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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | resdmdfsn 5901 | Restricting a relation to its domain without a set is the same as restricting the relation to the universe without this set. (Contributed by AV, 2-Dec-2018.) |
⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))) | ||
Theorem | resopab 5902* | Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | ||
Theorem | iss 5903 | A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (𝐴 ⊆ I ↔ 𝐴 = ( I ↾ dom 𝐴)) | ||
Theorem | resopab2 5904* | Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.) |
⊢ (𝐴 ⊆ 𝐵 → ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) | ||
Theorem | resmpt 5905* | Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.) |
⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
Theorem | resmpt3 5906* | Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.) |
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) | ||
Theorem | resmptf 5907 | Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
Theorem | resmptd 5908* | Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
Theorem | dfres2 5909* | Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.) |
⊢ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | ||
Theorem | mptss 5910* | Sufficient condition for inclusion among two functions in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
Theorem | elidinxp 5911* | Characterization of the elements of the intersection of the identity relation with a Cartesian product. (Contributed by Peter Mazsa, 9-Sep-2022.) |
⊢ (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝐶 = 〈𝑥, 𝑥〉) | ||
Theorem | elidinxpid 5912* | Characterization of the elements of the intersection of the identity relation with a Cartesian square. (Contributed by Peter Mazsa, 9-Sep-2022.) |
⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝐵 = 〈𝑥, 𝑥〉) | ||
Theorem | elrid 5913* | Characterization of the elements of a restricted identity relation. (Contributed by BJ, 28-Aug-2022.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) |
⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐴 = 〈𝑥, 𝑥〉) | ||
Theorem | idinxpres 5914 | The intersection of the identity relation with a cartesian product is the restriction of the identity relation to the intersection of the factors. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) Generalize statement from cartesian square (now idinxpresid 5915) to cartesian product. (Revised by BJ, 23-Dec-2023.) |
⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵)) | ||
Theorem | idinxpresid 5915 | The intersection of the identity relation with the cartesian square of a class is the restriction of the identity relation to that class. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) (Proof shortened by BJ, 23-Dec-2023.) |
⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴) | ||
Theorem | idssxp 5916 | A diagonal set as a subset of a Cartesian square. (Contributed by Thierry Arnoux, 29-Dec-2019.) (Proof shortened by BJ, 9-Sep-2022.) |
⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) | ||
Theorem | opabresid 5917* | The restricted identity relation expressed as an ordered-pair class abstraction. (Contributed by FL, 25-Apr-2012.) |
⊢ ( I ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | ||
Theorem | mptresid 5918* | The restricted identity relation expressed in maps-to notation. (Contributed by FL, 25-Apr-2012.) |
⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥) | ||
Theorem | opabresidOLD 5919* | Obsolete version of opabresid 5917 as of 26-Dec-2023. (Contributed by FL, 25-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴) | ||
Theorem | mptresidOLD 5920* | Obsolete version of mptresid 5918 as of 26-Dec-2023. (Contributed by FL, 25-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴) | ||
Theorem | dmresi 5921 | The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.) |
⊢ dom ( I ↾ 𝐴) = 𝐴 | ||
Theorem | restidsing 5922 | Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.) (Proof shortened by Peter Mazsa, 6-Oct-2022.) |
⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) | ||
Theorem | iresn0n0 5923 | The identity function restricted to a class 𝐴 is empty iff the class 𝐴 is empty. (Contributed by AV, 30-Jan-2024.) |
⊢ (𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅) | ||
Theorem | imaeq1 5924 | Equality theorem for image. (Contributed by NM, 14-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) | ||
Theorem | imaeq2 5925 | Equality theorem for image. (Contributed by NM, 14-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) | ||
Theorem | imaeq1i 5926 | Equality theorem for image. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 “ 𝐶) = (𝐵 “ 𝐶) | ||
Theorem | imaeq2i 5927 | Equality theorem for image. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 “ 𝐴) = (𝐶 “ 𝐵) | ||
Theorem | imaeq1d 5928 | Equality theorem for image. (Contributed by FL, 15-Dec-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) | ||
Theorem | imaeq2d 5929 | Equality theorem for image. (Contributed by FL, 15-Dec-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) | ||
Theorem | imaeq12d 5930 | Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷)) | ||
Theorem | dfima2 5931* | Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} | ||
Theorem | dfima3 5932* | Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | ||
Theorem | elimag 5933* | Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴)) | ||
Theorem | elima 5934* | Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴) | ||
Theorem | elima2 5935* | Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴)) | ||
Theorem | elima3 5936* | Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 〈𝑥, 𝐴〉 ∈ 𝐵)) | ||
Theorem | nfima 5937 | Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 “ 𝐵) | ||
Theorem | nfimad 5938 | Deduction version of bound-variable hypothesis builder nfima 5937. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵)) | ||
Theorem | imadmrn 5939 | The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.) |
⊢ (𝐴 “ dom 𝐴) = ran 𝐴 | ||
Theorem | imassrn 5940 | The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by NM, 31-Mar-1995.) |
⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴 | ||
Theorem | mptima 5941* | Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) | ||
Theorem | imai 5942 | Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.) |
⊢ ( I “ 𝐴) = 𝐴 | ||
Theorem | rnresi 5943 | The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.) |
⊢ ran ( I ↾ 𝐴) = 𝐴 | ||
Theorem | resiima 5944 | The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.) |
⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵) | ||
Theorem | ima0 5945 | Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.) |
⊢ (𝐴 “ ∅) = ∅ | ||
Theorem | 0ima 5946 | Image under the empty relation. (Contributed by FL, 11-Jan-2007.) |
⊢ (∅ “ 𝐴) = ∅ | ||
Theorem | csbima12 5947 | Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Revised by NM, 20-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) | ||
Theorem | imadisj 5948 | A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.) |
⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) | ||
Theorem | cnvimass 5949 | A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.) |
⊢ (◡𝐴 “ 𝐵) ⊆ dom 𝐴 | ||
Theorem | cnvimarndm 5950 | The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.) |
⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴 | ||
Theorem | imasng 5951* | The image of a singleton. (Contributed by NM, 8-May-2005.) |
⊢ (𝐴 ∈ 𝐵 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) | ||
Theorem | relimasn 5952* | The image of a singleton. (Contributed by NM, 20-May-1998.) |
⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) | ||
Theorem | elrelimasn 5953 | Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.) |
⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵)) | ||
Theorem | elimasn 5954 | Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴) | ||
Theorem | elimasng 5955 | Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) | ||
Theorem | elimasni 5956 | Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.) |
⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶) | ||
Theorem | args 5957* | Two ways to express the class of unique-valued arguments of 𝐹, which is the same as the domain of 𝐹 whenever 𝐹 is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg 𝐹 " for this class (for which we have no separate notation). Observe the resemblance to the alternate definition dffv4 6667 of function value, which is based on the idea in Quine's definition. (Contributed by NM, 8-May-2005.) |
⊢ {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} | ||
Theorem | eliniseg 5958 | Membership in an initial segment. The idiom (◡𝐴 “ {𝐵}), meaning {𝑥 ∣ 𝑥𝐴𝐵}, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ 𝐶 ∈ V ⇒ ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) | ||
Theorem | epini 5959 | Any set is equal to its preimage under the converse membership relation. (Contributed by Mario Carneiro, 9-Mar-2013.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (◡ E “ {𝐴}) = 𝐴 | ||
Theorem | iniseg 5960* | An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) |
⊢ (𝐵 ∈ 𝑉 → (◡𝐴 “ {𝐵}) = {𝑥 ∣ 𝑥𝐴𝐵}) | ||
Theorem | inisegn0 5961 | Nonemptiness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
⊢ (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅) | ||
Theorem | dffr3 5962* | Alternate definition of well-founded relation. Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 23-Jun-2015.) |
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)) | ||
Theorem | dfse2 5963* | Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.) |
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) | ||
Theorem | imass1 5964 | Subset theorem for image. (Contributed by NM, 16-Mar-2004.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶)) | ||
Theorem | imass2 5965 | Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵)) | ||
Theorem | ndmima 5966 | The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.) (Proof shortened by OpenAI, 3-Jul-2020.) |
⊢ (¬ 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅) | ||
Theorem | relcnv 5967 | A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.) |
⊢ Rel ◡𝐴 | ||
Theorem | relbrcnvg 5968 | When 𝑅 is a relation, the sethood assumptions on brcnv 5753 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) | ||
Theorem | eliniseg2 5969 | Eliminate the class existence constraint in eliniseg 5958. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.) |
⊢ (Rel 𝐴 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) | ||
Theorem | relbrcnv 5970 | When 𝑅 is a relation, the sethood assumptions on brcnv 5753 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ Rel 𝑅 ⇒ ⊢ (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴) | ||
Theorem | cotrg 5971* | Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 5972 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5972. (Revised by Richard Penner, 24-Dec-2019.) |
⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | ||
Theorem | cotr 5972* | Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. Special instance of cotrg 5971. (Contributed by NM, 27-Dec-1996.) |
⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | ||
Theorem | idrefALT 5973* | Alternate proof of idref 6908 not relying on definitions related to functions. Two ways to state that a relation is reflexive on a class. (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Revised by NM, 30-Mar-2016.) (Proof shortened by BJ, 28-Aug-2022.) The "proof modification is discouraged" tag is here only because this is an *ALT result. (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) | ||
Theorem | cnvsym 5974* | Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) | ||
Theorem | intasym 5975* | Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) | ||
Theorem | asymref 5976* | Two ways of saying a relation is antisymmetric and reflexive. ∪ ∪ 𝑅 is the field of a relation by relfld 6126. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) | ||
Theorem | asymref2 5977* | Two ways of saying a relation is antisymmetric and reflexive. (Contributed by NM, 6-May-2008.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) |
⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ (∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))) | ||
Theorem | intirr 5978* | Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥) | ||
Theorem | brcodir 5979* | Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(◡𝑅 ∘ 𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧))) | ||
Theorem | codir 5980* | Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.) |
⊢ ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)) | ||
Theorem | qfto 5981* | A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.) |
⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) | ||
Theorem | xpidtr 5982 | A Cartesian square is a transitive relation. (Contributed by FL, 31-Jul-2009.) |
⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) | ||
Theorem | trin2 5983 | The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.) |
⊢ (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ (𝑅 ∩ 𝑆)) | ||
Theorem | poirr2 5984 | A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) = ∅) | ||
Theorem | trinxp 5985 | The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a Cartesian square is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.) |
⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴))) | ||
Theorem | soirri 5986 | A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝑅 Or 𝑆 & ⊢ 𝑅 ⊆ (𝑆 × 𝑆) ⇒ ⊢ ¬ 𝐴𝑅𝐴 | ||
Theorem | sotri 5987 | A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝑅 Or 𝑆 & ⊢ 𝑅 ⊆ (𝑆 × 𝑆) ⇒ ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) | ||
Theorem | son2lpi 5988 | A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝑅 Or 𝑆 & ⊢ 𝑅 ⊆ (𝑆 × 𝑆) ⇒ ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) | ||
Theorem | sotri2 5989 | A transitivity relation. (Read 𝐴 ≤ 𝐵 and 𝐵 < 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.) |
⊢ 𝑅 Or 𝑆 & ⊢ 𝑅 ⊆ (𝑆 × 𝑆) ⇒ ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) | ||
Theorem | sotri3 5990 | A transitivity relation. (Read 𝐴 < 𝐵 and 𝐵 ≤ 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.) |
⊢ 𝑅 Or 𝑆 & ⊢ 𝑅 ⊆ (𝑆 × 𝑆) ⇒ ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶) | ||
Theorem | poleloe 5991 | Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))) | ||
Theorem | poltletr 5992 | Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ 𝐵(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶)) | ||
Theorem | somin1 5993 | Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴) | ||
Theorem | somincom 5994 | Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴)) | ||
Theorem | somin2 5995 | Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐵) | ||
Theorem | soltmin 5996 | Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑅𝐶))) | ||
Theorem | cnvopab 5997* | The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ◡{〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑦, 𝑥〉 ∣ 𝜑} | ||
Theorem | mptcnv 5998* | The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.) |
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷))) ⇒ ⊢ (𝜑 → ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ 𝐷)) | ||
Theorem | cnv0 5999 | The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5203, ax-nul 5210, ax-pr 5330. (Revised by KP, 25-Oct-2021.) |
⊢ ◡∅ = ∅ | ||
Theorem | cnvi 6000 | The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ◡ I = I |
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