ||
Theorem | iunn0 4981* | 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 4982* | Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.) |
⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} | ||
Theorem | iinrab 4983* | Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.) |
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑}) | ||
Theorem | iinrab2 4984* | Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.) |
⊢ (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} | ||
Theorem | iunin2 4985* | Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4974 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) |
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iunin1 4986* | Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4974 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) | ||
Theorem | iinun2 4987* | Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4975 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.) |
⊢ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iundif2 4988* | Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4975 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.) |
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iindif1 4989* | Indexed intersection of class difference with the subtrahend held constant. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶)) | ||
Theorem | 2iunin 4990* | Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.) |
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷) | ||
Theorem | iindif2 4991* | Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4974 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.) |
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶)) | ||
Theorem | iinin2 4992* | Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4975 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶)) | ||
Theorem | iinin1 4993* | Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4975 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵)) | ||
Theorem | iinvdif 4994* | The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.) |
⊢ ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | elriin 4995* | Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ (𝐵 ∈ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆)) | ||
Theorem | riin0 4996* | Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴) | ||
Theorem | riinn0 4997* | Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = ∩ 𝑥 ∈ 𝑋 𝑆) | ||
Theorem | riinrab 4998* | Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝜑}) = {𝑦 ∈ 𝐴 ∣ ∀𝑥 ∈ 𝑋 𝜑} | ||
Theorem | symdif0 4999 | Symmetric difference with the empty class. The empty class is the identity element for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.) |
⊢ (𝐴 △ ∅) = 𝐴 | ||
Theorem | symdifv 5000 | The symmetric difference with the universal class is the complement. (Contributed by Scott Fenton, 24-Apr-2012.) |
⊢ (𝐴 △ V) = (V ∖ 𝐴) |
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