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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | op1steq 7101* | Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.) |
⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) | ||
Theorem | el2xptp 7102* | A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.) |
⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | ||
Theorem | el2xptp0 7103 | A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.) |
⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st ‘𝐴)) = 𝑋 ∧ (2nd ‘(1st ‘𝐴)) = 𝑌 ∧ (2nd ‘𝐴) = 𝑍)) ↔ 𝐴 = 〈𝑋, 𝑌, 𝑍〉)) | ||
Theorem | 2nd1st 7104 | Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) | ||
Theorem | 1st2nd 7105 | Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.) |
⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | ||
Theorem | 1stdm 7106 | The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.) |
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) ∈ dom 𝑅) | ||
Theorem | 2ndrn 7107 | The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.) |
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅) | ||
Theorem | 1st2ndbr 7108 | Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.) |
⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴)) | ||
Theorem | releldm2 7109* | Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
⊢ (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵)) | ||
Theorem | reldm 7110* | An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))) | ||
Theorem | sbcopeq1a 7111 | Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3413 that avoids the existential quantifiers of copsexg 4882). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 = 〈𝑥, 𝑦〉 → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑)) | ||
Theorem | csbopeq1a 7112 | Equality theorem for substitution of a class 𝐴 for an ordered pair 〈𝑥, 𝑦〉 in 𝐵 (analogue of csbeq1a 3508). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵) | ||
Theorem | dfopab2 7113* | A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑} | ||
Theorem | dfoprab3s 7114* | A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)} | ||
Theorem | dfoprab3 7115* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.) |
⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} | ||
Theorem | dfoprab4 7116* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} | ||
Theorem | dfoprab4f 7117* | Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} | ||
Theorem | opiota 7118* | The property of a uniquely specified ordered pair. The proof uses properties of the ℩ description binder. (Contributed by Mario Carneiro, 21-May-2015.) |
⊢ 𝐼 = (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) & ⊢ 𝑋 = (1st ‘𝐼) & ⊢ 𝑌 = (2nd ‘𝐼) & ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐷 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝜒) ↔ (𝐶 = 𝑋 ∧ 𝐷 = 𝑌))) | ||
Theorem | dfxp3 7119* | Define the Cartesian product of three classes. Compare df-xp 5044. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) |
⊢ ((𝐴 × 𝐵) × 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)} | ||
Theorem | elopabi 7120* | A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.) |
⊢ (𝑥 = (1st ‘𝐴) → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = (2nd ‘𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝜒) | ||
Theorem | eloprabi 7121* | A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ (𝑥 = (1st ‘(1st ‘𝐴)) → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = (2nd ‘(1st ‘𝐴)) → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = (2nd ‘𝐴) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝐴 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → 𝜃) | ||
Theorem | mpt2mptsx 7122* | Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶) | ||
Theorem | mpt2mpts 7123* | Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.) |
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶) | ||
Theorem | dmmpt2ssx 7124* | The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | ||
Theorem | fmpt2x 7125* | Functionality, domain and codomain of a class given by the "maps to" notation, where 𝐵(𝑥) is not constant but depends on 𝑥. (Contributed by NM, 29-Dec-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷) | ||
Theorem | fmpt2 7126* | Functionality, domain and range of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:(𝐴 × 𝐵)⟶𝐷) | ||
Theorem | fnmpt2 7127* | Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → 𝐹 Fn (𝐴 × 𝐵)) | ||
Theorem | fnmpt2i 7128* | Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ 𝐶 ∈ V ⇒ ⊢ 𝐹 Fn (𝐴 × 𝐵) | ||
Theorem | dmmpt2 7129* | Domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ 𝐶 ∈ V ⇒ ⊢ dom 𝐹 = (𝐴 × 𝐵) | ||
Theorem | ovmpt2elrn 7130* | An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.) |
⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) ∈ 𝑀) | ||
Theorem | dmmpt2ga 7131* | Domain of an operation given by the "maps to" notation, closed form of dmmpt2 7129. (Contributed by Alexander van der Vekens, 10-Feb-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵)) | ||
Theorem | dmmpt2g 7132* | Domain of an operation given by the "maps to" notation, closed form of dmmpt2 7129. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Prove shortened by AV, 10-Feb-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵)) | ||
Theorem | mpt2exxg 7133* | Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆) → 𝐹 ∈ V) | ||
Theorem | mpt2exg 7134* | Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝐹 ∈ V) | ||
Theorem | mpt2exga 7135* | If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 12-Sep-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) | ||
Theorem | mpt2ex 7136* | If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by Mario Carneiro, 20-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V | ||
Theorem | el2mpt2csbcl 7137* | If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. (Contributed by AV, 21-May-2021.) |
⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷)))) | ||
Theorem | el2mpt2cl 7138* | If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. Using implicit substitution. (Contributed by AV, 21-May-2021.) |
⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸)) & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐶 = 𝐹 ∧ 𝐷 = 𝐺)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺)))) | ||
Theorem | fnmpt2ovd 7139* | A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.) |
⊢ (𝜑 → 𝑀 Fn (𝐴 × 𝐵)) & ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ⇒ ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷)) | ||
Theorem | offval22 7140* | The function operation expressed as a mapping, variation of offval2 6812. (Contributed by SO, 15-Jul-2018.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑅𝐷))) | ||
Theorem | brovpreldm 7141 | If a binary relation holds for the result of an operation, the operands are in the domain of the operation. (Contributed by AV, 31-Dec-2020.) |
⊢ (𝐷(𝐵𝐴𝐶)𝐸 → 〈𝐵, 𝐶〉 ∈ dom 𝐴) | ||
Theorem | bropopvvv 7142* | If a binary relation holds for the result of an operation which is a result of an operation, the involved classes are sets. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Proof shortened by AV, 3-Jan-2021.) |
⊢ 𝑂 = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {〈𝑓, 𝑝〉 ∣ 𝜑})) & ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜑 ↔ 𝜓)) & ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉𝑂𝐸)𝐵) = {〈𝑓, 𝑝〉 ∣ 𝜃}) ⇒ ⊢ (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) | ||
Theorem | bropfvvvvlem 7143* | Lemma for bropfvvvv 7144. (Contributed by AV, 31-Dec-2020.) (Revised by AV, 16-Jan-2021.) |
⊢ 𝑂 = (𝑎 ∈ 𝑈 ↦ (𝑏 ∈ 𝑉, 𝑐 ∈ 𝑊 ↦ {〈𝑑, 𝑒〉 ∣ 𝜑})) & ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (𝐵(𝑂‘𝐴)𝐶) = {〈𝑑, 𝑒〉 ∣ 𝜃}) ⇒ ⊢ ((〈𝐵, 𝐶〉 ∈ (𝑆 × 𝑇) ∧ 𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸) → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))) | ||
Theorem | bropfvvvv 7144* | If a binary relation holds for the result of an operation which is a function value, the involved classes are sets. (Contributed by AV, 31-Dec-2020.) (Revised by AV, 16-Jan-2021.) |
⊢ 𝑂 = (𝑎 ∈ 𝑈 ↦ (𝑏 ∈ 𝑉, 𝑐 ∈ 𝑊 ↦ {〈𝑑, 𝑒〉 ∣ 𝜑})) & ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (𝐵(𝑂‘𝐴)𝐶) = {〈𝑑, 𝑒〉 ∣ 𝜃}) & ⊢ (𝑎 = 𝐴 → 𝑉 = 𝑆) & ⊢ (𝑎 = 𝐴 → 𝑊 = 𝑇) & ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))) | ||
Theorem | ovmptss 7145* | If all the values of the mapping are subsets of a class 𝑋, then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋) | ||
Theorem | relmpt2opab 7146* | Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈𝑧, 𝑤〉 ∣ 𝜑}) ⇒ ⊢ Rel (𝐶𝐹𝐷) | ||
Theorem | fmpt2co 7147* | Composition of two functions. Variation of fmptco 6303 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ 𝐶) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅)) & ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐶 ↦ 𝑆)) & ⊢ (𝑧 = 𝑅 → 𝑆 = 𝑇) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇)) | ||
Theorem | oprabco 7148* | Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷) & ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶)) ⇒ ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹)) | ||
Theorem | oprab2co 7149* | Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑅) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 〈𝐶, 𝐷〉) & ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷)) ⇒ ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹)) | ||
Theorem | df1st2 7150* | An alternate possible definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} = (1st ↾ (V × V)) | ||
Theorem | df2nd2 7151* | An alternate possible definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V)) | ||
Theorem | 1stconst 7152 | The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.) |
⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴) | ||
Theorem | 2ndconst 7153 | The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.) |
⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵) | ||
Theorem | dfmpt2 7154* | Alternate definition for the "maps to" notation df-mpt2 6554 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐶 ∈ V ⇒ ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} | ||
Theorem | mpt2sn 7155* | An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.) |
⊢ 𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶) & ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) & ⊢ (𝑦 = 𝐵 → 𝐷 = 𝐸) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → 𝐹 = {〈〈𝐴, 𝐵〉, 𝐸〉}) | ||
Theorem | curry1 7156* | Composition with ◡(2nd ↾ ({𝐶} × V)) turns any binary operation 𝐹 with a constant first operand into a function 𝐺 of the second operand only. This transformation is called "currying." (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.) |
⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ⇒ ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))) | ||
Theorem | curry1val 7157 | The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ⇒ ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷)) | ||
Theorem | curry1f 7158 | Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.) |
⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ⇒ ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺:𝐵⟶𝐷) | ||
Theorem | curry2 7159* | Composition with ◡(1st ↾ (V × {𝐶})) turns any binary operation 𝐹 with a constant second operand into a function 𝐺 of the first operand only. This transformation is called "currying." (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.) |
⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) ⇒ ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))) | ||
Theorem | curry2f 7160 | Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.) |
⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) ⇒ ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → 𝐺:𝐴⟶𝐷) | ||
Theorem | curry2val 7161 | The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.) |
⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) ⇒ ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = (𝐷𝐹𝐶)) | ||
Theorem | cnvf1olem 7162 | Lemma for cnvf1o 7163. (Contributed by Mario Carneiro, 27-Apr-2014.) |
⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ◡{𝐵})) → (𝐶 ∈ ◡𝐴 ∧ 𝐵 = ∪ ◡{𝐶})) | ||
Theorem | cnvf1o 7163* | Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.) |
⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) | ||
Theorem | fparlem1 7164 | Lemma for fpar 7168. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (◡(1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V) | ||
Theorem | fparlem2 7165 | Lemma for fpar 7168. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (◡(2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦}) | ||
Theorem | fparlem3 7166* | Lemma for fpar 7168. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐹 Fn 𝐴 → (◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V))) | ||
Theorem | fparlem4 7167* | Lemma for fpar 7168. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐺 Fn 𝐵 → (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = ∪ 𝑦 ∈ 𝐵 ((V × {𝑦}) × (V × {(𝐺‘𝑦)}))) | ||
Theorem | fpar 7168* | Merge two functions in parallel. Use as the second argument of a composition with a (2-place) operation to build compound operations such as 𝑧 = ((√‘𝑥) + (abs‘𝑦)). (Contributed by NM, 17-Sep-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
⊢ 𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V))))) ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉)) | ||
Theorem | fsplit 7169 | A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 7168 in order to build compound functions such as 𝑦 = ((√‘𝑥) + (abs‘𝑥)). (Contributed by NM, 17-Sep-2007.) |
⊢ ◡(1st ↾ I ) = (𝑥 ∈ V ↦ 〈𝑥, 𝑥〉) | ||
Theorem | f2ndf 7170 | The 2nd (second member of an ordered pair) function restricted to a function 𝐹 is a function of 𝐹 into the codomain of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵) | ||
Theorem | fo2ndf 7171 | The 2nd (second member of an ordered pair) function restricted to a function 𝐹 is a function of 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹) | ||
Theorem | f1o2ndf1 7172 | The 2nd (second member of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function of 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹) | ||
Theorem | algrflem 7173 | Lemma for algrf 15124 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵) | ||
Theorem | frxp 7174* | A lexicographical ordering of two well-founded classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) (Proof shortened by Wolf Lammen, 4-Oct-2014.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))} ⇒ ⊢ ((𝑅 Fr 𝐴 ∧ 𝑆 Fr 𝐵) → 𝑇 Fr (𝐴 × 𝐵)) | ||
Theorem | xporderlem 7175* | Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))} ⇒ ⊢ (〈𝑎, 𝑏〉𝑇〈𝑐, 𝑑〉 ↔ (((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑)))) | ||
Theorem | poxp 7176* | A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))} ⇒ ⊢ ((𝑅 Po 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐴 × 𝐵)) | ||
Theorem | soxp 7177* | A lexicographical ordering of two strictly ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))} ⇒ ⊢ ((𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐴 × 𝐵)) | ||
Theorem | wexp 7178* | A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))} ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵)) | ||
Theorem | fnwelem 7179* | Lemma for fnwe 7180. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))} & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝑅 We 𝐵) & ⊢ (𝜑 → 𝑆 We 𝐴) & ⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V) & ⊢ 𝑄 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd ‘𝑢)𝑆(2nd ‘𝑣))))} & ⊢ 𝐺 = (𝑧 ∈ 𝐴 ↦ 〈(𝐹‘𝑧), 𝑧〉) ⇒ ⊢ (𝜑 → 𝑇 We 𝐴) | ||
Theorem | fnwe 7180* | A variant on lexicographic order, which sorts first by some function of the base set, and then by a "backup" well-ordering when the function value is equal on both elements. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))} & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝑅 We 𝐵) & ⊢ (𝜑 → 𝑆 We 𝐴) & ⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V) ⇒ ⊢ (𝜑 → 𝑇 We 𝐴) | ||
Theorem | fnse 7181* | Condition for the well-order in fnwe 7180 to be set-like. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))} & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝑅 Se 𝐵) & ⊢ (𝜑 → (◡𝐹 “ 𝑤) ∈ V) ⇒ ⊢ (𝜑 → 𝑇 Se 𝐴) | ||
In this section, the support of functions is defined and corresponding theorems are provided. Since basic properties (see suppval 7184) are based on the Axiom of Union (usage of dmexg 6989), these definition and theorems cannot be provided earlier. Until April 2019, the support of a function was represented by the expression (◡𝑅 “ (V ∖ {𝑍})) (see suppimacnv 7193). The theorems which are based on this representation and which are provided in previous sections could be moved into this section to have all related theorems in one section, although they do not depend on the Axiom of Union. This was possible because they are not used before. The current theorems differ from the original ones by requiring that the classes representing the "function" (or its "domain") and the "zero element" are sets. Actually, this does not cause any problem (until now). | ||
Syntax | csupp 7182 | Extend class definition to include the support of functions. |
class supp | ||
Definition | df-supp 7183* | Define the support of a function against a "zero" value. According to Wikipedia ("Support (mathematics)", 31-Mar-2019, https://en.wikipedia.org/wiki/Support_(mathematics)) "In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero." and "The notion of support also extends in a natural way to functions taking values in more general sets than R [the real numbers] and to other objects.". The following definition allows for such extensions, being applicable for any sets (which usually are functions) and any element (even not necessarily from the range of the function) regarded as "zero". (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.) |
⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}) | ||
Theorem | suppval 7184* | The value of the operation constructing the support of a function. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}}) | ||
Theorem | supp0prc 7185 | The support of a class is empty if either the class or the "zero" is a proper class. . (Contributed by AV, 28-May-2019.) |
⊢ (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅) | ||
Theorem | suppvalbr 7186* | The value of the operation constructing the support of a function expressed by binary relations. (Contributed by AV, 7-Apr-2019.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))}) | ||
Theorem | supp0 7187 | The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.) |
⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = ∅) | ||
Theorem | suppval1 7188* | The value of the operation constructing the support of a function. (Contributed by AV, 6-Apr-2019.) |
⊢ ((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋‘𝑖) ≠ 𝑍}) | ||
Theorem | suppvalfn 7189* | The value of the operation constructing the support of a function with a given domain. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 22-Apr-2019.) |
⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍}) | ||
Theorem | elsuppfn 7190 | An element of the support of a function with a given domain. (Contributed by AV, 27-May-2019.) |
⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆 ∈ 𝑋 ∧ (𝐹‘𝑆) ≠ 𝑍))) | ||
Theorem | cnvimadfsn 7191* | The support of functions "defined" by inverse images expressed by binary relations. (Contributed by AV, 7-Apr-2019.) |
⊢ (◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)} | ||
Theorem | suppimacnvss 7192 | The support of functions "defined" by inverse images is a subset of the support defined by df-supp 7183. (Contributed by AV, 7-Apr-2019.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (◡𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍)) | ||
Theorem | suppimacnv 7193 | Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = (◡𝑅 “ (V ∖ {𝑍}))) | ||
Theorem | frnsuppeq 7194 | Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.) |
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍})))) | ||
Theorem | suppssdm 7195 | The support of a function is a subset of the function's domain. (Contributed by AV, 30-May-2019.) |
⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹 | ||
Theorem | suppsnop 7196 | The support of a singleton of an ordered pair. (Contributed by AV, 12-Apr-2019.) |
⊢ 𝐹 = {〈𝑋, 𝑌〉} ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → (𝐹 supp 𝑍) = if(𝑌 = 𝑍, ∅, {𝑋})) | ||
Theorem | snopsuppss 7197 | The support of a singleton containing an ordered pair is a subset of the singleton containing the first element of the ordered pair, i.e. it is empty or the singleton itself. (Contributed by AV, 19-Jul-2019.) |
⊢ ({〈𝑋, 𝑌〉} supp 𝑍) ⊆ {𝑋} | ||
Theorem | fvn0elsupp 7198 | If the function value for a given argument is not empty, the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 2-Jul-2019.) (Revised by AV, 4-Apr-2020.) |
⊢ (((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺‘𝑋) ≠ ∅)) → 𝑋 ∈ (𝐺 supp ∅)) | ||
Theorem | fvn0elsuppb 7199 | The function value for a given argument is not empty iff the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 4-Apr-2020.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝐺 Fn 𝐵) → ((𝐺‘𝑋) ≠ ∅ ↔ 𝑋 ∈ (𝐺 supp ∅))) | ||
Theorem | rexsupp 7200* | Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by AV, 27-May-2019.) |
⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (∃𝑥 ∈ (𝐹 supp 𝑍)𝜑 ↔ ∃𝑥 ∈ 𝑋 ((𝐹‘𝑥) ≠ 𝑍 ∧ 𝜑))) |
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