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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | fvn0elsupp 8201 | 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 8202 | 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 8203* | 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))) | ||
| Theorem | ressuppss 8204 | The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.) |
| ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐹 ↾ 𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍)) | ||
| Theorem | suppun 8205 | The support of a class/function is a subset of the support of the union of this class/function with another class/function. (Contributed by AV, 4-Jun-2019.) |
| ⊢ (𝜑 → 𝐺 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍)) | ||
| Theorem | ressuppssdif 8206 | The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.) |
| ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵))) | ||
| Theorem | mptsuppdifd 8207* | The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})}) | ||
| Theorem | mptsuppd 8208* | The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019.) (Revised by AV, 28-May-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝑍}) | ||
| Theorem | extmptsuppeq 8209* | The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 30-Jun-2019.) |
| ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐵 ∖ 𝐴)) → 𝑋 = 𝑍) ⇒ ⊢ (𝜑 → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍)) | ||
| Theorem | suppfnss 8210* | The support of a function which has the same zero values (in its domain) as another function is a subset of the support of this other function. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 6-Jun-2022.) |
| ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → (∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))) | ||
| Theorem | funsssuppss 8211 | The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.) |
| ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)) | ||
| Theorem | fnsuppres 8212 | Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 28-May-2019.) |
| ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ (𝐹 ↾ 𝐵) = (𝐵 × {𝑍}))) | ||
| Theorem | fnsuppeq0 8213 | The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 28-May-2019.) |
| ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍}))) | ||
| Theorem | fczsupp0 8214 | The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.) |
| ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅ | ||
| Theorem | suppss 8215* | Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) (Proof shortened by SN, 5-Aug-2024.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍) ⇒ ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) | ||
| Theorem | suppssr 8216 | A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍) | ||
| Theorem | suppssrg 8217 | A function is zero outside its support. Version of suppssr 8216 avoiding ax-rep 5277 by assuming 𝐹 is a set rather than its domain 𝐴. (Contributed by SN, 5-May-2024.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍) | ||
| Theorem | suppssov1 8218* | Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.) (Proof shortened by SN, 11-Apr-2025.) |
| ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ 𝐿) & ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅) & ⊢ (𝜑 → 𝑌 ∈ 𝑊) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿) | ||
| Theorem | suppssov2 8219* | Formula building theorem for support restrictions: operator with right annihilator. (Contributed by SN, 11-Apr-2025.) |
| ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵) supp 𝑌) ⊆ 𝐿) & ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑣𝑂𝑌) = 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑊) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿) | ||
| Theorem | suppssof1 8220* | Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.) |
| ⊢ (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿) & ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍) & ⊢ (𝜑 → 𝐴:𝐷⟶𝑉) & ⊢ (𝜑 → 𝐵:𝐷⟶𝑅) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝐴 ∘f 𝑂𝐵) supp 𝑍) ⊆ 𝐿) | ||
| Theorem | suppss2 8221* | Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 22-Mar-2015.) (Revised by AV, 28-May-2019.) |
| ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊) | ||
| Theorem | suppsssn 8222* | Show that the support of a function is a subset of a singleton. (Contributed by AV, 21-Jul-2019.) |
| ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴 ∧ 𝑘 ≠ 𝑊) → 𝐵 = 𝑍) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ {𝑊}) | ||
| Theorem | suppssfv 8223* | Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.) |
| ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ 𝐿) & ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝐴)) supp 𝑍) ⊆ 𝐿) | ||
| Theorem | suppofssd 8224 | Condition for the support of a function operation to be a subset of the union of the supports of the left and right function terms. (Contributed by Steven Nguyen, 28-Aug-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → (𝑍𝑋𝑍) = 𝑍) ⇒ ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) | ||
| Theorem | suppofss1d 8225* | Condition for the support of a function operation to be a subset of the support of the left function term. (Contributed by Thierry Arnoux, 21-Jun-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍𝑋𝑥) = 𝑍) ⇒ ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍)) | ||
| Theorem | suppofss2d 8226* | Condition for the support of a function operation to be a subset of the support of the right function term. (Contributed by Thierry Arnoux, 21-Jun-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝑋𝑍) = 𝑍) ⇒ ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍)) | ||
| Theorem | suppco 8227 | The support of the composition of two functions is the inverse image by the inner function of the support of the outer function. (Contributed by AV, 30-May-2019.) Extract this statement from the proof of supp0cosupp0 8229. (Revised by SN, 15-Sep-2023.) |
| ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))) | ||
| Theorem | suppcoss 8228 | The support of the composition of two functions is a subset of the support of the inner function if the outer function preserves zero. Compare suppssfv 8223, which has a sethood condition on 𝐴 instead of 𝐵. (Contributed by SN, 25-May-2024.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍) ⇒ ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌)) | ||
| Theorem | supp0cosupp0 8229 | The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.) |
| ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)) | ||
| Theorem | imacosupp 8230 | The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.) |
| ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))) | ||
The following theorems are about maps-to operations (see df-mpo 7434) where the domain of the second argument depends on the domain of the first argument, especially when the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpox" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpox 7524, ovmpox 7583 and fmpox 8088). If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpoxop", and the maps-to operations are called "x-op maps-to operations" for short. | ||
| Theorem | opeliunxp2f 8231* | Membership in a union of Cartesian products, using bound-variable hypothesis for 𝐸 instead of distinct variable conditions as in opeliunxp2 5847. (Contributed by AV, 25-Oct-2020.) |
| ⊢ Ⅎ𝑥𝐸 & ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸) ⇒ ⊢ (〈𝐶, 𝐷〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)) | ||
| Theorem | mpoxeldm 8232* | If there is an element of the value of an operation given by a maps-to rule, then the first argument is an element of the first component of the domain and the second argument is an element of the second component of the domain depending on the first argument. (Contributed by AV, 25-Oct-2020.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) ⇒ ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷)) | ||
| Theorem | mpoxneldm 8233* | If the first argument of an operation given by a maps-to rule is not an element of the first component of the domain or the second argument is not an element of the second component of the domain depending on the first argument, then the value of the operation is the empty set. (Contributed by AV, 25-Oct-2020.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) ⇒ ⊢ ((𝑋 ∉ 𝐶 ∨ 𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷) → (𝑋𝐹𝑌) = ∅) | ||
| Theorem | mpoxopn0yelv 8234* | If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 𝐾 ∈ 𝑉)) | ||
| Theorem | mpoxopynvov0g 8235* | If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) ⇒ ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∉ 𝑉) → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) | ||
| Theorem | mpoxopxnop0 8236* | If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) ⇒ ⊢ (¬ 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅) | ||
| Theorem | mpoxopx0ov0 8237* | If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is the empty set, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) ⇒ ⊢ (∅𝐹𝐾) = ∅ | ||
| Theorem | mpoxopxprcov0 8238* | If the components of the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, are not sets, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) ⇒ ⊢ (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) | ||
| Theorem | mpoxopynvov0 8239* | If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) ⇒ ⊢ (𝐾 ∉ 𝑉 → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) | ||
| Theorem | mpoxopoveq 8240* | Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}) ⇒ ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) | ||
| Theorem | mpoxopovel 8241* | Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))) | ||
| Theorem | mpoxopoveqd 8242* | Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}) & ⊢ (𝜓 → (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) & ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑} = ∅) ⇒ ⊢ (𝜓 → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) | ||
| Theorem | brovex 8243* | A binary relation of the value of an operation given by the maps-to notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.) |
| ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) & ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Rel (𝑉𝑂𝐸)) ⇒ ⊢ (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) | ||
| Theorem | brovmpoex 8244* | A binary relation of the value of an operation given by the maps-to notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.) |
| ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {〈𝑧, 𝑤〉 ∣ 𝜑}) ⇒ ⊢ (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) | ||
| Theorem | sprmpod 8245* | The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 20-Jun-2019.) |
| ⊢ 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒)}) & ⊢ ((𝜑 ∧ 𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓)) & ⊢ (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V)) & ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃)) & ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜃} ∈ V) ⇒ ⊢ (𝜑 → (𝑉𝑀𝐸) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)}) | ||
| Syntax | ctpos 8246 | The transposition of a function. |
| class tpos 𝐹 | ||
| Definition | df-tpos 8247* | Define the transposition of a function, which is a function 𝐺 = tpos 𝐹 satisfying 𝐺(𝑥, 𝑦) = 𝐹(𝑦, 𝑥). (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) | ||
| Theorem | tposss 8248 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺) | ||
| Theorem | tposeq 8249 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺) | ||
| Theorem | tposeqd 8250 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
| ⊢ (𝜑 → 𝐹 = 𝐺) ⇒ ⊢ (𝜑 → tpos 𝐹 = tpos 𝐺) | ||
| Theorem | tposssxp 8251 | The transposition is a subset of a Cartesian product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) | ||
| Theorem | reltpos 8252 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ Rel tpos 𝐹 | ||
| Theorem | brtpos2 8253 | Value of the transposition at an ordered pair 〈𝐴, 𝐵〉. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))) | ||
| Theorem | brtpos0 8254 | The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). This allows to eliminate sethood hypotheses on 𝐴, 𝐵 in brtpos 8256. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴)) | ||
| Theorem | reldmtpos 8255 | Necessary and sufficient condition for dom tpos 𝐹 to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹) | ||
| Theorem | brtpos 8256 | The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (𝐶 ∈ 𝑉 → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) | ||
| Theorem | ottpos 8257 | The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ (𝐶 ∈ 𝑉 → (〈𝐴, 𝐵, 𝐶〉 ∈ tpos 𝐹 ↔ 〈𝐵, 𝐴, 𝐶〉 ∈ 𝐹)) | ||
| Theorem | relbrtpos 8258 | The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015.) |
| ⊢ (Rel 𝐹 → (〈𝐴, 𝐵〉tpos 𝐹𝐶 ↔ 〈𝐵, 𝐴〉𝐹𝐶)) | ||
| Theorem | dmtpos 8259 | The domain of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) | ||
| Theorem | rntpos 8260 | The range of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹) | ||
| Theorem | tposexg 8261 | The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V) | ||
| Theorem | ovtpos 8262 | The transposition swaps the arguments in a two-argument function. When 𝐹 is a matrix, which is to say a function from (1...𝑚) × (1...𝑛) to ℝ or some ring, tpos 𝐹 is the transposition of 𝐹, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴) | ||
| Theorem | tposfun 8263 | The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (Fun 𝐹 → Fun tpos 𝐹) | ||
| Theorem | dftpos2 8264* | Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))) | ||
| Theorem | dftpos3 8265* | Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 5691. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (Rel dom 𝐹 → tpos 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 〈𝑦, 𝑥〉𝐹𝑧}) | ||
| Theorem | dftpos4 8266* | Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) | ||
| Theorem | tpostpos 8267 | Value of the double transposition for a general class 𝐹. (Contributed by Mario Carneiro, 16-Sep-2015.) |
| ⊢ tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V)) | ||
| Theorem | tpostpos2 8268 | Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.) |
| ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹) | ||
| Theorem | tposfn2 8269 | The domain of a transposition. (Contributed by NM, 10-Sep-2015.) |
| ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴)) | ||
| Theorem | tposfo2 8270 | Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
| ⊢ (Rel 𝐴 → (𝐹:𝐴–onto→𝐵 → tpos 𝐹:◡𝐴–onto→𝐵)) | ||
| Theorem | tposf2 8271 | The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.) |
| ⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:◡𝐴⟶𝐵)) | ||
| Theorem | tposf12 8272 | Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.) |
| ⊢ (Rel 𝐴 → (𝐹:𝐴–1-1→𝐵 → tpos 𝐹:◡𝐴–1-1→𝐵)) | ||
| Theorem | tposf1o2 8273 | Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.) |
| ⊢ (Rel 𝐴 → (𝐹:𝐴–1-1-onto→𝐵 → tpos 𝐹:◡𝐴–1-1-onto→𝐵)) | ||
| Theorem | tposfo 8274 | The domain and codomain/range of a transposition. (Contributed by NM, 10-Sep-2015.) |
| ⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 → tpos 𝐹:(𝐵 × 𝐴)–onto→𝐶) | ||
| Theorem | tposf 8275 | The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.) |
| ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → tpos 𝐹:(𝐵 × 𝐴)⟶𝐶) | ||
| Theorem | tposfn 8276 | Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| ⊢ (𝐹 Fn (𝐴 × 𝐵) → tpos 𝐹 Fn (𝐵 × 𝐴)) | ||
| Theorem | tpos0 8277 | Transposition of the empty set. (Contributed by NM, 10-Sep-2015.) |
| ⊢ tpos ∅ = ∅ | ||
| Theorem | tposco 8278 | Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| ⊢ tpos (𝐹 ∘ 𝐺) = (𝐹 ∘ tpos 𝐺) | ||
| Theorem | tpossym 8279* | Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| ⊢ (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))) | ||
| Theorem | tposeqi 8280 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ 𝐹 = 𝐺 ⇒ ⊢ tpos 𝐹 = tpos 𝐺 | ||
| Theorem | tposex 8281 | A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ 𝐹 ∈ V ⇒ ⊢ tpos 𝐹 ∈ V | ||
| Theorem | nftpos 8282 | Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ Ⅎ𝑥𝐹 ⇒ ⊢ Ⅎ𝑥tpos 𝐹 | ||
| Theorem | tposoprab 8283* | Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⇒ ⊢ tpos 𝐹 = {〈〈𝑦, 𝑥〉, 𝑧〉 ∣ 𝜑} | ||
| Theorem | tposmpo 8284* | Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ tpos 𝐹 = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶) | ||
| Theorem | tposconst 8285 | The transposition of a constant operation using the relation representation. (Contributed by SO, 11-Jul-2018.) |
| ⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = ((𝐵 × 𝐴) × {𝐶}) | ||
| Syntax | ccur 8286 | Extend class notation to include the currying function. |
| class curry 𝐴 | ||
| Syntax | cunc 8287 | Extend class notation to include the uncurrying function. |
| class uncurry 𝐴 | ||
| Definition | df-cur 8288* | Define the currying of 𝐹, which splits a function of two arguments into a function of the first argument, producing a function over the second argument. (Contributed by Mario Carneiro, 7-Jan-2017.) |
| ⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐹𝑧}) | ||
| Definition | df-unc 8289* | Define the uncurrying of 𝐹, which takes a function producing functions, and transforms it into a two-argument function. (Contributed by Mario Carneiro, 7-Jan-2017.) |
| ⊢ uncurry 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐹‘𝑥)𝑧} | ||
| Theorem | mpocurryd 8290* | The currying of an operation given in maps-to notation, splitting the operation (function of two arguments) into a function of the first argument, producing a function over the second argument. (Contributed by AV, 27-Oct-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ≠ ∅) ⇒ ⊢ (𝜑 → curry 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶))) | ||
| Theorem | mpocurryvald 8291* | The value of a curried operation given in maps-to notation is a function over the second argument of the original operation. (Contributed by AV, 27-Oct-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ≠ ∅) & ⊢ (𝜑 → 𝑌 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) ⇒ ⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶)) | ||
| Theorem | fvmpocurryd 8292* | The value of the value of a curried operation given in maps-to notation is the operation value of the original operation. (Contributed by AV, 27-Oct-2019.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑌) ⇒ ⊢ (𝜑 → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵)) | ||
| Syntax | cund 8293 | Extend class notation with undefined value function. |
| class Undef | ||
| Definition | df-undef 8294 | Define the undefined value function, whose value at set 𝑠 is guaranteed not to be a member of 𝑠 (see pwuninel 8296). (Contributed by NM, 15-Sep-2011.) |
| ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠) | ||
| Theorem | pwuninel2 8295 | Proof of pwuninel 8296 under the assumption that the union of the given class is a set, avoiding ax-pr 5430 and ax-un 7751. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
| ⊢ (∪ 𝐴 ∈ 𝑉 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴) | ||
| Theorem | pwuninel 8296 | The powerclass of the union of a class does not belong to that class. This theorem provides a way of constructing a new set that does not belong to a given set. See also pwuninel2 8295. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
| ⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴 | ||
| Theorem | undefval 8297 | Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 8299 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.) |
| ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆) | ||
| Theorem | undefnel2 8298 | The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.) |
| ⊢ (𝑆 ∈ 𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆) | ||
| Theorem | undefnel 8299 | The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.) |
| ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) ∉ 𝑆) | ||
| Theorem | undefne0 8300 | The undefined value generated from a set is not empty. (Contributed by NM, 3-Sep-2018.) |
| ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) ≠ ∅) | ||
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