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Type | Label | Description |
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Statement | ||
Theorem | xpider 8101 | 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 8102* | 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 8103* | 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 8104 | 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 8105 | 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 8106 | 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 8107 | 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 8108* | A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.) |
⊢ (𝑅 Er 𝑋 → Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥) | ||
Theorem | qsel 8109 | 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 8110 | Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4693. (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 8111* | 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 8112* | 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 8113* | 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 8114* | 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 8115* | 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 8116* | 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 8117* | 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 8118* | 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 8119* | 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 8120* | 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 8121* | 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 8122* | 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 8123* | 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 8124 | Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.) (Revised by AV, 12-Jul-2022.) |
⊢ 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 | brecop2OLD 8125 | Obsolete version of brecop2 8124 as of 12-Jul-2022. (Contributed by NM, 13-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∼ ∈ 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 8126* | Lemma for erov 8128 and eroprf 8129. (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 8127* | Lemma for erov 8128 and eroprf 8129. (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 8128* | 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 8129* | 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 8130* | 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 8131* | 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 8132* | 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 8133* | 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 8134* | 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 8135* | 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 | eceqoveq 8136* | 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 8137* | 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 8138* | 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 8139* | 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 8140 | 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 8141 | 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 8142* | 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 8152). 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 8143* | 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 8151). 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 8142). See mapsspm 8174 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 8144* | 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 8145* | 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 8146* | 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 8147 | 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 8148 | Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ ↑pm Fn (V × V) | ||
Theorem | reldmmap 8149 | Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.) |
⊢ Rel dom ↑𝑚 | ||
Theorem | mapvalg 8150* | 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 8151* | 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 8152* | 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 8153 | 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 8154 | Deduction form of elmapg 8153. (Contributed by BJ, 11-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) | ||
Theorem | mapdm0 8155 | The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux, 3-Dec-2021.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵 ↑𝑚 ∅) = {∅}) | ||
Theorem | elpmg 8156 | 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 8157 | 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 8158 | 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 8159 | 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 8160 | 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 8161 | 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 8162 | 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 8163 | A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.) |
⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → 𝐴 Fn 𝐶) | ||
Theorem | elmapfun 8164 | 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 8165 | 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 8166 | 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 8167 | 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 8168 | 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 8169 | Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶𝐴) | ||
Theorem | mapval2 8170* | 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 8171 | 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 8172 | 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 8173 | 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 8174 | 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 8175 | 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 8176 | 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 | mapfvd 8177 | The value of a function that maps from 𝐵 to 𝐴. (Contributed by AV, 2-Feb-2023.) |
⊢ 𝑀 = (𝐴 ↑𝑚 𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐴) | ||
Theorem | fvmptmap 8178* | Special case of fvmpt 6542 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 8179 | 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.) (Proof shortened by AV, 14-Jul-2022.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = 1o) | ||
Theorem | map0eOLD 8180 | Obsolete proof of map0e 8179 as of 14-Jul-2022. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = 1o) | ||
Theorem | map0b 8181 | 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 8182 | 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 | mapsnd 8183* | The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 ↑𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {〈𝐵, 𝑦〉}}) | ||
Theorem | map0 8184 | 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 8185* | The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Proof shortened by AV, 17-Jul-2022.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ↑𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {〈𝐵, 𝑦〉}} | ||
Theorem | mapss 8186 | 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 8187* | 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 8188* | 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 8189 | 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 8190* | 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 8191* | 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 8192* | Explicit bijection in the reverse of mapsnf1o2 8191. (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 8193* | 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 8194 | Extend class notation to include infinite Cartesian products. |
class X𝑥 ∈ 𝐴 𝐵 | ||
Definition | df-ixp 8195* | 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 8196* | Eliminate the expression {𝑥 ∣ 𝑥 ∈ 𝐴} in df-ixp 8195, 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 8197* | 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 8198* | Membership in an infinite Cartesian product. See df-ixp 8195 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 8199* | 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 8200* | 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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