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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | ofreq 7701 | Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| ⊢ (𝑅 = 𝑆 → ∘r 𝑅 = ∘r 𝑆) | ||
| Theorem | ofexg 7702 | A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.) |
| ⊢ (𝐴 ∈ 𝑉 → ( ∘f 𝑅 ↾ 𝐴) ∈ V) | ||
| Theorem | nfof 7703 | Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.) |
| ⊢ Ⅎ𝑥𝑅 ⇒ ⊢ Ⅎ𝑥 ∘f 𝑅 | ||
| Theorem | nfofr 7704 | Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| ⊢ Ⅎ𝑥𝑅 ⇒ ⊢ Ⅎ𝑥 ∘r 𝑅 | ||
| Theorem | ofrfvalg 7705* | Value of a relation applied to two functions. Originally part of ofrfval 7707, this version assumes the functions are sets rather than their domains, avoiding ax-rep 5279. (Contributed by SN, 5-Aug-2024.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝐴 ∩ 𝐵) = 𝑆 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) ⇒ ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷)) | ||
| Theorem | offval 7706* | Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝐴 ∩ 𝐵) = 𝑆 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷))) | ||
| Theorem | ofrfval 7707* | Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝐴 ∩ 𝐵) = 𝑆 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) ⇒ ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷)) | ||
| Theorem | ofval 7708 | Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝐴 ∩ 𝐵) = 𝑆 & ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) & ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷)) | ||
| Theorem | ofrval 7709 | Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝐴 ∩ 𝐵) = 𝑆 & ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) & ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷) | ||
| Theorem | offn 7710 | The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝐴 ∩ 𝐵) = 𝑆 ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝑆) | ||
| Theorem | offun 7711 | The function operation produces a function. (Contributed by SN, 23-Jul-2024.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → Fun (𝐹 ∘f 𝑅𝐺)) | ||
| Theorem | offval2f 7712* | The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶)) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) | ||
| Theorem | ofmresval 7713 | Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝐴) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹( ∘f 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹 ∘f 𝑅𝐺)) | ||
| Theorem | fnfvof 7714 | Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.) |
| ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) | ||
| Theorem | off 7715* | The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.) |
| ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝐴 ∩ 𝐵) = 𝐶 ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈) | ||
| Theorem | ofres 7716 | Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝐴 ∩ 𝐵) = 𝐶 ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘f 𝑅(𝐺 ↾ 𝐶))) | ||
| Theorem | offval2 7717* | The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶)) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) | ||
| Theorem | ofrfval2 7718* | The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶)) ⇒ ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝐴 𝐵𝑅𝐶)) | ||
| Theorem | offvalfv 7719* | The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐴) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) | ||
| Theorem | ofmpteq 7720* | Value of a pointwise operation on two functions defined using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∘f 𝑅(𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) | ||
| Theorem | coof 7721* | The composition of a homomorphism with a function operation. (Contributed by SN, 20-May-2025.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐻 Fn 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑏𝑅𝑐) ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐))) ⇒ ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = ((𝐻 ∘ 𝐹) ∘f 𝑆(𝐻 ∘ 𝐺))) | ||
| Theorem | ofco 7722 | The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐵) & ⊢ (𝜑 → 𝐻:𝐷⟶𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐷 ∈ 𝑋) & ⊢ (𝐴 ∩ 𝐵) = 𝐶 ⇒ ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻))) | ||
| Theorem | offveq 7723* | Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐴) & ⊢ (𝜑 → 𝐻 Fn 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥)) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = 𝐻) | ||
| Theorem | offveqb 7724* | Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐴) & ⊢ (𝜑 → 𝐻 Fn 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶) ⇒ ⊢ (𝜑 → (𝐻 = (𝐹 ∘f 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶))) | ||
| Theorem | ofc1 7725 | Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (((𝐴 × {𝐵}) ∘f 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶)) | ||
| Theorem | ofc2 7726 | Right operation by a constant. (Contributed by NM, 7-Oct-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵)) | ||
| Theorem | ofc12 7727 | Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) ⇒ ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)})) | ||
| Theorem | caofref 7728* | Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥𝑅𝑥) ⇒ ⊢ (𝜑 → 𝐹 ∘r 𝑅𝐹) | ||
| Theorem | caofinvl 7729* | Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑁:𝑆⟶𝑆) & ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑁‘𝑥)𝑅𝑥) = 𝐵) ⇒ ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝐴 × {𝐵})) | ||
| Theorem | caofid0l 7730* | Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝑥) ⇒ ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = 𝐹) | ||
| Theorem | caofid0r 7731* | Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝑥) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐵})) = 𝐹) | ||
| Theorem | caofid1 7732* | Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶})) | ||
| Theorem | caofid2 7733* | Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝐶) ⇒ ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅𝐹) = (𝐴 × {𝐶})) | ||
| Theorem | caofcom 7734* | Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥)) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐺 ∘f 𝑅𝐹)) | ||
| Theorem | caofidlcan 7735* | Transfer a cancellation/identity law to the function operation. (Contributed by SN, 16-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑥𝑅𝑦) = 𝑦 ↔ 𝑥 = 0 )) ⇒ ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) = 𝐺 ↔ 𝐹 = (𝐴 × { 0 }))) | ||
| Theorem | caofrss 7736* | Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦 → 𝑥𝑇𝑦)) ⇒ ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 → 𝐹 ∘r 𝑇𝐺)) | ||
| Theorem | caofass 7737* | Transfer an associative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐻:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧))) ⇒ ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘f 𝑇𝐻) = (𝐹 ∘f 𝑂(𝐺 ∘f 𝑃𝐻))) | ||
| Theorem | caoftrn 7738* | Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐻:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧)) ⇒ ⊢ (𝜑 → ((𝐹 ∘r 𝑅𝐺 ∧ 𝐺 ∘r 𝑇𝐻) → 𝐹 ∘r 𝑈𝐻)) | ||
| Theorem | caofdi 7739* | Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐾) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐻:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧))) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑇(𝐺 ∘f 𝑅𝐻)) = ((𝐹 ∘f 𝑇𝐺) ∘f 𝑂(𝐹 ∘f 𝑇𝐻))) | ||
| Theorem | caofdir 7740* | Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐾) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐻:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧))) ⇒ ⊢ (𝜑 → ((𝐺 ∘f 𝑅𝐻) ∘f 𝑇𝐹) = ((𝐺 ∘f 𝑇𝐹) ∘f 𝑂(𝐻 ∘f 𝑇𝐹))) | ||
| Theorem | caonncan 7741* | Transfer nncan 11538-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
| ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐴:𝐼⟶𝑆) & ⊢ (𝜑 → 𝐵:𝐼⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦) ⇒ ⊢ (𝜑 → (𝐴 ∘f 𝑀(𝐴 ∘f 𝑀𝐵)) = 𝐵) | ||
| Syntax | crpss 7742 | Extend class notation to include the reified proper subset relation. |
| class [⊊] | ||
| Definition | df-rpss 7743* | Define a relation which corresponds to proper subsethood df-pss 3971 on sets. This allows to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss 7748. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
| ⊢ [⊊] = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊊ 𝑦} | ||
| Theorem | relrpss 7744 | The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
| ⊢ Rel [⊊] | ||
| Theorem | brrpssg 7745 | The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
| ⊢ (𝐵 ∈ 𝑉 → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) | ||
| Theorem | brrpss 7746 | The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
| ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵) | ||
| Theorem | porpss 7747 | Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
| ⊢ [⊊] Po 𝐴 | ||
| Theorem | sorpss 7748* | Express strict ordering under proper subsets, i.e. the notion of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
| ⊢ ( [⊊] Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) | ||
| Theorem | sorpssi 7749 | Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
| ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) | ||
| Theorem | sorpssun 7750 | A chain of sets is closed under binary union. (Contributed by Mario Carneiro, 16-May-2015.) |
| ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∪ 𝐶) ∈ 𝐴) | ||
| Theorem | sorpssin 7751 | A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.) |
| ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∩ 𝐶) ∈ 𝐴) | ||
| Theorem | sorpssuni 7752* | In a chain of sets, a maximal element is the union of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
| ⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣 ↔ ∪ 𝑌 ∈ 𝑌)) | ||
| Theorem | sorpssint 7753* | In a chain of sets, a minimal element is the intersection of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
| ⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 ↔ ∩ 𝑌 ∈ 𝑌)) | ||
| Theorem | sorpsscmpl 7754* | The componentwise complement of a chain of sets is also a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
| ⊢ ( [⊊] Or 𝑌 → [⊊] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌}) | ||
| Axiom | ax-un 7755* |
Axiom of Union. An axiom of Zermelo-Fraenkel set theory. It states
that a set 𝑦 exists that includes the union of a
given set 𝑥
i.e. the collection of all members of the members of 𝑥. The
variant axun2 7757 states that the union itself exists. A
version with the
standard abbreviation for union is uniex2 7758. A version using class
notation is uniex 7761.
The union of a class df-uni 4908 should not be confused with the union of two classes df-un 3956. Their relationship is shown in unipr 4924. (Contributed by NM, 23-Dec-1993.) |
| ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | ||
| Theorem | zfun 7756* | Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.) Use ax-un 7755 instead. (New usage is discouraged.) |
| ⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
| Theorem | axun2 7757* | A variant of the Axiom of Union ax-un 7755. For any set 𝑥, there exists a set 𝑦 whose members are exactly the members of the members of 𝑥 i.e. the union of 𝑥. Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
| ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) | ||
| Theorem | uniex2 7758* | The Axiom of Union using the standard abbreviation for union. Given any set 𝑥, its union 𝑦 exists. (Contributed by NM, 4-Jun-2006.) |
| ⊢ ∃𝑦 𝑦 = ∪ 𝑥 | ||
| Theorem | vuniex 7759 | The union of a setvar is a set. (Contributed by BJ, 3-May-2021.) (Revised by BJ, 6-Apr-2024.) |
| ⊢ ∪ 𝑥 ∈ V | ||
| Theorem | uniexg 7760 | The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent 𝐴 ∈ 𝑉 instead of 𝐴 ∈ V to make the theorem more general and thus shorten some proofs; obviously the universal class constant V is one possible substitution for class variable 𝑉. (Contributed by NM, 25-Nov-1994.) |
| ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | ||
| Theorem | uniex 7761 | The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 3494), then the union of 𝐴 is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ∪ 𝐴 ∈ V | ||
| Theorem | uniexd 7762 | Deduction version of the ZF Axiom of Union in class notation. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∪ 𝐴 ∈ V) | ||
| Theorem | unexg 7763 | The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.) Prove unexg 7763 first and then unex 7764 and unexb 7767 from it. (Revised by BJ, 21-Jul-2025.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | ||
| Theorem | unex 7764 | The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∪ 𝐵) ∈ V | ||
| Theorem | unexOLD 7765 | Obsolete version of unex 7764 as of 21-Jul-2025. (Contributed by NM, 1-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∪ 𝐵) ∈ V | ||
| Theorem | tpex 7766 | An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.) |
| ⊢ {𝐴, 𝐵, 𝐶} ∈ V | ||
| Theorem | unexb 7767 | Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.) |
| ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) | ||
| Theorem | unexbOLD 7768 | Obsolete version of unexb 7767 as of 21-Jul-2025. (Contributed by NM, 11-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) | ||
| Theorem | unexgOLD 7769 | Obsolete version of unexg 7763 as of 21-Jul-2025. (Contributed by NM, 18-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | ||
| Theorem | xpexg 7770 | The Cartesian product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. See also xpexgALT 8006. (Contributed by NM, 14-Aug-1994.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) | ||
| Theorem | xpexd 7771 | The Cartesian product of two sets is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V) | ||
| Theorem | 3xpexg 7772 | The Cartesian product of three sets is a set. (Contributed by Alexander van der Vekens, 21-Feb-2018.) |
| ⊢ (𝑉 ∈ 𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V) | ||
| Theorem | xpex 7773 | The Cartesian product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 × 𝐵) ∈ V | ||
| Theorem | unexd 7774 | The union of two sets is a set. (Contributed by SN, 16-Jul-2024.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) | ||
| Theorem | sqxpexg 7775 | The Cartesian square of a set is a set. (Contributed by AV, 13-Jan-2020.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | ||
| Theorem | abnexg 7776* | Sufficient condition for a class abstraction to be a proper class. The class 𝐹 can be thought of as an expression in 𝑥 and the abstraction appearing in the statement as the class of values 𝐹 as 𝑥 varies through 𝐴. Assuming the antecedents, if that class is a set, then so is the "domain" 𝐴. The converse holds without antecedent, see abrexexg 7985. Note that the second antecedent ∀𝑥 ∈ 𝐴𝑥 ∈ 𝐹 cannot be translated to 𝐴 ⊆ 𝐹 since 𝐹 may depend on 𝑥. In applications, one may take 𝐹 = {𝑥} or 𝐹 = 𝒫 𝑥 (see snnex 7778 and pwnex 7779 respectively, proved from abnex 7777, which is a consequence of abnexg 7776 with 𝐴 = V). (Contributed by BJ, 2-Dec-2021.) |
| ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ 𝑊 → 𝐴 ∈ V)) | ||
| Theorem | abnex 7777* | Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 7778 and pwnex 7779. See the comment of abnexg 7776. (Contributed by BJ, 2-May-2021.) |
| ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V) | ||
| Theorem | snnex 7778* | The class of all singletons is a proper class. See also pwnex 7779. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) (Proof shortened by BJ, 5-Dec-2021.) |
| ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V | ||
| Theorem | pwnex 7779* | The class of all power sets is a proper class. See also snnex 7778. (Contributed by BJ, 2-May-2021.) |
| ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V | ||
| Theorem | difex2 7780 | If the subtrahend of a class difference exists, then the minuend exists iff the difference exists. (Contributed by NM, 12-Nov-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
| ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐵) ∈ V)) | ||
| Theorem | difsnexi 7781 | If the difference of a class and a singleton is a set, the class itself is a set. (Contributed by AV, 15-Jan-2019.) |
| ⊢ ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V) | ||
| Theorem | uniuni 7782* | Expression for double union that moves union into a class abstraction. (Contributed by FL, 28-May-2007.) |
| ⊢ ∪ ∪ 𝐴 = ∪ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)} | ||
| Theorem | uniexr 7783 | Converse of the Axiom of Union. Note that it does not require ax-un 7755. (Contributed by NM, 11-Nov-2003.) |
| ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V) | ||
| Theorem | uniexb 7784 | The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.) |
| ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | ||
| Theorem | pwexr 7785 | Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5365. (Contributed by NM, 11-Nov-2003.) |
| ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V) | ||
| Theorem | pwexb 7786 | The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.) |
| ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | ||
| Theorem | elpwpwel 7787 | A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.) |
| ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵) | ||
| Theorem | eldifpw 7788 | Membership in a power class difference. (Contributed by NM, 25-Mar-2007.) |
| ⊢ 𝐶 ∈ V ⇒ ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵)) | ||
| Theorem | elpwun 7789 | Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.) |
| ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵) | ||
| Theorem | pwuncl 7790 | Power classes are closed under union. (Contributed by AV, 27-Feb-2024.) |
| ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ 𝒫 𝑋) | ||
| Theorem | iunpw 7791* | An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 ↔ 𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥) | ||
| Theorem | fr3nr 7792 | A well-founded relation has no 3-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 10-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
| ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵)) | ||
| Theorem | epne3 7793 | A well-founded class contains no 3-cycle loops. (Contributed by NM, 19-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
| ⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐵)) | ||
| Theorem | dfwe2 7794* | Alternate definition of well-ordering. Definition 6.24(2) of [TakeutiZaring] p. 30. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
| ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | ||
| Theorem | epweon 7795 | The membership relation well-orders the class of ordinal numbers. This proof does not require the axiom of regularity. Proposition 4.8(g) of [Mendelson] p. 244. For a shorter proof requiring ax-un 7755, see epweonALT 7796. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7755. (Revised by BTernaryTau, 30-Nov-2024.) |
| ⊢ E We On | ||
| Theorem | epweonALT 7796 | Alternate proof of epweon 7795, shorter but requiring ax-un 7755. (Contributed by NM, 1-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ E We On | ||
| Theorem | ordon 7797 | The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.) |
| ⊢ Ord On | ||
| Theorem | onprc 7798 | No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 7797), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.) |
| ⊢ ¬ On ∈ V | ||
| Theorem | ssorduni 7799 | The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. Lemma 2.7 of [Schloeder] p. 4. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
| ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | ||
| Theorem | ssonuni 7800 | The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. Lemma 2.7 of [Schloeder] p. 4. (Contributed by NM, 1-Nov-2003.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On)) | ||
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