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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ectocld 7701* | Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ 𝑆 = (𝐵 / 𝑅) & ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → 𝜑) ⇒ ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓) | ||
Theorem | ectocl 7702* | Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ 𝑆 = (𝐵 / 𝑅) & ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 ∈ 𝐵 → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝑆 → 𝜓) | ||
Theorem | elqsn0 7703 | A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.) |
⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅) | ||
Theorem | ecelqsdm 7704 | Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.) |
⊢ ((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → 𝐵 ∈ 𝐴) | ||
Theorem | xpider 7705 | A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝐴 × 𝐴) Er 𝐴 | ||
Theorem | iiner 7706* | The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵) | ||
Theorem | riiner 7707* | The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ (∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵) | ||
Theorem | erinxp 7708 | A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵) | ||
Theorem | ecinxp 7709 | Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.) |
⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴))) | ||
Theorem | qsinxp 7710 | Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝐴 / 𝑅) = (𝐴 / (𝑅 ∩ (𝐴 × 𝐴)))) | ||
Theorem | qsdisj 7711 | Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐵 ∈ (𝐴 / 𝑅)) & ⊢ (𝜑 → 𝐶 ∈ (𝐴 / 𝑅)) ⇒ ⊢ (𝜑 → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅)) | ||
Theorem | qsdisj2 7712* | A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.) |
⊢ (𝑅 Er 𝑋 → Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥) | ||
Theorem | qsel 7713 | If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ ((𝑅 Er 𝑋 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅) | ||
Theorem | uniinqs 7714 | Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4393. (Contributed by FL, 25-May-2007.) (Proof shortened by Mario Carneiro, 11-Jul-2014.) |
⊢ 𝑅 Er 𝑋 ⇒ ⊢ ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → ∪ (𝐵 ∩ 𝐶) = (∪ 𝐵 ∩ ∪ 𝐶)) | ||
Theorem | qliftlem 7715* | 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅)) | ||
Theorem | qliftrel 7716* | 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) ⇒ ⊢ (𝜑 → 𝐹 ⊆ ((𝑋 / 𝑅) × 𝑌)) | ||
Theorem | qliftel 7717* | Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) ⇒ ⊢ (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶𝑅𝑥 ∧ 𝐷 = 𝐴))) | ||
Theorem | qliftel1 7718* | Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅𝐹𝐴) | ||
Theorem | qliftfun 7719* | The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵))) | ||
Theorem | qliftfund 7720* | The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
Theorem | qliftfuns 7721* | The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) ⇒ ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴))) | ||
Theorem | qliftf 7722* | The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) ⇒ ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:(𝑋 / 𝑅)⟶𝑌)) | ||
Theorem | qliftval 7723* | The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝑋 ∈ V) & ⊢ (𝑥 = 𝐶 → 𝐴 = 𝐵) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵) | ||
Theorem | ecoptocl 7724* | Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.) |
⊢ 𝑆 = ((𝐵 × 𝐶) / 𝑅) & ⊢ ([〈𝑥, 𝑦〉]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝑆 → 𝜓) | ||
Theorem | 2ecoptocl 7725* | Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.) |
⊢ 𝑆 = ((𝐶 × 𝐷) / 𝑅) & ⊢ ([〈𝑥, 𝑦〉]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ ([〈𝑧, 𝑤〉]𝑅 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → 𝜑) ⇒ ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝜒) | ||
Theorem | 3ecoptocl 7726* | Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.) |
⊢ 𝑆 = ((𝐷 × 𝐷) / 𝑅) & ⊢ ([〈𝑥, 𝑦〉]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ ([〈𝑧, 𝑤〉]𝑅 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ ([〈𝑣, 𝑢〉]𝑅 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ (𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐷)) → 𝜑) ⇒ ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 𝜃) | ||
Theorem | brecop 7727* | Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.) |
⊢ ∼ ∈ V & ⊢ ∼ Er (𝐺 × 𝐺) & ⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ ) & ⊢ ≤ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))} & ⊢ ((((𝑧 ∈ 𝐺 ∧ 𝑤 ∈ 𝐺) ∧ (𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺)) ∧ ((𝑣 ∈ 𝐺 ∧ 𝑢 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))) → (([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) → (𝜑 ↔ 𝜓))) ⇒ ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([〈𝐴, 𝐵〉] ∼ ≤ [〈𝐶, 𝐷〉] ∼ ↔ 𝜓)) | ||
Theorem | brecop2 7728 | Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.) |
⊢ ∼ ∈ V & ⊢ dom ∼ = (𝐺 × 𝐺) & ⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ ) & ⊢ 𝑅 ⊆ (𝐻 × 𝐻) & ⊢ ≤ ⊆ (𝐺 × 𝐺) & ⊢ ¬ ∅ ∈ 𝐺 & ⊢ dom + = (𝐺 × 𝐺) & ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([〈𝐴, 𝐵〉] ∼ 𝑅[〈𝐶, 𝐷〉] ∼ ↔ (𝐴 + 𝐷) ≤ (𝐵 + 𝐶))) ⇒ ⊢ ([〈𝐴, 𝐵〉] ∼ 𝑅[〈𝐶, 𝐷〉] ∼ ↔ (𝐴 + 𝐷) ≤ (𝐵 + 𝐶)) | ||
Theorem | eroveu 7729* | Lemma for erov 7731 and eroprf 7732. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ 𝐽 = (𝐴 / 𝑅) & ⊢ 𝐾 = (𝐵 / 𝑆) & ⊢ (𝜑 → 𝑇 ∈ 𝑍) & ⊢ (𝜑 → 𝑅 Er 𝑈) & ⊢ (𝜑 → 𝑆 Er 𝑉) & ⊢ (𝜑 → 𝑇 Er 𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → 𝐵 ⊆ 𝑉) & ⊢ (𝜑 → 𝐶 ⊆ 𝑊) & ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶) & ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) | ||
Theorem | erovlem 7730* | Lemma for erov 7731 and eroprf 7732. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ 𝐽 = (𝐴 / 𝑅) & ⊢ 𝐾 = (𝐵 / 𝑆) & ⊢ (𝜑 → 𝑇 ∈ 𝑍) & ⊢ (𝜑 → 𝑅 Er 𝑈) & ⊢ (𝜑 → 𝑆 Er 𝑉) & ⊢ (𝜑 → 𝑇 Er 𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → 𝐵 ⊆ 𝑉) & ⊢ (𝜑 → 𝐶 ⊆ 𝑊) & ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶) & ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢))) & ⊢ ⨣ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)} ⇒ ⊢ (𝜑 → ⨣ = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))) | ||
Theorem | erov 7731* | The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ 𝐽 = (𝐴 / 𝑅) & ⊢ 𝐾 = (𝐵 / 𝑆) & ⊢ (𝜑 → 𝑇 ∈ 𝑍) & ⊢ (𝜑 → 𝑅 Er 𝑈) & ⊢ (𝜑 → 𝑆 Er 𝑉) & ⊢ (𝜑 → 𝑇 Er 𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → 𝐵 ⊆ 𝑉) & ⊢ (𝜑 → 𝐶 ⊆ 𝑊) & ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶) & ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢))) & ⊢ ⨣ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)} & ⊢ (𝜑 → 𝑅 ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ([𝑃]𝑅 ⨣ [𝑄]𝑆) = [(𝑃 + 𝑄)]𝑇) | ||
Theorem | eroprf 7732* | Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ 𝐽 = (𝐴 / 𝑅) & ⊢ 𝐾 = (𝐵 / 𝑆) & ⊢ (𝜑 → 𝑇 ∈ 𝑍) & ⊢ (𝜑 → 𝑅 Er 𝑈) & ⊢ (𝜑 → 𝑆 Er 𝑉) & ⊢ (𝜑 → 𝑇 Er 𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → 𝐵 ⊆ 𝑉) & ⊢ (𝜑 → 𝐶 ⊆ 𝑊) & ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶) & ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢))) & ⊢ ⨣ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)} & ⊢ (𝜑 → 𝑅 ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) & ⊢ 𝐿 = (𝐶 / 𝑇) ⇒ ⊢ (𝜑 → ⨣ :(𝐽 × 𝐾)⟶𝐿) | ||
Theorem | erov2 7733* | The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐽 = (𝐴 / ∼ ) & ⊢ ⨣ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((𝑥 = [𝑝] ∼ ∧ 𝑦 = [𝑞] ∼ ) ∧ 𝑧 = [(𝑝 + 𝑞)] ∼ )} & ⊢ (𝜑 → ∼ ∈ 𝑋) & ⊢ (𝜑 → ∼ Er 𝑈) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → + :(𝐴 × 𝐴)⟶𝐴) & ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴))) → ((𝑟 ∼ 𝑠 ∧ 𝑡 ∼ 𝑢) → (𝑟 + 𝑡) ∼ (𝑠 + 𝑢))) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ([𝑃] ∼ ⨣ [𝑄] ∼ ) = [(𝑃 + 𝑄)] ∼ ) | ||
Theorem | eroprf2 7734* | Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐽 = (𝐴 / ∼ ) & ⊢ ⨣ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((𝑥 = [𝑝] ∼ ∧ 𝑦 = [𝑞] ∼ ) ∧ 𝑧 = [(𝑝 + 𝑞)] ∼ )} & ⊢ (𝜑 → ∼ ∈ 𝑋) & ⊢ (𝜑 → ∼ Er 𝑈) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → + :(𝐴 × 𝐴)⟶𝐴) & ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴))) → ((𝑟 ∼ 𝑠 ∧ 𝑡 ∼ 𝑢) → (𝑟 + 𝑡) ∼ (𝑠 + 𝑢))) ⇒ ⊢ (𝜑 → ⨣ :(𝐽 × 𝐽)⟶𝐽) | ||
Theorem | ecopoveq 7735* | This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation ∼ (specified by the hypothesis) in terms of its operation 𝐹. (Contributed by NM, 16-Aug-1995.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} ⇒ ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (〈𝐴, 𝐵〉 ∼ 〈𝐶, 𝐷〉 ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))) | ||
Theorem | ecopovsym 7736* | Assuming the operation 𝐹 is commutative, show that the relation ∼, specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} & ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥) ⇒ ⊢ (𝐴 ∼ 𝐵 → 𝐵 ∼ 𝐴) | ||
Theorem | ecopovtrn 7737* | Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation ∼, specified by the first hypothesis, is transitive. (Contributed by NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} & ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥) & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧)) ⇒ ⊢ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → 𝐴 ∼ 𝐶) | ||
Theorem | ecopover 7738* | Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation ∼, specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (Proof shortened by AV, 1-May-2021.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} & ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥) & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧)) ⇒ ⊢ ∼ Er (𝑆 × 𝑆) | ||
Theorem | ecopoverOLD 7739* | Obsolete proof of ecopover 7738 as of 1-May-2021. Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation ∼, specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} & ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥) & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧)) ⇒ ⊢ ∼ Er (𝑆 × 𝑆) | ||
Theorem | eceqoveq 7740* | Equality of equivalence relation in terms of an operation. (Contributed by NM, 15-Feb-1996.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ Er (𝑆 × 𝑆) & ⊢ dom + = (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (〈𝐴, 𝐵〉 ∼ 〈𝐶, 𝐷〉 ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))) ⇒ ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ([〈𝐴, 𝐵〉] ∼ = [〈𝐶, 𝐷〉] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))) | ||
Theorem | ecovcom 7741* | Lemma used to transfer a commutative law via an equivalence relation. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
⊢ 𝐶 = ((𝑆 × 𝑆) / ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = [〈𝐷, 𝐺〉] ∼ ) & ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) = [〈𝐻, 𝐽〉] ∼ ) & ⊢ 𝐷 = 𝐻 & ⊢ 𝐺 = 𝐽 ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
Theorem | ecovass 7742* | Lemma used to transfer an associative law via an equivalence relation. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
⊢ 𝐷 = ((𝑆 × 𝑆) / ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = [〈𝐺, 𝐻〉] ∼ ) & ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑣, 𝑢〉] ∼ ) = [〈𝑁, 𝑄〉] ∼ ) & ⊢ (((𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([〈𝐺, 𝐻〉] ∼ + [〈𝑣, 𝑢〉] ∼ ) = [〈𝐽, 𝐾〉] ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑁 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑁, 𝑄〉] ∼ ) = [〈𝐿, 𝑀〉] ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) & ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑁 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆)) & ⊢ 𝐽 = 𝐿 & ⊢ 𝐾 = 𝑀 ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | ||
Theorem | ecovdi 7743* | Lemma used to transfer a distributive law via an equivalence relation. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
⊢ 𝐷 = ((𝑆 × 𝑆) / ∼ ) & ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑣, 𝑢〉] ∼ ) = [〈𝑀, 𝑁〉] ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ · [〈𝑀, 𝑁〉] ∼ ) = [〈𝐻, 𝐽〉] ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ · [〈𝑧, 𝑤〉] ∼ ) = [〈𝑊, 𝑋〉] ∼ ) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ · [〈𝑣, 𝑢〉] ∼ ) = [〈𝑌, 𝑍〉] ∼ ) & ⊢ (((𝑊 ∈ 𝑆 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ∈ 𝑆 ∧ 𝑍 ∈ 𝑆)) → ([〈𝑊, 𝑋〉] ∼ + [〈𝑌, 𝑍〉] ∼ ) = [〈𝐾, 𝐿〉] ∼ ) & ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑊 ∈ 𝑆 ∧ 𝑋 ∈ 𝑆)) & ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑌 ∈ 𝑆 ∧ 𝑍 ∈ 𝑆)) & ⊢ 𝐻 = 𝐾 & ⊢ 𝐽 = 𝐿 ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) | ||
Syntax | cmap 7744 | Extend the definition of a class to include the mapping operation. (Read for 𝐴 ↑𝑚 𝐵, "the set of all functions that map from 𝐵 to 𝐴.) |
class ↑𝑚 | ||
Syntax | cpm 7745 | Extend the definition of a class to include the partial mapping operation. (Read for 𝐴 ↑pm 𝐵, "the set of all partial functions that map from 𝐵 to 𝐴.) |
class ↑pm | ||
Definition | df-map 7746* | Define the mapping operation or set exponentiation. The set of all functions that map from 𝐵 to 𝐴 is written (𝐴 ↑𝑚 𝐵) (see mapval 7756). Many authors write 𝐴 followed by 𝐵 as a superscript for this operation and rely on context to avoid confusion other exponentiation operations (e.g., Definition 10.42 of [TakeutiZaring] p. 95). Other authors show 𝐵 as a prefixed superscript, which is read "𝐴 pre 𝐵 " (e.g., definition of [Enderton] p. 52). Definition 8.21 of [Eisenberg] p. 125 uses the notation Map(𝐵, 𝐴) for our (𝐴 ↑𝑚 𝐵). The up-arrow is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976). We adopt the first case of his notation (simple exponentiation) and subscript it with m to distinguish it from other kinds of exponentiation. (Contributed by NM, 8-Dec-2003.) |
⊢ ↑𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥}) | ||
Definition | df-pm 7747* | Define the partial mapping operation. A partial function from 𝐵 to 𝐴 is a function from a subset of 𝐵 to 𝐴. The set of all partial functions from 𝐵 to 𝐴 is written (𝐴 ↑pm 𝐵) (see pmvalg 7755). A notation for this operation apparently does not appear in the literature. We use ↑pm to distinguish it from the less general set exponentiation operation ↑𝑚 (df-map 7746) . See mapsspm 7777 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.) |
⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) | ||
Theorem | mapprc 7748* | When 𝐴 is a proper class, the class of all functions mapping 𝐴 to 𝐵 is empty. Exercise 4.41 of [Mendelson] p. 255. (Contributed by NM, 8-Dec-2003.) |
⊢ (¬ 𝐴 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = ∅) | ||
Theorem | pmex 7749* | The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵))} ∈ V) | ||
Theorem | mapex 7750* | The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V) | ||
Theorem | fnmap 7751 | Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ ↑𝑚 Fn (V × V) | ||
Theorem | fnpm 7752 | Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ ↑pm Fn (V × V) | ||
Theorem | reldmmap 7753 | Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.) |
⊢ Rel dom ↑𝑚 | ||
Theorem | mapvalg 7754* | The value of set exponentiation. (𝐴 ↑𝑚 𝐵) is the set of all functions that map from 𝐵 to 𝐴. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑𝑚 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}) | ||
Theorem | pmvalg 7755* | The value of the partial mapping operation. (𝐴 ↑pm 𝐵) is the set of all partial functions that map from 𝐵 to 𝐴. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓}) | ||
Theorem | mapval 7756* | The value of set exponentiation (inference version). (𝐴 ↑𝑚 𝐵) is the set of all functions that map from 𝐵 to 𝐴. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ↑𝑚 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴} | ||
Theorem | elmapg 7757 | Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) | ||
Theorem | elmapd 7758 | Deduction form of elmapg 7757. (Contributed by BJ, 11-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) | ||
Theorem | elpmg 7759 | The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝐶 ∧ 𝐶 ⊆ (𝐵 × 𝐴)))) | ||
Theorem | elpm2g 7760 | The predicate "is a partial function." (Contributed by NM, 31-Dec-2013.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))) | ||
Theorem | elpm2r 7761 | Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐹 ∈ (𝐴 ↑pm 𝐵)) | ||
Theorem | elpmi 7762 | A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.) |
⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) | ||
Theorem | pmfun 7763 | A partial function is a function. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → Fun 𝐹) | ||
Theorem | elmapex 7764 | Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) | ||
Theorem | elmapi 7765 | A mapping is a function, forward direction only with superfluous antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → 𝐴:𝐶⟶𝐵) | ||
Theorem | elmapfn 7766 | A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.) |
⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → 𝐴 Fn 𝐶) | ||
Theorem | elmapfun 7767 | A mapping is always a function. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → Fun 𝐴) | ||
Theorem | elmapssres 7768 | A restricted mapping is a mapping. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ ((𝐴 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑𝑚 𝐷)) | ||
Theorem | fpmg 7769 | A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 ∈ (𝐵 ↑pm 𝐴)) | ||
Theorem | pmss12g 7770 | Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.) |
⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (𝐴 ↑pm 𝐵) ⊆ (𝐶 ↑pm 𝐷)) | ||
Theorem | pmresg 7771 | Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵)) | ||
Theorem | elmap 7772 | Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶𝐴) | ||
Theorem | mapval2 7773* | Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ↑𝑚 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) | ||
Theorem | elpm 7774 | The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (𝐵 × 𝐴))) | ||
Theorem | elpm2 7775 | The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) | ||
Theorem | fpm 7776 | A total function is a partial function. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ∈ (𝐵 ↑pm 𝐴)) | ||
Theorem | mapsspm 7777 | Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ (𝐴 ↑𝑚 𝐵) ⊆ (𝐴 ↑pm 𝐵) | ||
Theorem | pmsspw 7778 | Partial maps are a subset of the power set of the Cartesian product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) | ||
Theorem | mapsspw 7779 | Set exponentiation is a subset of the power set of the Cartesian product of its arguments. (Contributed by NM, 8-Dec-2006.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ↑𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) | ||
Theorem | fvmptmap 7780* | Special case of fvmpt 6191 for operator theorems. (Contributed by NM, 27-Nov-2007.) |
⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V & ⊢ 𝑅 ∈ V & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑𝑚 𝐷) ↦ 𝐵) ⇒ ⊢ (𝐴:𝐷⟶𝑅 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | map0e 7781 | Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = 1𝑜) | ||
Theorem | map0b 7782 | Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ≠ ∅ → (∅ ↑𝑚 𝐴) = ∅) | ||
Theorem | map0g 7783 | Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅))) | ||
Theorem | map0 7784 | Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐴 ↑𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)) | ||
Theorem | mapsn 7785* | The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ↑𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {〈𝐵, 𝑦〉}} | ||
Theorem | mapss 7786 | Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) | ||
Theorem | fdiagfn 7787* | Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) ⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐵⟶(𝐵 ↑𝑚 𝐼)) | ||
Theorem | fvdiagfn 7788* | Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (𝐼 × {𝑋})) | ||
Theorem | mapsnconst 7789 | Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝑆 = {𝑋} & ⊢ 𝐵 ∈ V & ⊢ 𝑋 ∈ V ⇒ ⊢ (𝐹 ∈ (𝐵 ↑𝑚 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) | ||
Theorem | mapsncnv 7790* | Expression for the inverse of the canonical map between a set and its set of singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝑆 = {𝑋} & ⊢ 𝐵 ∈ V & ⊢ 𝑋 ∈ V & ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)) ⇒ ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) | ||
Theorem | mapsnf1o2 7791* | Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝑆 = {𝑋} & ⊢ 𝐵 ∈ V & ⊢ 𝑋 ∈ V & ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)) ⇒ ⊢ 𝐹:(𝐵 ↑𝑚 𝑆)–1-1-onto→𝐵 | ||
Theorem | mapsnf1o3 7792* | Explicit bijection in the reverse of mapsnf1o2 7791. (Contributed by Stefan O'Rear, 24-Mar-2015.) |
⊢ 𝑆 = {𝑋} & ⊢ 𝐵 ∈ V & ⊢ 𝑋 ∈ V & ⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) ⇒ ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑𝑚 𝑆) | ||
Theorem | ralxpmap 7793* | Quantification over functions in terms of quantification over values and punctured functions. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ (𝑓 = (𝑔 ∪ {〈𝐽, 𝑦〉}) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐽 ∈ 𝑇 → (∀𝑓 ∈ (𝑆 ↑𝑚 𝑇)𝜑 ↔ ∀𝑦 ∈ 𝑆 ∀𝑔 ∈ (𝑆 ↑𝑚 (𝑇 ∖ {𝐽}))𝜓)) | ||
Syntax | cixp 7794 | Extend class notation to include infinite Cartesian products. |
class X𝑥 ∈ 𝐴 𝐵 | ||
Definition | df-ixp 7795* | Definition of infinite Cartesian product of [Enderton] p. 54. Enderton uses a bold "X" with 𝑥 ∈ 𝐴 written underneath or as a subscript, as does Stoll p. 47. Some books use a capital pi, but we will reserve that notation for products of numbers. Usually 𝐵 represents a class expression containing 𝑥 free and thus can be thought of as 𝐵(𝑥). Normally, 𝑥 is not free in 𝐴, although this is not a requirement of the definition. (Contributed by NM, 28-Sep-2006.) |
⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | ||
Theorem | dfixp 7796* | Eliminate the expression {𝑥 ∣ 𝑥 ∈ 𝐴} in df-ixp 7795, under the assumption that 𝐴 and 𝑥 are disjoint. This way, we can say that 𝑥 is bound in X𝑥 ∈ 𝐴𝐵 even if it appears free in 𝐴. (Contributed by Mario Carneiro, 12-Aug-2016.) |
⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | ||
Theorem | ixpsnval 7797* | The value of an infinite Cartesian product with a singleton. (Contributed by AV, 3-Dec-2018.) |
⊢ (𝑋 ∈ 𝑉 → X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)}) | ||
Theorem | elixp2 7798* | Membership in an infinite Cartesian product. See df-ixp 7795 for discussion of the notation. (Contributed by NM, 28-Sep-2006.) |
⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | ||
Theorem | fvixp 7799* | Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) ⇒ ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷) | ||
Theorem | ixpfn 7800* | A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.) |
⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴) |
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