Home | Metamath
Proof Explorer Theorem List (p. 90 of 460) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28853) |
Hilbert Space Explorer
(28854-30376) |
Users' Mathboxes
(30377-45962) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | f1opwfi 8901* | A one-to-one mapping induces a one-to-one mapping on finite subsets. (Contributed by Mario Carneiro, 25-Jan-2015.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐹 “ 𝑏)):(𝒫 𝐴 ∩ Fin)–1-1-onto→(𝒫 𝐵 ∩ Fin)) | ||
Theorem | fissuni 8902* | A finite subset of a union is covered by finitely many elements. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)𝐴 ⊆ ∪ 𝑐) | ||
Theorem | fipreima 8903* | Given a finite subset 𝐴 of the range of a function, there exists a finite subset of the domain whose image is 𝐴. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑐 ∈ (𝒫 𝐵 ∩ Fin)(𝐹 “ 𝑐) = 𝐴) | ||
Theorem | finsschain 8904* | A finite subset of the union of a superset chain is a subset of some element of the chain. A useful preliminary result for alexsub 22796 and others. (Contributed by Jeff Hankins, 25-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 18-May-2015.) |
⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝐵 ∈ Fin ∧ 𝐵 ⊆ ∪ 𝐴)) → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧) | ||
Theorem | indexfi 8905* | If for every element of a finite indexing set 𝐴 there exists a corresponding element of another set 𝐵, then there exists a finite subset of 𝐵 consisting only of those elements which are indexed by 𝐴. Proven without the Axiom of Choice, unlike indexdom 35515. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑐 ∈ Fin (𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑)) | ||
Syntax | cfsupp 8906 | Extend class definition to include the predicate to be a finitely supported function. |
class finSupp | ||
Definition | df-fsupp 8907* | Define the property of a function to be finitely supported (in relation to a given zero). (Contributed by AV, 23-May-2019.) |
⊢ finSupp = {〈𝑟, 𝑧〉 ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)} | ||
Theorem | relfsupp 8908 | The property of a function to be finitely supported is a relation. (Contributed by AV, 7-Jun-2019.) |
⊢ Rel finSupp | ||
Theorem | relprcnfsupp 8909 | A proper class is never finitely supported. (Contributed by AV, 7-Jun-2019.) |
⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 finSupp 𝑍) | ||
Theorem | isfsupp 8910 | The property of a class to be a finitely supported function (in relation to a given zero). (Contributed by AV, 23-May-2019.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) | ||
Theorem | funisfsupp 8911 | The property of a function to be finitely supported (in relation to a given zero). (Contributed by AV, 23-May-2019.) |
⊢ ((Fun 𝑅 ∧ 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (𝑅 supp 𝑍) ∈ Fin)) | ||
Theorem | fsuppimp 8912 | Implications of a class being a finitely supported function (in relation to a given zero). (Contributed by AV, 26-May-2019.) |
⊢ (𝑅 finSupp 𝑍 → (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)) | ||
Theorem | fsuppimpd 8913 | A finitely supported function is a function with a finite support. (Contributed by AV, 6-Jun-2019.) |
⊢ (𝜑 → 𝐹 finSupp 𝑍) ⇒ ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) | ||
Theorem | fisuppfi 8914 | A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (◡𝐹 “ 𝐶) ∈ Fin) | ||
Theorem | fdmfisuppfi 8915 | The support of a function with a finite domain is always finite. (Contributed by AV, 27-Apr-2019.) |
⊢ (𝜑 → 𝐹:𝐷⟶𝑅) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) | ||
Theorem | fdmfifsupp 8916 | A function with a finite domain is always finitely supported. (Contributed by AV, 25-May-2019.) |
⊢ (𝜑 → 𝐹:𝐷⟶𝑅) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐹 finSupp 𝑍) | ||
Theorem | fsuppmptdm 8917* | A mapping with a finite domain is finitely supported. (Contributed by AV, 7-Jun-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑌) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐹 finSupp 𝑍) | ||
Theorem | fndmfisuppfi 8918 | The support of a function with a finite domain is always finite. (Contributed by AV, 25-May-2019.) |
⊢ (𝜑 → 𝐹 Fn 𝐷) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) | ||
Theorem | fndmfifsupp 8919 | A function with a finite domain is always finitely supported. (Contributed by AV, 25-May-2019.) |
⊢ (𝜑 → 𝐹 Fn 𝐷) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐹 finSupp 𝑍) | ||
Theorem | suppeqfsuppbi 8920 | If two functions have the same support, one function is finitely supported iff the other one is finitely supported. (Contributed by AV, 30-Jun-2019.) |
⊢ (((𝐹 ∈ 𝑈 ∧ Fun 𝐹) ∧ (𝐺 ∈ 𝑉 ∧ Fun 𝐺)) → ((𝐹 supp 𝑍) = (𝐺 supp 𝑍) → (𝐹 finSupp 𝑍 ↔ 𝐺 finSupp 𝑍))) | ||
Theorem | suppssfifsupp 8921 | If the support of a function is a subset of a finite set, the function is finitely supported. (Contributed by AV, 15-Jul-2019.) |
⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → 𝐺 finSupp 𝑍) | ||
Theorem | fsuppsssupp 8922 | If the support of a function is a subset of the support of a finitely supported function, the function is finitely supported. (Contributed by AV, 2-Jul-2019.) (Proof shortened by AV, 15-Jul-2019.) |
⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ (𝐹 finSupp 𝑍 ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))) → 𝐺 finSupp 𝑍) | ||
Theorem | fsuppxpfi 8923 | The cartesian product of two finitely supported functions is finite. (Contributed by AV, 17-Jul-2019.) |
⊢ ((𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍) → ((𝐹 supp 𝑍) × (𝐺 supp 𝑍)) ∈ Fin) | ||
Theorem | fczfsuppd 8924 | A constant function with value zero is finitely supported. (Contributed by AV, 30-Jun-2019.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍) | ||
Theorem | fsuppun 8925 | The union of two finitely supported functions is finitely supported (but not necessarily a function!). (Contributed by AV, 3-Jun-2019.) |
⊢ (𝜑 → 𝐹 finSupp 𝑍) & ⊢ (𝜑 → 𝐺 finSupp 𝑍) ⇒ ⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) ∈ Fin) | ||
Theorem | fsuppunfi 8926 | The union of the support of two finitely supported functions is finite. (Contributed by AV, 1-Jul-2019.) |
⊢ (𝜑 → 𝐹 finSupp 𝑍) & ⊢ (𝜑 → 𝐺 finSupp 𝑍) ⇒ ⊢ (𝜑 → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ∈ Fin) | ||
Theorem | fsuppunbi 8927 | If the union of two classes/functions is a function, this union is finitely supported iff the two functions are finitely supported. (Contributed by AV, 18-Jun-2019.) |
⊢ (𝜑 → Fun (𝐹 ∪ 𝐺)) ⇒ ⊢ (𝜑 → ((𝐹 ∪ 𝐺) finSupp 𝑍 ↔ (𝐹 finSupp 𝑍 ∧ 𝐺 finSupp 𝑍))) | ||
Theorem | 0fsupp 8928 | The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019.) |
⊢ (𝑍 ∈ 𝑉 → ∅ finSupp 𝑍) | ||
Theorem | snopfsupp 8929 | A singleton containing an ordered pair is a finitely supported function. (Contributed by AV, 19-Jul-2019.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {〈𝑋, 𝑌〉} finSupp 𝑍) | ||
Theorem | funsnfsupp 8930 | Finite support for a function extended by a singleton. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by AV, 19-Jul-2019.) |
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹)) → ((𝐹 ∪ {〈𝑋, 𝑌〉}) finSupp 𝑍 ↔ 𝐹 finSupp 𝑍)) | ||
Theorem | fsuppres 8931 | The restriction of a finitely supported function is finitely supported. (Contributed by AV, 14-Jul-2019.) |
⊢ (𝜑 → 𝐹 finSupp 𝑍) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍) | ||
Theorem | ressuppfi 8932 | If the support of the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finite, the support of the function itself is finite. (Contributed by AV, 22-Apr-2019.) |
⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) & ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵)) & ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) | ||
Theorem | resfsupp 8933 | If the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finitely supported, the function itself is finitely supported. (Contributed by AV, 27-May-2019.) |
⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) & ⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵)) & ⊢ (𝜑 → 𝐺 finSupp 𝑍) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐹 finSupp 𝑍) | ||
Theorem | resfifsupp 8934 | The restriction of a function to a finite set is finitely supported. (Contributed by AV, 12-Dec-2019.) |
⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍) | ||
Theorem | frnfsuppbi 8935 | Two ways of saying that a function with known codomain is finitely supported. (Contributed by AV, 8-Jul-2019.) |
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 finSupp 𝑍 ↔ (◡𝐹 “ (𝑆 ∖ {𝑍})) ∈ Fin))) | ||
Theorem | fsuppmptif 8936* | A function mapping an argument to either a value of a finitely supported function or zero is finitely supported. (Contributed by AV, 6-Jun-2019.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 finSupp 𝑍) ⇒ ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍) | ||
Theorem | sniffsupp 8937* | A function mapping all but one arguments to zero is finitely supported. (Contributed by AV, 8-Jul-2019.) |
⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 0 ∈ 𝑊) & ⊢ 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) ⇒ ⊢ (𝜑 → 𝐹 finSupp 0 ) | ||
Theorem | fsuppcolem 8938 | Lemma for fsuppco 8939. Formula building theorem for finite supports: rearranging the index set. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) & ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) ⇒ ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin) | ||
Theorem | fsuppco 8939 | The composition of a 1-1 function with a finitely supported function is finitely supported. (Contributed by AV, 28-May-2019.) |
⊢ (𝜑 → 𝐹 finSupp 𝑍) & ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍) | ||
Theorem | fsuppco2 8940 | The composition of a function which maps the zero to zero with a finitely supported function is finitely supported. This is not only a special case of fsuppcor 8941 because it does not require that the "zero" is an element of the range of the finitely supported function. (Contributed by AV, 6-Jun-2019.) |
⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 finSupp 𝑍) & ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍) | ||
Theorem | fsuppcor 8941 | The composition of a function which maps the zero of the range of a finitely supported function to the zero of its range with this finitely supported function is finitely supported. (Contributed by AV, 6-Jun-2019.) |
⊢ (𝜑 → 0 ∈ 𝑊) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐷) & ⊢ (𝜑 → 𝐶 ⊆ 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 finSupp 𝑍) & ⊢ (𝜑 → (𝐺‘𝑍) = 0 ) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 ) | ||
Theorem | mapfienlem1 8942* | Lemma 1 for mapfien 8945. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.) |
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} & ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} & ⊢ 𝑊 = (𝐺‘𝑍) & ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊) | ||
Theorem | mapfienlem2 8943* | Lemma 2 for mapfien 8945. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.) |
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} & ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} & ⊢ 𝑊 = (𝐺‘𝑍) & ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) finSupp 𝑍) | ||
Theorem | mapfienlem3 8944* | Lemma 3 for mapfien 8945. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.) |
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} & ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} & ⊢ 𝑊 = (𝐺‘𝑍) & ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ 𝑆) | ||
Theorem | mapfien 8945* | A bijection of the base sets induces a bijection on the set of finitely supported functions. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.) |
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} & ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} & ⊢ 𝑊 = (𝐺‘𝑍) & ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹))):𝑆–1-1-onto→𝑇) | ||
Theorem | mapfien2 8946* | Equinumerousity relation for sets of finitely supported functions. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.) |
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 0 } & ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} & ⊢ (𝜑 → 𝐴 ≈ 𝐶) & ⊢ (𝜑 → 𝐵 ≈ 𝐷) & ⊢ (𝜑 → 0 ∈ 𝐵) & ⊢ (𝜑 → 𝑊 ∈ 𝐷) ⇒ ⊢ (𝜑 → 𝑆 ≈ 𝑇) | ||
Syntax | cfi 8947 | Extend class notation with the function whose value is the class of finite intersections of the elements of a given set. |
class fi | ||
Definition | df-fi 8948* | Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 8951). (Contributed by FL, 27-Apr-2008.) |
⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}) | ||
Theorem | fival 8949* | The set of all the finite intersections of the elements of 𝐴. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥}) | ||
Theorem | elfi 8950* | Specific properties of an element of (fi‘𝐵). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥)) | ||
Theorem | elfi2 8951* | The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥)) | ||
Theorem | elfir 8952 | Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.) |
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ∈ (fi‘𝐵)) | ||
Theorem | intrnfi 8953 | Sufficient condition for the intersection of the range of a function to be in the set of finite intersections. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ ran 𝐹 ∈ (fi‘𝐵)) | ||
Theorem | iinfi 8954* | An indexed intersection of elements of 𝐶 is an element of the finite intersections of 𝐶. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ ((𝐶 ∈ 𝑉 ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (fi‘𝐶)) | ||
Theorem | inelfi 8955 | The intersection of two sets is a finite intersection. (Contributed by Thierry Arnoux, 6-Jan-2017.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∩ 𝐵) ∈ (fi‘𝑋)) | ||
Theorem | ssfii 8956 | Any element of a set 𝐴 is the intersection of a finite subset of 𝐴. (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) | ||
Theorem | fi0 8957 | The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ (fi‘∅) = ∅ | ||
Theorem | fieq0 8958 | A set is empty iff the class of all the finite intersections of that set is empty. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅)) | ||
Theorem | fiin 8959 | The elements of (fi‘𝐶) are closed under finite intersection. (Contributed by Mario Carneiro, 24-Nov-2013.) |
⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴 ∩ 𝐵) ∈ (fi‘𝐶)) | ||
Theorem | dffi2 8960* | The set of finite intersections is the smallest set that contains 𝐴 and is closed under pairwise intersection. (Contributed by Mario Carneiro, 24-Nov-2013.) |
⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∩ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)}) | ||
Theorem | fiss 8961 | Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵)) | ||
Theorem | inficl 8962* | A set which is closed under pairwise intersection is closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴)) | ||
Theorem | fipwuni 8963 | The set of finite intersections of a set is contained in the powerset of the union of the elements of 𝐴. (Contributed by Mario Carneiro, 24-Nov-2013.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
⊢ (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴 | ||
Theorem | fisn 8964 | A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (fi‘{𝐴}) = {𝐴} | ||
Theorem | fiuni 8965 | The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴)) | ||
Theorem | fipwss 8966 | If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
⊢ (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋) | ||
Theorem | elfiun 8967* | A finite intersection of elements taken from a union of collections. (Contributed by Jeff Hankins, 15-Nov-2009.) (Proof shortened by Mario Carneiro, 26-Nov-2013.) |
⊢ ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐾) → (𝐴 ∈ (fi‘(𝐵 ∪ 𝐶)) ↔ (𝐴 ∈ (fi‘𝐵) ∨ 𝐴 ∈ (fi‘𝐶) ∨ ∃𝑥 ∈ (fi‘𝐵)∃𝑦 ∈ (fi‘𝐶)𝐴 = (𝑥 ∩ 𝑦)))) | ||
Theorem | dffi3 8968* | The set of finite intersections can be "constructed" inductively by iterating binary intersection ω-many times. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ 𝑅 = (𝑢 ∈ V ↦ ran (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧))) ⇒ ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∪ (rec(𝑅, 𝐴) “ ω)) | ||
Theorem | fifo 8969* | Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴)) | ||
Theorem | marypha1lem 8970* | Core induction for Philip Hall's marriage theorem. (Contributed by Stefan O'Rear, 19-Feb-2015.) |
⊢ (𝐴 ∈ Fin → (𝑏 ∈ Fin → ∀𝑐 ∈ 𝒫 (𝐴 × 𝑏)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑒 ∈ 𝒫 𝑐𝑒:𝐴–1-1→V))) | ||
Theorem | marypha1 8971* | (Philip) Hall's marriage theorem, sufficiency: a finite relation contains an injection if there is no subset of its domain which would be forced to violate the pigeonhole principle. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐶 ⊆ (𝐴 × 𝐵)) & ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑)) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→𝐵) | ||
Theorem | marypha2lem1 8972* | Lemma for marypha2 8976. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⇒ ⊢ 𝑇 ⊆ (𝐴 × ∪ ran 𝐹) | ||
Theorem | marypha2lem2 8973* | Lemma for marypha2 8976. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⇒ ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} | ||
Theorem | marypha2lem3 8974* | Lemma for marypha2 8976. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐺 ⊆ 𝑇 ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹‘𝑥))) | ||
Theorem | marypha2lem4 8975* | Lemma for marypha2 8976. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ (𝐹 “ 𝑋)) | ||
Theorem | marypha2 8976* | Version of marypha1 8971 using a functional family of sets instead of a relation. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶Fin) & ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ ∪ (𝐹 “ 𝑑)) ⇒ ⊢ (𝜑 → ∃𝑔(𝑔:𝐴–1-1→V ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥))) | ||
Syntax | csup 8977 | Extend class notation to include supremum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers. |
class sup(𝐴, 𝐵, 𝑅) | ||
Syntax | cinf 8978 | Extend class notation to include infimum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers. |
class inf(𝐴, 𝐵, 𝑅) | ||
Definition | df-sup 8979* | Define the supremum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the supremum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals whose square is less than 2; in this case the supremum is defined as the square root of 2 per sqrtval 14686. See dfsup2 8981 for alternate definition not requiring dummy variables. (Contributed by NM, 22-May-1999.) |
⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))} | ||
Definition | df-inf 8980 | Define the infimum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the infimum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.) |
⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) | ||
Theorem | dfsup2 8981 | Quantifier-free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.) |
⊢ sup(𝐵, 𝐴, 𝑅) = ∪ (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))) | ||
Theorem | supeq1 8982 | Equality theorem for supremum. (Contributed by NM, 22-May-1999.) |
⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)) | ||
Theorem | supeq1d 8983 | Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)) | ||
Theorem | supeq1i 8984 | Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ 𝐵 = 𝐶 ⇒ ⊢ sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅) | ||
Theorem | supeq2 8985 | Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅)) | ||
Theorem | supeq3 8986 | Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.) |
⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆)) | ||
Theorem | supeq123d 8987 | Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ (𝜑 → 𝐴 = 𝐷) & ⊢ (𝜑 → 𝐵 = 𝐸) & ⊢ (𝜑 → 𝐶 = 𝐹) ⇒ ⊢ (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹)) | ||
Theorem | nfsup 8988 | Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝑅 ⇒ ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) | ||
Theorem | supmo 8989* | Any class 𝐵 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by NM, 5-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) | ||
Theorem | supexd 8990 | A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V) | ||
Theorem | supeu 8991* | A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) | ||
Theorem | supval2 8992* | Alternate expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Thierry Arnoux, 24-Sep-2017.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))) | ||
Theorem | eqsup 8993* | Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶)) | ||
Theorem | eqsupd 8994* | Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) | ||
Theorem | supcl 8995* | A supremum belongs to its base class (closure law). See also supub 8996 and suplub 8997. (Contributed by NM, 12-Oct-2004.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) | ||
Theorem | supub 8996* |
A supremum is an upper bound. See also supcl 8995 and suplub 8997.
This proof demonstrates how to expand an iota-based definition (df-iota 6297) using riotacl2 7144. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)) | ||
Theorem | suplub 8997* | A supremum is the least upper bound. See also supcl 8995 and supub 8996. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧)) | ||
Theorem | suplub2 8998* | Bidirectional form of suplub 8997. (Contributed by Mario Carneiro, 6-Sep-2014.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧)) | ||
Theorem | supnub 8999* | An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝐶𝑅𝑧) → ¬ 𝐶𝑅sup(𝐵, 𝐴, 𝑅))) | ||
Theorem | supex 9000 | A supremum is a set. (Contributed by NM, 22-May-1999.) |
⊢ 𝑅 Or 𝐴 ⇒ ⊢ sup(𝐵, 𝐴, 𝑅) ∈ V |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |