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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cnvcnv 6201 | The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.) (Proof shortened by BJ, 26-Nov-2021.) |
⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) | ||
Theorem | cnvcnv2 6202 | The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.) |
⊢ ◡◡𝐴 = (𝐴 ↾ V) | ||
Theorem | cnvcnvss 6203 | The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.) |
⊢ ◡◡𝐴 ⊆ 𝐴 | ||
Theorem | cnvrescnv 6204 | Two ways to express the corestriction of a class. (Contributed by BJ, 28-Dec-2023.) |
⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵)) | ||
Theorem | cnveqb 6205 | Equality theorem for converse. (Contributed by FL, 19-Sep-2011.) |
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵)) | ||
Theorem | cnveq0 6206 | A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.) |
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅)) | ||
Theorem | dfrel3 6207 | Alternate definition of relation. (Contributed by NM, 14-May-2008.) |
⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅) | ||
Theorem | elid 6208* | Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6207 after elrid 6054. (Contributed by BJ, 28-Aug-2022.) |
⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) | ||
Theorem | dmresv 6209 | The domain of a universal restriction. (Contributed by NM, 14-May-2008.) |
⊢ dom (𝐴 ↾ V) = dom 𝐴 | ||
Theorem | rnresv 6210 | The range of a universal restriction. (Contributed by NM, 14-May-2008.) |
⊢ ran (𝐴 ↾ V) = ran 𝐴 | ||
Theorem | dfrn4 6211 | Range defined in terms of image. (Contributed by NM, 14-May-2008.) |
⊢ ran 𝐴 = (𝐴 “ V) | ||
Theorem | csbrn 6212 | Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.) |
⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵 | ||
Theorem | rescnvcnv 6213 | The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | ||
Theorem | cnvcnvres 6214 | The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.) |
⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵) | ||
Theorem | imacnvcnv 6215 | The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.) |
⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵) | ||
Theorem | dmsnn0 6216 | The domain of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) | ||
Theorem | rnsnn0 6217 | The range of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) |
⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅) | ||
Theorem | dmsn0 6218 | The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.) |
⊢ dom {∅} = ∅ | ||
Theorem | cnvsn0 6219 | The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ ◡{∅} = ∅ | ||
Theorem | dmsn0el 6220 | The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.) |
⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅) | ||
Theorem | relsn2 6221 | A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.) Make hypothesis an antecedent. (Revised by BJ, 12-Feb-2022.) |
⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ dom {𝐴} ≠ ∅)) | ||
Theorem | dmsnopg 6222 | The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) | ||
Theorem | dmsnopss 6223 | The domain of a singleton of an ordered pair is a subset of the singleton of the first member (with no sethood assumptions on 𝐵). (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ dom {〈𝐴, 𝐵〉} ⊆ {𝐴} | ||
Theorem | dmpropg 6224 | The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶}) | ||
Theorem | dmsnop 6225 | The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐵 ∈ V ⇒ ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} | ||
Theorem | dmprop 6226 | The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.) |
⊢ 𝐵 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶} | ||
Theorem | dmtpop 6227 | The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.) |
⊢ 𝐵 ∈ V & ⊢ 𝐷 ∈ V & ⊢ 𝐹 ∈ V ⇒ ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = {𝐴, 𝐶, 𝐸} | ||
Theorem | cnvcnvsn 6228 | Double converse of a singleton of an ordered pair. (Unlike cnvsn 6235, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ◡◡{〈𝐴, 𝐵〉} = ◡{〈𝐵, 𝐴〉} | ||
Theorem | dmsnsnsn 6229 | The domain of the singleton of the singleton of a singleton. (Contributed by NM, 15-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ dom {{{𝐴}}} = {𝐴} | ||
Theorem | rnsnopg 6230 | The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, 𝐵〉} = {𝐵}) | ||
Theorem | rnpropg 6231 | The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐶, 𝐷}) | ||
Theorem | cnvsng 6232 | Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) (Proof shortened by BJ, 12-Feb-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) | ||
Theorem | rnsnop 6233 | The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ran {〈𝐴, 𝐵〉} = {𝐵} | ||
Theorem | op1sta 6234 | Extract the first member of an ordered pair. (See op2nda 6237 to extract the second member, op1stb 5477 for an alternate version, and op1st 8007 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∪ dom {〈𝐴, 𝐵〉} = 𝐴 | ||
Theorem | cnvsn 6235 | Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof shortened by BJ, 12-Feb-2022.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} | ||
Theorem | op2ndb 6236 | Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5477 to extract the first member, op2nda 6237 for an alternate version, and op2nd 8008 for the preferred version.) (Contributed by NM, 25-Nov-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = 𝐵 | ||
Theorem | op2nda 6237 | Extract the second member of an ordered pair. (See op1sta 6234 to extract the first member, op2ndb 6236 for an alternate version, and op2nd 8008 for the preferred version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∪ ran {〈𝐴, 𝐵〉} = 𝐵 | ||
Theorem | opswap 6238 | Swap the members of an ordered pair. (Contributed by NM, 14-Dec-2008.) (Revised by Mario Carneiro, 30-Aug-2015.) |
⊢ ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉 | ||
Theorem | cnvresima 6239 | An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) |
⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴) | ||
Theorem | resdm2 6240 | A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.) |
⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴 | ||
Theorem | resdmres 6241 | Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵) | ||
Theorem | resresdm 6242 | A restriction by an arbitrary set is a restriction by its domain. (Contributed by AV, 16-Nov-2020.) |
⊢ (𝐹 = (𝐸 ↾ 𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹)) | ||
Theorem | imadmres 6243 | The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵) | ||
Theorem | resdmss 6244 | Subset relationship for the domain of a restriction. (Contributed by Scott Fenton, 9-Aug-2024.) |
⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵 | ||
Theorem | resdifdi 6245 | Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.) |
⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶)) | ||
Theorem | resdifdir 6246 | Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.) |
⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶)) | ||
Theorem | mptpreima 6247* | The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (◡𝐹 “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} | ||
Theorem | mptiniseg 6248* | Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝐶 ∈ 𝑉 → (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶}) | ||
Theorem | dmmpt 6249 | The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} | ||
Theorem | dmmptss 6250* | The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ dom 𝐹 ⊆ 𝐴 | ||
Theorem | dmmptg 6251* | The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) | ||
Theorem | rnmpt0f 6252* | The range of a function in maps-to notation is empty if and only if its domain is empty. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅)) | ||
Theorem | rnmptn0 6253* | The range of a function in maps-to notation is nonempty if the domain is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐴 ≠ ∅) ⇒ ⊢ (𝜑 → ran 𝐹 ≠ ∅) | ||
Theorem | dfco2 6254* | Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.) |
⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) | ||
Theorem | dfco2a 6255* | Generalization of dfco2 6254, where 𝐶 can have any value between dom 𝐴 ∩ ran 𝐵 and V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))) | ||
Theorem | coundi 6256 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶)) | ||
Theorem | coundir 6257 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶)) | ||
Theorem | cores 6258 | Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵)) | ||
Theorem | resco 6259 | Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.) |
⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶)) | ||
Theorem | imaco 6260 | Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) (Proof shortened by Wolf Lammen, 16-May-2025.) |
⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶)) | ||
Theorem | rnco 6261 | The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) | ||
Theorem | rnco2 6262 | The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.) |
⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵) | ||
Theorem | dmco 6263 | The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.) |
⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴) | ||
Theorem | coeq0 6264 | A composition of two relations is empty iff there is no overlap between the range of the second and the domain of the first. Useful in combination with coundi 6256 and coundir 6257 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅) | ||
Theorem | coiun 6265* | Composition with an indexed union. (Contributed by NM, 21-Dec-2008.) |
⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) | ||
Theorem | cocnvcnv1 6266 | A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.) |
⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) | ||
Theorem | cocnvcnv2 6267 | A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.) |
⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵) | ||
Theorem | cores2 6268 | Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.) |
⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵)) | ||
Theorem | co02 6269 | Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.) |
⊢ (𝐴 ∘ ∅) = ∅ | ||
Theorem | co01 6270 | Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
⊢ (∅ ∘ 𝐴) = ∅ | ||
Theorem | coi1 6271 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴) | ||
Theorem | coi2 6272 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴) | ||
Theorem | coires1 6273 | Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.) |
⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵) | ||
Theorem | coass 6274 | Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.) |
⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶)) | ||
Theorem | relcnvtrg 6275 | General form of relcnvtr 6276. (Contributed by Peter Mazsa, 17-Oct-2023.) |
⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅 ∘ 𝑆) ⊆ 𝑇 ↔ (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇)) | ||
Theorem | relcnvtr 6276 | A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Peter Mazsa, 17-Oct-2023.) |
⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅)) | ||
Theorem | relssdmrn 6277 | A relation is included in the Cartesian product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.) (Proof shortened by SN, 23-Dec-2024.) |
⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) | ||
Theorem | relssdmrnOLD 6278 | Obsolete version of relssdmrn 6277 as of 23-Dec-2024. (Contributed by NM, 3-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) | ||
Theorem | resssxp 6279 | If the 𝑅-image of a class 𝐴 is a subclass of 𝐵, then the restriction of 𝑅 to 𝐴 is a subset of the Cartesian product of 𝐴 and 𝐵. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) | ||
Theorem | cnvssrndm 6280 | The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴) | ||
Theorem | cossxp 6281 | Composition as a subset of the Cartesian product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) | ||
Theorem | relrelss 6282 | Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.) |
⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V)) | ||
Theorem | unielrel 6283 | The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.) |
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∪ 𝐴 ∈ ∪ 𝑅) | ||
Theorem | relfld 6284 | The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.) |
⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) | ||
Theorem | relresfld 6285 | Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.) |
⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅) | ||
Theorem | relcoi2 6286 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.) |
⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = 𝑅) | ||
Theorem | relcoi1 6287 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) (Proof shortened by OpenAI, 3-Jul-2020.) |
⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅) | ||
Theorem | unidmrn 6288 | The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.) |
⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴) | ||
Theorem | relcnvfld 6289 | if 𝑅 is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.) |
⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) | ||
Theorem | dfdm2 6290 | Alternate definition of domain df-dm 5692 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴) | ||
Theorem | unixp 6291 | The double class union of a nonempty Cartesian product is the union of it members. (Contributed by NM, 17-Sep-2006.) |
⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵)) | ||
Theorem | unixp0 6292 | A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.) |
⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅) | ||
Theorem | unixpid 6293 | Field of a Cartesian square. (Contributed by FL, 10-Oct-2009.) |
⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴 | ||
Theorem | ressn 6294 | Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) | ||
Theorem | cnviin 6295* | The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.) |
⊢ (𝐴 ≠ ∅ → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵) | ||
Theorem | cnvpo 6296 | The converse of a partial order is a partial order. (Contributed by NM, 15-Jun-2005.) |
⊢ (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴) | ||
Theorem | cnvso 6297 | The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.) |
⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | ||
Theorem | xpco 6298 | Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.) |
⊢ (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶)) | ||
Theorem | xpcoid 6299 | Composition of two Cartesian squares. (Contributed by Thierry Arnoux, 14-Jan-2018.) |
⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴) | ||
Theorem | elsnxp 6300* | Membership in a Cartesian product with a singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑍 = 〈𝑋, 𝑦〉)) |
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