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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rnuni 6001* | The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro, 29-May-2015.) |
⊢ ran ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ran 𝑥 | ||
Theorem | imaundi 6002 | Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.) |
⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) | ||
Theorem | imaundir 6003 | The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.) |
⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶)) | ||
Theorem | dminss 6004 | An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by NM, 11-Aug-2004.) |
⊢ (dom 𝑅 ∩ 𝐴) ⊆ (◡𝑅 “ (𝑅 “ 𝐴)) | ||
Theorem | imainss 6005 | An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.) |
⊢ ((𝑅 “ 𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵))) | ||
Theorem | inimass 6006 | The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶)) | ||
Theorem | inimasn 6007 | The intersection of the image of singleton. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∩ 𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶}))) | ||
Theorem | cnvxp 6008 | The converse of a Cartesian product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | ||
Theorem | xp0 6009 | The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.) |
⊢ (𝐴 × ∅) = ∅ | ||
Theorem | xpnz 6010 | The Cartesian product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.) |
⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) | ||
Theorem | xpeq0 6011 | At least one member of an empty Cartesian product is empty. (Contributed by NM, 27-Aug-2006.) |
⊢ ((𝐴 × 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)) | ||
Theorem | xpdisj1 6012 | Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅) | ||
Theorem | xpdisj2 6013 | Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ∅) | ||
Theorem | xpsndisj 6014 | Cartesian products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.) |
⊢ (𝐵 ≠ 𝐷 → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅) | ||
Theorem | difxp 6015 | Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.) |
⊢ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) = (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵))) | ||
Theorem | difxp1 6016 | Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.) |
⊢ ((𝐴 ∖ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∖ (𝐵 × 𝐶)) | ||
Theorem | difxp2 6017 | Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.) |
⊢ (𝐴 × (𝐵 ∖ 𝐶)) = ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶)) | ||
Theorem | djudisj 6018* | Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐶) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝐷)) = ∅) | ||
Theorem | xpdifid 6019* | The set of distinct couples in a Cartesian product. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) = ((𝐴 × 𝐵) ∖ I ) | ||
Theorem | resdisj 6020 | A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ↾ 𝐴) ↾ 𝐵) = ∅) | ||
Theorem | rnxp 6021 | The range of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.) |
⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) | ||
Theorem | dmxpss 6022 | The domain of a Cartesian product is included in its first factor. (Contributed by NM, 19-Mar-2007.) |
⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | ||
Theorem | rnxpss 6023 | The range of a Cartesian product is included in its second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | ||
Theorem | rnxpid 6024 | The range of a Cartesian square. (Contributed by FL, 17-May-2010.) |
⊢ ran (𝐴 × 𝐴) = 𝐴 | ||
Theorem | ssxpb 6025 | A Cartesian product subclass relationship is equivalent to the conjunction of the analogous relationships for the factors. (Contributed by NM, 17-Dec-2008.) |
⊢ ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷))) | ||
Theorem | xp11 6026 | The Cartesian product of nonempty classes is a one-to-one "function" of its two "arguments". In other words, two Cartesian products, at least one with nonempty factors, are equal if and only if their respective factors are equal. (Contributed by NM, 31-May-2008.) |
⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ((𝐴 × 𝐵) = (𝐶 × 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | xpcan 6027 | Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.) |
⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | xpcan2 6028 | Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.) |
⊢ (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | ssrnres 6029 | Two ways to express surjectivity of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product): the LHS expresses inclusion in the range of the restricted relation, while the RHS expresses equality with the range of the restricted and corestricted relation. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵) | ||
Theorem | rninxp 6030* | Two ways to express surjectivity of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) | ||
Theorem | dminxp 6031* | Two ways to express totality of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (Contributed by NM, 17-Jan-2006.) |
⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦) | ||
Theorem | imainrect 6032 | Image by a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (Contributed by Stefan O'Rear, 19-Feb-2015.) |
⊢ ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵) | ||
Theorem | xpima 6033 | Direct image by a Cartesian product. (Contributed by Thierry Arnoux, 4-Feb-2017.) |
⊢ ((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵) | ||
Theorem | xpima1 6034 | Direct image by a Cartesian product (case of empty intersection with the domain). (Contributed by Thierry Arnoux, 16-Dec-2017.) |
⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅) | ||
Theorem | xpima2 6035 | Direct image by a Cartesian product (case of nonempty intersection with the domain). (Contributed by Thierry Arnoux, 16-Dec-2017.) |
⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵) | ||
Theorem | xpimasn 6036 | Direct image of a singleton by a Cartesian product. (Contributed by Thierry Arnoux, 14-Jan-2018.) (Proof shortened by BJ, 6-Apr-2019.) |
⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) | ||
Theorem | sossfld 6037 | The base set of a strict order is contained in the field of the relation, except possibly for one element (note that ∅ Or {𝐵}). (Contributed by Mario Carneiro, 27-Apr-2015.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅)) | ||
Theorem | sofld 6038 | The base set of a nonempty strict order is the same as the field of the relation. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 = (dom 𝑅 ∪ ran 𝑅)) | ||
Theorem | cnvcnv3 6039* | The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.) |
⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} | ||
Theorem | dfrel2 6040 | Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.) |
⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | ||
Theorem | dfrel4v 6041* | A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6718 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) |
⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) | ||
Theorem | dfrel4 6042* | A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6718 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑦𝑅 ⇒ ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) | ||
Theorem | cnvcnv 6043 | 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 6044 | The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.) |
⊢ ◡◡𝐴 = (𝐴 ↾ V) | ||
Theorem | cnvcnvss 6045 | 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 6046 | Two ways to express the corestriction of a class. (Contributed by BJ, 28-Dec-2023.) |
⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵)) | ||
Theorem | cnveqb 6047 | Equality theorem for converse. (Contributed by FL, 19-Sep-2011.) |
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵)) | ||
Theorem | cnveq0 6048 | A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.) |
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅)) | ||
Theorem | dfrel3 6049 | Alternate definition of relation. (Contributed by NM, 14-May-2008.) |
⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅) | ||
Theorem | elid 6050* | Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6049 after elrid 5907. (Contributed by BJ, 28-Aug-2022.) |
⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) | ||
Theorem | dmresv 6051 | The domain of a universal restriction. (Contributed by NM, 14-May-2008.) |
⊢ dom (𝐴 ↾ V) = dom 𝐴 | ||
Theorem | rnresv 6052 | The range of a universal restriction. (Contributed by NM, 14-May-2008.) |
⊢ ran (𝐴 ↾ V) = ran 𝐴 | ||
Theorem | dfrn4 6053 | Range defined in terms of image. (Contributed by NM, 14-May-2008.) |
⊢ ran 𝐴 = (𝐴 “ V) | ||
Theorem | csbrn 6054 | 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 6055 | 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 6056 | The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.) |
⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵) | ||
Theorem | imacnvcnv 6057 | The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.) |
⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵) | ||
Theorem | dmsnn0 6058 | 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 6059 | 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 6060 | The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.) |
⊢ dom {∅} = ∅ | ||
Theorem | cnvsn0 6061 | The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ ◡{∅} = ∅ | ||
Theorem | dmsn0el 6062 | 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 6063 | 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 6064 | 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 6065 | 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 6066 | 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 6067 | 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 6068 | 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 6069 | 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 6070 | Double converse of a singleton of an ordered pair. (Unlike cnvsn 6077, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ◡◡{〈𝐴, 𝐵〉} = ◡{〈𝐵, 𝐴〉} | ||
Theorem | dmsnsnsn 6071 | 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 6072 | 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 6073 | 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 6074 | 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 6075 | 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 6076 | Extract the first member of an ordered pair. (See op2nda 6079 to extract the second member, op1stb 5355 for an alternate version, and op1st 7688 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∪ dom {〈𝐴, 𝐵〉} = 𝐴 | ||
Theorem | cnvsn 6077 | 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 6078 | Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5355 to extract the first member, op2nda 6079 for an alternate version, and op2nd 7689 for the preferred version.) (Contributed by NM, 25-Nov-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = 𝐵 | ||
Theorem | op2nda 6079 | Extract the second member of an ordered pair. (See op1sta 6076 to extract the first member, op2ndb 6078 for an alternate version, and op2nd 7689 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 6080 | 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 6081 | An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) |
⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴) | ||
Theorem | resdm2 6082 | A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.) |
⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴 | ||
Theorem | resdmres 6083 | Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵) | ||
Theorem | resresdm 6084 | 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 6085 | The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵) | ||
Theorem | mptpreima 6086* | 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 6087* | 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 6088 | 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 6089* | 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 6090* | 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 | relco 6091 | A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) |
⊢ Rel (𝐴 ∘ 𝐵) | ||
Theorem | dfco2 6092* | 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 6093* | Generalization of dfco2 6092, 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 6094 | 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 6095 | 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 6096 | 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 6097 | Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.) |
⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶)) | ||
Theorem | imaco 6098 | Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) |
⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶)) | ||
Theorem | rnco 6099 | 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 6100 | The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.) |
⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵) |
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