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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elixpconst 8901* | Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by NM, 12-Apr-2008.) |
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ 𝐹:𝐴⟶𝐵) | ||
Theorem | ixpconstg 8902* | Infinite Cartesian product of a constant 𝐵. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑m 𝐴)) | ||
Theorem | ixpconst 8903* | Infinite Cartesian product of a constant 𝐵. (Contributed by NM, 28-Sep-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑m 𝐴) | ||
Theorem | ixpeq1 8904* | Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) |
⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) | ||
Theorem | ixpeq1d 8905* | Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) | ||
Theorem | ss2ixp 8906 | Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpeq2 8907 | Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpeq2dva 8908* | Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpeq2dv 8909* | Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | cbvixp 8910* | Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 | ||
Theorem | cbvixpv 8911* | Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 | ||
Theorem | nfixpw 8912* | Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8913 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2371. (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfixp 8913 | Bound-variable hypothesis builder for indexed Cartesian product. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfixpw 8912 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfixp1 8914 | The index variable in an indexed Cartesian product is not free. (Contributed by Jeff Madsen, 19-Jun-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵 | ||
Theorem | ixpprc 8915* | A cartesian product of proper-class many sets is empty, because any function in the cartesian product has to be a set with domain 𝐴, which is not possible for a proper class domain. (Contributed by Mario Carneiro, 25-Jan-2015.) |
⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 = ∅) | ||
Theorem | ixpf 8916* | A member of an infinite Cartesian product maps to the indexed union of the product argument. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.) |
⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | uniixp 8917* | The union of an infinite Cartesian product is included in a Cartesian product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 24-Jun-2015.) |
⊢ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | ixpexg 8918* | The existence of an infinite Cartesian product. 𝑥 is normally a free-variable parameter in 𝐵. Remark in Enderton p. 54. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 25-Jan-2015.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ∈ V) | ||
Theorem | ixpin 8919* | The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) |
⊢ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpiin 8920* | The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) |
⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpint 8921* | The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) |
⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦) | ||
Theorem | ixp0x 8922 | An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.) |
⊢ X𝑥 ∈ ∅ 𝐴 = {∅} | ||
Theorem | ixpssmap2g 8923* | An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 8924 avoids ax-rep 5285. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴)) | ||
Theorem | ixpssmapg 8924* | An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴)) | ||
Theorem | 0elixp 8925 | Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.) |
⊢ ∅ ∈ X𝑥 ∈ ∅ 𝐴 | ||
Theorem | ixpn0 8926 | The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 10480. (Contributed by Mario Carneiro, 22-Jun-2016.) |
⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) | ||
Theorem | ixp0 8927 | The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 10480. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.) |
⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅) | ||
Theorem | ixpssmap 8928* | An infinite Cartesian product is a subset of set exponentiation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.) |
⊢ 𝐵 ∈ V ⇒ ⊢ X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴) | ||
Theorem | resixp 8929* | Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.) |
⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶) | ||
Theorem | undifixp 8930* | Union of two projections of a cartesian product. (Contributed by FL, 7-Nov-2011.) |
⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ∪ 𝐺) ∈ X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | mptelixpg 8931* | Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.) |
⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾)) | ||
Theorem | resixpfo 8932* | Restriction of elements of an infinite Cartesian product creates a surjection, if the original Cartesian product is nonempty. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ 𝐹 = (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↦ (𝑓 ↾ 𝐵)) ⇒ ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥 ∈ 𝐴 𝐶–onto→X𝑥 ∈ 𝐵 𝐶) | ||
Theorem | elixpsn 8933* | Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩})) | ||
Theorem | ixpsnf1o 8934* | A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥})) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴) | ||
Theorem | mapsnf1o 8935* | A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥})) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐴–1-1-onto→(𝐴 ↑m {𝐼})) | ||
Theorem | boxriin 8936* | A rectangular subset of a rectangular set can be recovered as the relative intersection of single-axis restrictions. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → X𝑥 ∈ 𝐼 𝐴 = (X𝑥 ∈ 𝐼 𝐵 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))) | ||
Theorem | boxcutc 8937* | The relative complement of a box set restricted on one axis. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐶 ⊆ 𝐵) → (X𝑘 ∈ 𝐴 𝐵 ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑋, 𝐶, 𝐵)) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑋, (𝐵 ∖ 𝐶), 𝐵)) | ||
Syntax | cen 8938 | Extend class definition to include the equinumerosity relation ("approximately equals" symbol) |
class ≈ | ||
Syntax | cdom 8939 | Extend class definition to include the dominance relation (curly "less than or equal to") |
class ≼ | ||
Syntax | csdm 8940 | Extend class definition to include the strict dominance relation (curly less-than) |
class ≺ | ||
Syntax | cfn 8941 | Extend class definition to include the class of all finite sets. |
class Fin | ||
Definition | df-en 8942* | Define the equinumerosity relation. Definition of [Enderton] p. 129. We define ≈ to be a binary relation rather than a connective, so its arguments must be sets to be meaningful. This is acceptable because we do not consider equinumerosity for proper classes. We derive the usual definition as bren 8951. (Contributed by NM, 28-Mar-1998.) |
⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | ||
Definition | df-dom 8943* | Define the dominance relation. For an alternate definition see dfdom2 8976. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 8958 and domen 8959. (Contributed by NM, 28-Mar-1998.) |
⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | ||
Definition | df-sdom 8944 | Define the strict dominance relation. Alternate possible definitions are derived as brsdom 8973 and brsdom2 9099. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.) |
⊢ ≺ = ( ≼ ∖ ≈ ) | ||
Definition | df-fin 8945* | Define the (proper) class of all finite sets. Similar to Definition 10.29 of [TakeutiZaring] p. 91, whose "Fin(a)" corresponds to our "𝑎 ∈ Fin". This definition is meaningful whether or not we accept the Axiom of Infinity ax-inf2 9638. If we accept Infinity, we can also express 𝐴 ∈ Fin by 𝐴 ≺ ω (Theorem isfinite 9649.) (Contributed by NM, 22-Aug-2008.) |
⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} | ||
Theorem | relen 8946 | Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.) |
⊢ Rel ≈ | ||
Theorem | reldom 8947 | Dominance is a relation. (Contributed by NM, 28-Mar-1998.) |
⊢ Rel ≼ | ||
Theorem | relsdom 8948 | Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.) |
⊢ Rel ≺ | ||
Theorem | encv 8949 | If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | breng 8950* | Equinumerosity relation. This variation of bren 8951 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 8951. (Revised by BTernaryTau, 23-Sep-2024.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) | ||
Theorem | bren 8951* | Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) Extract breng 8950 as an intermediate result. (Revised by BTernaryTau, 23-Sep-2024.) |
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) | ||
Theorem | brenOLD 8952* | Obsolete version of bren 8951 as of 23-Sep-2024. (Contributed by NM, 15-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) | ||
Theorem | brdom2g 8953* | Dominance relation. This variation of brdomg 8954 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 8954. (Revised by BTernaryTau, 29-Nov-2024.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | ||
Theorem | brdomg 8954* | Dominance relation. (Contributed by NM, 15-Jun-1998.) Extract brdom2g 8953 as an intermediate result. (Revised by BTernaryTau, 29-Nov-2024.) |
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | ||
Theorem | brdomgOLD 8955* | Obsolete version of brdomg 8954 as of 29-Nov-2024. (Contributed by NM, 15-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | ||
Theorem | brdomi 8956* | Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.) Avoid ax-un 7727. (Revised by BTernaryTau, 29-Nov-2024.) |
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) | ||
Theorem | brdomiOLD 8957* | Obsolete version of brdomi 8956 as of 29-Nov-2024. (Contributed by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) | ||
Theorem | brdom 8958* | Dominance relation. (Contributed by NM, 15-Jun-1998.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) | ||
Theorem | domen 8959* | Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) | ||
Theorem | domeng 8960* | Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.) |
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))) | ||
Theorem | ctex 8961 | A countable set is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.) |
⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | ||
Theorem | f1oen4g 8962 | The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 8969 does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.) |
⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | ||
Theorem | f1dom4g 8963 | The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 8970 does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.) |
⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) | ||
Theorem | f1oen3g 8964 | The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 8969 does not require the Axiom of Replacement nor the Axiom of Power Sets. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | ||
Theorem | f1dom3g 8965 | The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 8970 does not require the Axiom of Replacement nor the Axiom of Power Sets. (Contributed by BTernaryTau, 9-Sep-2024.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) | ||
Theorem | f1oen2g 8966 | The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 8969 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | ||
Theorem | f1dom2g 8967 | The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 8970 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.) (Proof shortened by BTernaryTau, 25-Sep-2024.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) | ||
Theorem | f1dom2gOLD 8968 | Obsolete version of f1dom2g 8967 as of 25-Sep-2024. (Contributed by Mario Carneiro, 24-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) | ||
Theorem | f1oeng 8969 | The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | ||
Theorem | f1domg 8970 | The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.) |
⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵)) | ||
Theorem | f1oen 8971 | The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐴 ≈ 𝐵) | ||
Theorem | f1dom 8972 | The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵) | ||
Theorem | brsdom 8973 | Strict dominance relation, meaning "𝐵 is strictly greater in size than 𝐴". Definition of [Mendelson] p. 255. (Contributed by NM, 25-Jun-1998.) |
⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) | ||
Theorem | isfi 8974* | Express "𝐴 is finite". Definition 10.29 of [TakeutiZaring] p. 91 (whose "Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.) |
⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | ||
Theorem | enssdom 8975 | Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.) |
⊢ ≈ ⊆ ≼ | ||
Theorem | dfdom2 8976 | Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.) |
⊢ ≼ = ( ≺ ∪ ≈ ) | ||
Theorem | endom 8977 | Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.) |
⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | ||
Theorem | sdomdom 8978 | Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.) |
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | ||
Theorem | sdomnen 8979 | Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.) |
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | ||
Theorem | brdom2 8980 | Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97. (Contributed by NM, 17-Jun-1998.) |
⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | ||
Theorem | bren2 8981 | Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.) |
⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) | ||
Theorem | enrefg 8982 | Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) | ||
Theorem | enref 8983 | Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝐴 ≈ 𝐴 | ||
Theorem | eqeng 8984 | Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵)) | ||
Theorem | domrefg 8985 | Dominance is reflexive. (Contributed by NM, 18-Jun-1998.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝐴) | ||
Theorem | en2d 8986* | Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑋)) & ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝑌)) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) ⇒ ⊢ (𝜑 → 𝐴 ≈ 𝐵) | ||
Theorem | en3d 8987* | Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) & ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴)) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))) ⇒ ⊢ (𝜑 → 𝐴 ≈ 𝐵) | ||
Theorem | en2i 8988* | Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ V) & ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ V) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) ⇒ ⊢ 𝐴 ≈ 𝐵 | ||
Theorem | en3i 8989* | Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) & ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) ⇒ ⊢ 𝐴 ≈ 𝐵 | ||
Theorem | dom2lem 8990* | A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵) | ||
Theorem | dom2d 8991* | A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))) ⇒ ⊢ (𝜑 → (𝐵 ∈ 𝑅 → 𝐴 ≼ 𝐵)) | ||
Theorem | dom3d 8992* | A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐴 ≼ 𝐵) | ||
Theorem | dom2 8993* | A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. 𝐶 and 𝐷 can be read 𝐶(𝑥) and 𝐷(𝑦), as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.) |
⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)) ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐴 ≼ 𝐵) | ||
Theorem | dom3 8994* | A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. 𝐶 and 𝐷 can be read 𝐶(𝑥) and 𝐷(𝑦), as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.) |
⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ≼ 𝐵) | ||
Theorem | idssen 8995 | Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ I ⊆ ≈ | ||
Theorem | domssl 8996 | If 𝐴 is a subset of 𝐵 and 𝐶 dominates 𝐵, then 𝐶 also dominates 𝐴. (Contributed by BTernaryTau, 7-Dec-2024.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | ||
Theorem | domssr 8997 | If 𝐶 is a superset of 𝐵 and 𝐵 dominates 𝐴, then 𝐶 also dominates 𝐴. (Contributed by BTernaryTau, 7-Dec-2024.) |
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐴 ≼ 𝐵) → 𝐴 ≼ 𝐶) | ||
Theorem | ssdomg 8998 | A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) | ||
Theorem | ener 8999 | Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof shortened by AV, 1-May-2021.) |
⊢ ≈ Er V | ||
Theorem | ensymb 9000 | Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴) |
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