P. 89 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 8801-8900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iiner 8801*	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 8802*	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 8803	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 8804	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 8805	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 8806	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 8807*	A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.)
⊢ (𝑅 Er 𝑋 → Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥)
Theorem	qsel 8808	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 8809	Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4930. (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 8810*	Lemma for theorems about a function lift. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
⊢ 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 8811*	𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
⊢ 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 8812*	Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
⊢ 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 8813*	Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
⊢ 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 8814*	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.) (Revised by AV, 3-Aug-2024.)
⊢ 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 8815*	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.) (Revised by AV, 3-Aug-2024.)
⊢ 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 8816*	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.) (Revised by AV, 3-Aug-2024.)
⊢ 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 8817*	The domain and codomain of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
⊢ 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 8818*	The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
⊢ 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 8819*	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 8820*	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 8821*	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 8822*	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 8823	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	eroveu 8824*	Lemma for erov 8826 and eroprf 8827. (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 8825*	Lemma for erov 8826 and eroprf 8827. (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 8826*	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 8827*	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 8828*	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 8829*	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 8830*	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 8831*	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 8832*	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 8833*	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 8834*	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 8835*	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 8836*	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 8837*	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)))
2.4.27  The mapping operation
Syntax	cmap 8838	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 8839	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 8840*	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 8850). 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 8841*	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 8849). 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 8840). See mapsspm 8888 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 8842*	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 8843*	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 8844*	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 8845	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 8846	Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
⊢ ↑pm Fn (V × V)
Theorem	reldmmap 8847	Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
⊢ Rel dom ↑m
Theorem	mapvalg 8848*	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 8849*	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 8850*	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 8851	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 8852	Deduction form of elmapg 8851. (Contributed by BJ, 11-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑m 𝐵) ↔ 𝐶:𝐵⟶𝐴))
Theorem	elmapdd 8853	Deduction associated with elmapd 8852. (Contributed by SN, 29-Jul-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶:𝐵⟶𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑m 𝐵))
Theorem	mapdm0 8854	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 8855	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 8856	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 8857	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 8858	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 8859	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 8860	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 8861	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	mapfset 8862*	If 𝐵 is a set, the value of the set exponentiation (𝐵 ↑m 𝐴) is the class of all functions from 𝐴 to 𝐵. Generalisation of mapvalg 8848 (which does not require ax-rep 5281) to arbitrary domains. Note that the class {𝑓 ∣ 𝑓:𝐴⟶𝐵} can only contain set-functions, as opposed to arbitrary class-functions. When 𝐴 is a proper class, there can be no set-functions on it, so the above class is empty (see also fsetdmprc0 8867), hence a set. In this case, both sides of the equality in this theorem are the empty set. (Contributed by AV, 8-Aug-2024.)
⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑m 𝐴))
Theorem	mapssfset 8863*	The value of the set exponentiation (𝐵 ↑m 𝐴) is a subset of the class of functions from 𝐴 to 𝐵. (Contributed by AV, 10-Aug-2024.)
⊢ (𝐵 ↑m 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}
Theorem	mapfoss 8864*	The value of the set exponentiation (𝐵 ↑m 𝐴) is a superset of the set of all functions from 𝐴 onto 𝐵. (Contributed by AV, 7-Aug-2024.)
⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴)
Theorem	fsetsspwxp 8865*	The class of all functions from 𝐴 into 𝐵 is a subclass of the power class of the cartesion product of 𝐴 and 𝐵. (Contributed by AV, 13-Sep-2024.)
⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
Theorem	fset0 8866	The set of functions from the empty set is the singleton containing the empty set. (Contributed by AV, 13-Sep-2024.)
⊢ {𝑓 ∣ 𝑓:∅⟶𝐵} = {∅}
Theorem	fsetdmprc0 8867*	The set of functions with a proper class as domain is empty. (Contributed by AV, 22-Aug-2024.)
⊢ (𝐴 ∉ V → {𝑓 ∣ 𝑓 Fn 𝐴} = ∅)
Theorem	fsetex 8868*	The set of functions between two classes exists if the codomain exists. Generalization of mapex 8844 to arbitrary domains. (Contributed by AV, 14-Aug-2024.)
⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)
Theorem	f1setex 8869*	The set of injections between two classes exists if the codomain exists. (Contributed by AV, 14-Aug-2024.)
⊢ (𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ∈ V)
Theorem	fosetex 8870*	The set of surjections between two classes exists (without any precondition). (Contributed by AV, 8-Aug-2024.)
⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ∈ V
Theorem	f1osetex 8871*	The set of bijections between two classes exists. (Contributed by AV, 30-Mar-2024.) (Revised by AV, 8-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ∈ V
Theorem	fsetfcdm 8872*	The class of functions with a given domain and a given codomain is mapped, through evaluation at a point of the domain, into the codomain. (Contributed by AV, 15-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵}    &   ⊢ 𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋))    ⇒   ⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵)
Theorem	fsetfocdm 8873*	The class of functions with a given domain that is a set and a given codomain is mapped, through evaluation at a point of the domain, onto the codomain. (Contributed by AV, 15-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵}    &   ⊢ 𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑆:𝐹–onto→𝐵)
Theorem	fsetprcnex 8874*	The class of all functions from a nonempty set 𝐴 into a proper class 𝐵 is not a set. If one of the preconditions is not fufilled, then {𝑓 ∣ 𝑓:𝐴⟶𝐵} is a set, see fsetdmprc0 8867 for 𝐴 ∉ V, fset0 8866 for 𝐴 = ∅, and fsetex 8868 for 𝐵 ∈ V, see also fsetexb 8876. (Contributed by AV, 14-Sep-2024.) (Proof shortened by BJ, 15-Sep-2024.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)
Theorem	fsetcdmex 8875*	The class of all functions from a nonempty set 𝐴 into a class 𝐵 is a set iff 𝐵 is a set . (Contributed by AV, 15-Sep-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∈ V ↔ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V))
Theorem	fsetexb 8876*	The class of all functions from a class 𝐴 into a class 𝐵 is a set iff 𝐵 is a set or 𝐴 is not a set or 𝐴 is empty. (Contributed by AV, 15-Sep-2024.)
⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V ↔ (𝐴 ∉ V ∨ 𝐴 = ∅ ∨ 𝐵 ∈ V))
Theorem	elmapfn 8877	A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐴 Fn 𝐶)
Theorem	elmapfun 8878	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 8879	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 8880	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 8881	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 8882	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 8883	Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐹 ∈ (𝐴 ↑m 𝐵) ↔ 𝐹:𝐵⟶𝐴)
Theorem	mapval2 8884*	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 8885	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 8886	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 8887	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 8888	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 8889	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 8890	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 8891	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	elmapresaun 8892	fresaun 6762 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)))
Theorem	fvmptmap 8893*	Special case of fvmpt 6998 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 8894	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	map0b 8895	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 8896	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	0map0sn0 8897	The set of mappings of the empty set to the empty set is the singleton containing the empty set. (Contributed by AV, 31-Mar-2024.)
⊢ (∅ ↑m ∅) = {∅}
Theorem	mapsnd 8898*	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 8899	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 8900*	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⟩}}