P. 67 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iinerm 6601*	The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵)
Theorem	riinerm 6602*	The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵)
Theorem	erinxp 6603	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 6604	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 6605	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	qsel 6606	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	qliftlem 6607*	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 6608*	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 6609*	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 6610*	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 6611*	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 6612*	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 6613*	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 6614*	The domain and codomain 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 6615*	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 6616*	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 6617*	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 6618*	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 6619*	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	eroveu 6620*	Lemma for eroprf 6622. (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 6621*	Lemma for eroprf 6622. (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	eroprf 6622*	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	eroprf2 6623*	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 6624*	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 6625*	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 6626*	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 6627*	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.)
⊢ ∼ = {⟨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	ecopovsymg 6628*	Assuming the operation 𝐹 is commutative, show that the relation ∼, specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    ⇒   ⊢ (𝐴 ∼ 𝐵 → 𝐵 ∼ 𝐴)
Theorem	ecopovtrng 6629*	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 Jim Kingdon, 1-Sep-2019.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))    ⇒   ⊢ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → 𝐴 ∼ 𝐶)
Theorem	ecopoverg 6630*	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 Jim Kingdon, 1-Sep-2019.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))    ⇒   ⊢ ∼ Er (𝑆 × 𝑆)
Theorem	th3qlem1 6631*	Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption. (Contributed by NM, 3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
⊢ ∼ Er 𝑆    &   ⊢ (((𝑦 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → ((𝑦 ∼ 𝑤 ∧ 𝑧 ∼ 𝑣) → (𝑦 + 𝑧) ∼ (𝑤 + 𝑣)))    ⇒   ⊢ ((𝐴 ∈ (𝑆 / ∼ ) ∧ 𝐵 ∈ (𝑆 / ∼ )) → ∃*𝑥∃𝑦∃𝑧((𝐴 = [𝑦] ∼ ∧ 𝐵 = [𝑧] ∼ ) ∧ 𝑥 = [(𝑦 + 𝑧)] ∼ ))
Theorem	th3qlem2 6632*	Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ ∈ V    &   ⊢ ∼ Er (𝑆 × 𝑆)    &   ⊢ ((((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) ∧ ((𝑠 ∈ 𝑆 ∧ 𝑓 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆))) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓⟩ ∼ ⟨𝑔, ℎ⟩) → (⟨𝑤, 𝑣⟩ + ⟨𝑠, 𝑓⟩) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩)))    ⇒   ⊢ ((𝐴 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝐵 ∈ ((𝑆 × 𝑆) / ∼ )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))
Theorem	th3qcor 6633*	Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)
⊢ ∼ ∈ V    &   ⊢ ∼ Er (𝑆 × 𝑆)    &   ⊢ ((((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) ∧ ((𝑠 ∈ 𝑆 ∧ 𝑓 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆))) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓⟩ ∼ ⟨𝑔, ℎ⟩) → (⟨𝑤, 𝑣⟩ + ⟨𝑠, 𝑓⟩) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩)))    &   ⊢ 𝐺 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))}    ⇒   ⊢ Fun 𝐺
Theorem	th3q 6634*	Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
⊢ ∼ ∈ V    &   ⊢ ∼ Er (𝑆 × 𝑆)    &   ⊢ ((((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) ∧ ((𝑠 ∈ 𝑆 ∧ 𝑓 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆))) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓⟩ ∼ ⟨𝑔, ℎ⟩) → (⟨𝑤, 𝑣⟩ + ⟨𝑠, 𝑓⟩) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩)))    &   ⊢ 𝐺 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))}    ⇒   ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([⟨𝐴, 𝐵⟩] ∼ 𝐺[⟨𝐶, 𝐷⟩] ∼ ) = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ )
Theorem	oviec 6635*	Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See iset.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.)
⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐻 ∈ (𝑆 × 𝑆))    &   ⊢ (((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → 𝐾 ∈ (𝑆 × 𝑆))    &   ⊢ (((𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → 𝐿 ∈ (𝑆 × 𝑆))    &   ⊢ ∼ ∈ V    &   ⊢ ∼ Er (𝑆 × 𝑆)    &   ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}    &   ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → (𝜑 ↔ 𝜓))    &   ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → (𝜑 ↔ 𝜒))    &   ⊢ + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝐽))}    &   ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → 𝐽 = 𝐾)    &   ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → 𝐽 = 𝐿)    &   ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝐽 = 𝐻)    &   ⊢ ⨣ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑄) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ∼ ∧ 𝑦 = [⟨𝑐, 𝑑⟩] ∼ ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ + ⟨𝑐, 𝑑⟩)] ∼ ))}    &   ⊢ 𝑄 = ((𝑆 × 𝑆) / ∼ )    &   ⊢ ((((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) ∧ ((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆))) → ((𝜓 ∧ 𝜒) → 𝐾 ∼ 𝐿))    ⇒   ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([⟨𝐴, 𝐵⟩] ∼ ⨣ [⟨𝐶, 𝐷⟩] ∼ ) = [𝐻] ∼ )
Theorem	ecovcom 6636*	Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6637 instead. (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	ecovicom 6637*	Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.)
⊢ 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⟩] ∼ )    &   ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐷 = 𝐻)    &   ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐺 = 𝐽)    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	ecovass 6638*	Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6639 will be more useful. (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	ecoviass 6639*	Lemma used to transfer an associative law via an equivalence relation. (Contributed by Jim Kingdon, 16-Sep-2019.)
⊢ 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))    &   ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → 𝐽 = 𝐿)    &   ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → 𝐾 = 𝑀)    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Theorem	ecovdi 6640*	Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6641 will be more helpful. (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)))
Theorem	ecovidi 6641*	Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.)
⊢ 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))    &   ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → 𝐻 = 𝐾)    &   ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → 𝐽 = 𝐿)    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
2.6.26  The mapping operation
Syntax	cmap 6642	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 6643	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 6644*	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 6654). 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 6645*	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 6653). 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 6644) . See mapsspm 6676 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 6646*	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 6647*	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 6648*	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 6649	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 6650	Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
⊢ ↑pm Fn (V × V)
Theorem	reldmmap 6651	Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
⊢ Rel dom ↑𝑚
Theorem	mapvalg 6652*	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 6653*	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 6654*	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 6655	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 6656	Deduction form of elmapg 6655. (Contributed by BJ, 11-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴))
Theorem	mapdm0 6657	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 6658	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 6659	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 6660	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 6661	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 6662	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 6663	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 6664	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 6665	A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → 𝐴 Fn 𝐶)
Theorem	elmapfun 6666	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 6667	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 6668	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 6669	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 6670	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 6671	Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐹 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶𝐴)
Theorem	mapval2 6672*	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 6673	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 6674	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 6675	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 6676	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 6677	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 6678	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 6679*	Special case of fvmpt 5589 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 6680	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 6681	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 6682	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 6683	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 6684*	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 6685	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 6686*	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 6687*	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 6688	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 6689*	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 6690*	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 6691*	Explicit bijection in the reverse of mapsnf1o2 6690. (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)
2.6.27  Infinite Cartesian products
Syntax	cixp 6692	Extend class notation to include infinite Cartesian products.
class X𝑥 ∈ 𝐴 𝐵
Definition	df-ixp 6693*	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 6694*	Eliminate the expression {𝑥 ∣ 𝑥 ∈ 𝐴} in df-ixp 6693, 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 6695*	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 6696*	Membership in an infinite Cartesian product. See df-ixp 6693 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 6697*	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 6698*	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)
Theorem	elixp 6699*	Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
Theorem	elixpconst 6700*	Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by NM, 12-Apr-2008.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ 𝐹:𝐴⟶𝐵)