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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfrn4 6201 | Range defined in terms of image. (Contributed by NM, 14-May-2008.) |
⊢ ran 𝐴 = (𝐴 “ V) | ||
Theorem | csbrn 6202 | 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 6203 | 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 6204 | The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.) |
⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵) | ||
Theorem | imacnvcnv 6205 | The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.) |
⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵) | ||
Theorem | dmsnn0 6206 | 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 6207 | 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 6208 | The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.) |
⊢ dom {∅} = ∅ | ||
Theorem | cnvsn0 6209 | The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ ◡{∅} = ∅ | ||
Theorem | dmsn0el 6210 | 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 6211 | 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 6212 | 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 6213 | 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 6214 | 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 6215 | 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 6216 | 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 6217 | 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 6218 | Double converse of a singleton of an ordered pair. (Unlike cnvsn 6225, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩} | ||
Theorem | dmsnsnsn 6219 | 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 6220 | 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 6221 | 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 6222 | 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 6223 | 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 6224 | Extract the first member of an ordered pair. (See op2nda 6227 to extract the second member, op1stb 5471 for an alternate version, and op1st 7985 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴 | ||
Theorem | cnvsn 6225 | 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 6226 | Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5471 to extract the first member, op2nda 6227 for an alternate version, and op2nd 7986 for the preferred version.) (Contributed by NM, 25-Nov-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = 𝐵 | ||
Theorem | op2nda 6227 | Extract the second member of an ordered pair. (See op1sta 6224 to extract the first member, op2ndb 6226 for an alternate version, and op2nd 7986 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 6228 | 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 6229 | An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) |
⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴) | ||
Theorem | resdm2 6230 | A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.) |
⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴 | ||
Theorem | resdmres 6231 | Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵) | ||
Theorem | resresdm 6232 | 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 6233 | The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵) | ||
Theorem | resdmss 6234 | Subset relationship for the domain of a restriction. (Contributed by Scott Fenton, 9-Aug-2024.) |
⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵 | ||
Theorem | resdifdi 6235 | Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.) |
⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶)) | ||
Theorem | resdifdir 6236 | Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.) |
⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶)) | ||
Theorem | mptpreima 6237* | 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 6238* | 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 6239 | 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 6240* | 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 6241* | 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 6242* | 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 6243* | 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 6244* | 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 6245* | Generalization of dfco2 6244, 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 6246 | 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 6247 | 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 6248 | 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 6249 | Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.) |
⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶)) | ||
Theorem | imaco 6250 | Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) |
⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶)) | ||
Theorem | rnco 6251 | 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 6252 | The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.) |
⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵) | ||
Theorem | dmco 6253 | 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 6254 | 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 6246 and coundir 6247 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅) | ||
Theorem | coiun 6255* | Composition with an indexed union. (Contributed by NM, 21-Dec-2008.) |
⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) | ||
Theorem | cocnvcnv1 6256 | 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 6257 | 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 6258 | 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 6259 | Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.) |
⊢ (𝐴 ∘ ∅) = ∅ | ||
Theorem | co01 6260 | Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
⊢ (∅ ∘ 𝐴) = ∅ | ||
Theorem | coi1 6261 | 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 6262 | 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 6263 | 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 6264 | 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 6265 | General form of relcnvtr 6266. (Contributed by Peter Mazsa, 17-Oct-2023.) |
⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅 ∘ 𝑆) ⊆ 𝑇 ↔ (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇)) | ||
Theorem | relcnvtr 6266 | 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 6267 | 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 6268 | Obsolete version of relssdmrn 6267 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 6269 | 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 6270 | 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 6271 | 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 6272 | 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 6273 | 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 6274 | The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.) |
⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) | ||
Theorem | relresfld 6275 | Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.) |
⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅) | ||
Theorem | relcoi2 6276 | 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 6277 | 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 6278 | 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 6279 | 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 6280 | Alternate definition of domain df-dm 5686 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴) | ||
Theorem | unixp 6281 | 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 6282 | A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.) |
⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅) | ||
Theorem | unixpid 6283 | Field of a Cartesian square. (Contributed by FL, 10-Oct-2009.) |
⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴 | ||
Theorem | ressn 6284 | 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 6285* | 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 6286 | The converse of a partial order is a partial order. (Contributed by NM, 15-Jun-2005.) |
⊢ (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴) | ||
Theorem | cnvso 6287 | 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 6288 | Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.) |
⊢ (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶)) | ||
Theorem | xpcoid 6289 | Composition of two Cartesian squares. (Contributed by Thierry Arnoux, 14-Jan-2018.) |
⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴) | ||
Theorem | elsnxp 6290* | 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⟩)) | ||
Theorem | reu3op 6291* | There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 1-Jul-2023.) |
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 𝜒 ∧ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))) | ||
Theorem | reuop 6292* | There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 23-Jun-2023.) |
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓 ↔ 𝜒)) & ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜃)) ⇒ ⊢ (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝜃 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))) | ||
Theorem | opreu2reurex 6293* | There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 24-Jun-2023.) (Revised by AV, 1-Jul-2023.) |
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒)) ⇒ ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒)) | ||
Theorem | opreu2reu 6294* | If there is a unique ordered pair fulfilling a wff, then there is a double restricted unique existential qualification fulfilling a corresponding wff. (Contributed by AV, 25-Jun-2023.) (Revised by AV, 2-Jul-2023.) |
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒)) ⇒ ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 → ∃!𝑎 ∈ 𝐴 ∃!𝑏 ∈ 𝐵 𝜒) | ||
Theorem | dfpo2 6295 | Quantifier-free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝑅 Po 𝐴 ↔ ((𝑅 ∩ ( I ↾ 𝐴)) = ∅ ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅)) | ||
Theorem | csbcog 6296 | Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∘ ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | snres0 6297 | Condition for restriction of a singleton to be empty. (Contributed by Scott Fenton, 9-Aug-2024.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴 ∈ 𝐶) | ||
Theorem | imaindm 6298 | The image is unaffected by intersection with the domain. (Contributed by Scott Fenton, 17-Dec-2021.) |
⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅)) | ||
Syntax | cpred 6299 | The predecessors symbol. |
class Pred(𝑅, 𝐴, 𝑋) | ||
Definition | df-pred 6300 | Define the predecessor class of a binary relation. This is the class of all elements 𝑦 of 𝐴 such that 𝑦𝑅𝑋 (see elpred 6317). (Contributed by Scott Fenton, 29-Jan-2011.) |
⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) |
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