P. 81 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 8001-8100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opabex2 8001*	Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)
Theorem	opabn1stprc 8002*	An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wff. (Contributed by AV, 27-Dec-2020.)
⊢ (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)
Theorem	opiota 8003*	The property of a uniquely specified ordered pair. The proof uses properties of the ℩ description binder. (Contributed by Mario Carneiro, 21-May-2015.)
⊢ 𝐼 = (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))    &   ⊢ 𝑋 = (1st ‘𝐼)    &   ⊢ 𝑌 = (2nd ‘𝐼)    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐷 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝜒) ↔ (𝐶 = 𝑋 ∧ 𝐷 = 𝑌)))
Theorem	cnvoprab 8004*	The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) (Proof shortened by Thierry Arnoux, 20-Feb-2022.)
⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜑))    &   ⊢ (𝜓 → 𝑎 ∈ (V × V))    ⇒   ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
Theorem	dfxp3 8005*	Define the Cartesian product of three classes. Compare df-xp 5630. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
⊢ ((𝐴 × 𝐵) × 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)}
Theorem	elopabi 8006*	A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
⊢ (𝑥 = (1st ‘𝐴) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = (2nd ‘𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)
Theorem	eloprabi 8007*	A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ (𝑥 = (1st ‘(1st ‘𝐴)) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = (2nd ‘(1st ‘𝐴)) → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = (2nd ‘𝐴) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)
Theorem	mpomptsx 8008*	Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
Theorem	mpompts 8009*	Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
Theorem	dmmpossx 8010*	The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
Theorem	fmpox 8011*	Functionality, domain and codomain of a class given by the maps-to notation, where 𝐵(𝑥) is not constant but depends on 𝑥. (Contributed by NM, 29-Dec-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷)
Theorem	fmpo 8012*	Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:(𝐴 × 𝐵)⟶𝐷)
Theorem	fnmpo 8013*	Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → 𝐹 Fn (𝐴 × 𝐵))
Theorem	fnmpoi 8014*	Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ 𝐹 Fn (𝐴 × 𝐵)
Theorem	dmmpo 8015*	Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ dom 𝐹 = (𝐴 × 𝐵)
Theorem	ovmpoelrn 8016*	An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.)
⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) ∈ 𝑀)
Theorem	dmmpoga 8017*	Domain of an operation given by the maps-to notation, closed form of dmmpo 8015. (Contributed by Alexander van der Vekens, 10-Feb-2019.) (Proof shortened by Lammen, 29-May-2024.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵))
Theorem	dmmpog 8018*	Domain of an operation given by the maps-to notation, closed form of dmmpo 8015. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵))
Theorem	mpoexxg 8019*	Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆) → 𝐹 ∈ V)
Theorem	mpoexg 8020*	Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝐹 ∈ V)
Theorem	mpoexga 8021*	If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by NM, 12-Sep-2011.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V)
Theorem	mpoexw 8022*	Weak version of mpoex 8023 that holds without ax-rep 5224. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
Theorem	mpoex 8023*	If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by Mario Carneiro, 20-Dec-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
Theorem	mptmpoopabbrd 8024*	The operation value of a function value of a collection of ordered pairs of elements related in two ways. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable condition on 𝐷, 𝑓, ℎ to remove hypotheses; avoid ax-rep 5224. (Revised by SN, 7-Apr-2025.)
⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))    &   ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝜏 ↔ 𝜃))    &   ⊢ (𝑔 = 𝐺 → (𝜒 ↔ 𝜏))    &   ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)}))    ⇒   ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})
Theorem	mptmpoopabbrdOLD 8025*	Obsolete version of mptmpoopabbrd 8024 as of 7-Apr-2025. (Contributed by Alexander van Vekens, 8-Nov-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))    &   ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝜏 ↔ 𝜃))    &   ⊢ (𝑔 = 𝐺 → (𝜒 ↔ 𝜏))    &   ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)}))    ⇒   ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})
Theorem	mptmpoopabovd 8026*	The operation value of a function value of a collection of ordered pairs of related elements. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable condition on 𝐷, 𝑓, ℎ to remove hypotheses. (Revised by SN, 13-Dec-2024.)
⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))    &   ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ∧ 𝑓(𝐷‘𝑔)ℎ)}))    ⇒   ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)})
Theorem	el2mpocsbcl 8027*	If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. (Contributed by AV, 21-May-2021.)
⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷))))
Theorem	el2mpocl 8028*	If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. Using implicit substitution. (Contributed by AV, 21-May-2021.)
⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸))    &   ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐶 = 𝐹 ∧ 𝐷 = 𝐺))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺))))
Theorem	fnmpoovd 8029*	A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.) (Revised by AV, 3-Jul-2022.)
⊢ (𝜑 → 𝑀 Fn (𝐴 × 𝐵))    &   ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))
Theorem	offval22 8030*	The function operation expressed as a mapping, variation of offval2 7642. (Contributed by SO, 15-Jul-2018.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷))    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑅𝐷)))
Theorem	brovpreldm 8031	If a binary relation holds for the result of an operation, the operands are in the domain of the operation. (Contributed by AV, 31-Dec-2020.)
⊢ (𝐷(𝐵𝐴𝐶)𝐸 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐴)
Theorem	bropopvvv 8032*	If a binary relation holds for the result of an operation which is a result of an operation, the involved classes are sets. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Proof shortened by AV, 3-Jan-2021.)
⊢ 𝑂 = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜑}))    &   ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜑 ↔ 𝜓))    &   ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉𝑂𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ 𝜃})    ⇒   ⊢ (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
Theorem	bropfvvvvlem 8033*	Lemma for bropfvvvv 8034. (Contributed by AV, 31-Dec-2020.) (Revised by AV, 16-Jan-2021.)
⊢ 𝑂 = (𝑎 ∈ 𝑈 ↦ (𝑏 ∈ 𝑉, 𝑐 ∈ 𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}))    &   ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (𝐵(𝑂‘𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})    ⇒   ⊢ ((⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) ∧ 𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸) → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))
Theorem	bropfvvvv 8034*	If a binary relation holds for the result of an operation which is a function value, the involved classes are sets. (Contributed by AV, 31-Dec-2020.) (Revised by AV, 16-Jan-2021.)
⊢ 𝑂 = (𝑎 ∈ 𝑈 ↦ (𝑏 ∈ 𝑉, 𝑐 ∈ 𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}))    &   ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (𝐵(𝑂‘𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})    &   ⊢ (𝑎 = 𝐴 → 𝑉 = 𝑆)    &   ⊢ (𝑎 = 𝐴 → 𝑊 = 𝑇)    &   ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))
Theorem	ovmptss 8035*	If all the values of the mapping are subsets of a class 𝑋, then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)
Theorem	relmpoopab 8036*	Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨𝑧, 𝑤⟩ ∣ 𝜑})    ⇒   ⊢ Rel (𝐶𝐹𝐷)
Theorem	fmpoco 8037*	Composition of two functions. Variation of fmptco 7074 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐶 ↦ 𝑆))    &   ⊢ (𝑧 = 𝑅 → 𝑆 = 𝑇)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))
Theorem	oprabco 8038*	Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶))    ⇒   ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹))
Theorem	oprab2co 8039*	Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑅)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑆)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨𝐶, 𝐷⟩)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷))    ⇒   ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹))
Theorem	df1st2 8040*	An alternate possible definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))
Theorem	df2nd2 8041*	An alternate possible definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
Theorem	1stconst 8042	The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)
Theorem	2ndconst 8043	The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵)
Theorem	dfmpo 8044*	Alternate definition for the maps-to notation df-mpo 7363 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐶 ∈ V    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
Theorem	mposn 8045*	An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.)
⊢ 𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶)    &   ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (𝑦 = 𝐵 → 𝐷 = 𝐸)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})
Theorem	curry1 8046*	Composition with ◡(2nd ↾ ({𝐶} × V)) turns any binary operation 𝐹 with a constant first operand into a function 𝐺 of the second operand only. This transformation is called "currying". (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)
⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))    ⇒   ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
Theorem	curry1val 8047	The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))    ⇒   ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷))
Theorem	curry1f 8048	Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)
⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))    ⇒   ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺:𝐵⟶𝐷)
Theorem	curry2 8049*	Composition with ◡(1st ↾ (V × {𝐶})) turns any binary operation 𝐹 with a constant second operand into a function 𝐺 of the first operand only. This transformation is called "currying". (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.)
⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))    ⇒   ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)))
Theorem	curry2f 8050	Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))    ⇒   ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → 𝐺:𝐴⟶𝐷)
Theorem	curry2val 8051	The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))    ⇒   ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = (𝐷𝐹𝐶))
Theorem	cnvf1olem 8052	Lemma for cnvf1o 8053. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ◡{𝐵})) → (𝐶 ∈ ◡𝐴 ∧ 𝐵 = ∪ ◡{𝐶}))
Theorem	cnvf1o 8053*	Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴)
Theorem	fparlem1 8054	Lemma for fpar 8058. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (◡(1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
Theorem	fparlem2 8055	Lemma for fpar 8058. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (◡(2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})
Theorem	fparlem3 8056*	Lemma for fpar 8058. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐹 Fn 𝐴 → (◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)))
Theorem	fparlem4 8057*	Lemma for fpar 8058. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐺 Fn 𝐵 → (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = ∪ 𝑦 ∈ 𝐵 ((V × {𝑦}) × (V × {(𝐺‘𝑦)})))
Theorem	fpar 8058*	Merge two functions in parallel. Use as the second argument of a composition with a binary operation to build compound functions such as (𝑥 ∈ (0[,)+∞), 𝑦 ∈ ℝ ↦ ((√‘𝑥) + (sin‘𝑦))), see also ex-fpar 30537. (Contributed by NM, 17-Sep-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
⊢ 𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))
Theorem	fsplit 8059	A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 8058 in order to build compound functions such as (𝑥 ∈ (0[,)+∞) ↦ ((√‘𝑥) + (sin‘𝑥))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5521 with df-id 5519. (Revised by BJ, 31-Dec-2023.)
⊢ ◡(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
Theorem	fsplitfpar 8060*	Merge two functions with a common argument in parallel. Combination of fsplit 8059 and fpar 8058. (Contributed by AV, 3-Jan-2024.)
⊢ 𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   ⊢ 𝑆 = (◡(1st ↾ I ) ↾ 𝐴)    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
Theorem	offsplitfpar 8061	Express the function operation map ∘f by the functions defined in fsplit 8059 and fpar 8058. (Contributed by AV, 4-Jan-2024.)
⊢ 𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   ⊢ 𝑆 = (◡(1st ↾ I ) ↾ 𝐴)    ⇒   ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻 ∘ 𝑆)) = (𝐹 ∘f + 𝐺))
Theorem	f2ndf 8062	The 2nd (second component of an ordered pair) function restricted to a function 𝐹 is a function from 𝐹 into the codomain of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)
Theorem	fo2ndf 8063	The 2nd (second component of an ordered pair) function restricted to a function 𝐹 is a function from 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹)
Theorem	f1o2ndf1 8064	The 2nd (second component of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function from 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)
Theorem	opco1 8065	Value of an operation precomposed with the projection on the first component. (Contributed by Mario Carneiro, 28-May-2014.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴(𝐹 ∘ 1st )𝐵) = (𝐹‘𝐴))
Theorem	opco2 8066	Value of an operation precomposed with the projection on the second component. (Contributed by BJ, 27-Oct-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴(𝐹 ∘ 2nd )𝐵) = (𝐹‘𝐵))
Theorem	opco1i 8067	Inference form of opco1 8065. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵)
Theorem	frxp 8068*	A lexicographical ordering of two well-founded classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) (Proof shortened by Wolf Lammen, 4-Oct-2014.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}    ⇒   ⊢ ((𝑅 Fr 𝐴 ∧ 𝑆 Fr 𝐵) → 𝑇 Fr (𝐴 × 𝐵))
Theorem	xporderlem 8069*	Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}    ⇒   ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ (((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))
Theorem	poxp 8070*	A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}    ⇒   ⊢ ((𝑅 Po 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐴 × 𝐵))
Theorem	soxp 8071*	A lexicographical ordering of two strictly ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}    ⇒   ⊢ ((𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐴 × 𝐵))
Theorem	wexp 8072*	A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵))
Theorem	fnwelem 8073*	Lemma for fnwe 8074. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))}    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 We 𝐵)    &   ⊢ (𝜑 → 𝑆 We 𝐴)    &   ⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V)    &   ⊢ 𝑄 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd ‘𝑢)𝑆(2nd ‘𝑣))))}    &   ⊢ 𝐺 = (𝑧 ∈ 𝐴 ↦ ⟨(𝐹‘𝑧), 𝑧⟩)    ⇒   ⊢ (𝜑 → 𝑇 We 𝐴)
Theorem	fnwe 8074*	A variant on lexicographic order, which sorts first by some function of the base set, and then by a "backup" well-ordering when the function value is equal on both elements. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))}    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 We 𝐵)    &   ⊢ (𝜑 → 𝑆 We 𝐴)    &   ⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V)    ⇒   ⊢ (𝜑 → 𝑇 We 𝐴)
Theorem	fnse 8075*	Condition for the well-order in fnwe 8074 to be set-like. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))}    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 Se 𝐵)    &   ⊢ (𝜑 → (◡𝐹 “ 𝑤) ∈ V)    ⇒   ⊢ (𝜑 → 𝑇 Se 𝐴)
Theorem	fvproj 8076*	Value of a function on ordered pairs with values expressed as ordered pairs. Note that 𝐹 and 𝐺 are the projections of 𝐻 to the first and second coordinate respectively. (Contributed by Thierry Arnoux, 30-Dec-2019.)
⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
Theorem	fimaproj 8077*	Image of a cartesian product for a function on ordered pairs with values expressed as ordered pairs. Note that 𝐹 and 𝐺 are the projections of 𝐻 to the first and second coordinate respectively. (Contributed by Thierry Arnoux, 30-Dec-2019.)
⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))
Theorem	ralxpes 8078*	A version of ralxp 5790 with explicit substitution. (Contributed by Scott Fenton, 21-Aug-2024.)
⊢ (∀𝑥 ∈ (𝐴 × 𝐵)[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜑)
Theorem	ralxp3f 8079*	Restricted for all over a triple Cartesian product. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
Theorem	ralxp3 8080*	Restricted for all over a triple Cartesian product. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
Theorem	ralxp3es 8081*	Restricted for-all over a triple Cartesian product with explicit substitution. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)[(1st ‘(1st ‘𝑥)) / 𝑦][(2nd ‘(1st ‘𝑥)) / 𝑧][(2nd ‘𝑥) / 𝑤]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜑)
2.4.9  Induction on Cartesian products
Theorem	frpoins3xpg 8082*	Special case of founded partial induction over a Cartesian product. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒) → 𝜑))    &   ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑦 = 𝑌 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝜏)
Theorem	frpoins3xp3g 8083*	Special case of founded partial recursion over a triple Cartesian product. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (∀𝑤∀𝑡∀𝑢(⟨𝑤, 𝑡, 𝑢⟩ ∈ Pred(𝑅, ((𝐴 × 𝐵) × 𝐶), ⟨𝑥, 𝑦, 𝑧⟩) → 𝜃) → 𝜑))    &   ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑡 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = 𝑢 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑦 = 𝑌 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑧 = 𝑍 → (𝜂 ↔ 𝜁))    ⇒   ⊢ (((𝑅 Fr ((𝐴 × 𝐵) × 𝐶) ∧ 𝑅 Po ((𝐴 × 𝐵) × 𝐶) ∧ 𝑅 Se ((𝐴 × 𝐵) × 𝐶)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶)) → 𝜁)
2.4.10  Ordering on Cartesian products
Theorem	xpord2lem 8084*	Lemma for Cartesian product ordering. Calculate the value of the Cartesian product relation. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ ((𝑎𝑅𝑐 ∨ 𝑎 = 𝑐) ∧ (𝑏𝑆𝑑 ∨ 𝑏 = 𝑑) ∧ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑑))))
Theorem	poxp2 8085*	Another way of partially ordering a Cartesian product of two classes. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜑 → 𝑅 Po 𝐴)    &   ⊢ (𝜑 → 𝑆 Po 𝐵)    ⇒   ⊢ (𝜑 → 𝑇 Po (𝐴 × 𝐵))
Theorem	frxp2 8086*	Another way of giving a well-founded order to a Cartesian product of two classes. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜑 → 𝑅 Fr 𝐴)    &   ⊢ (𝜑 → 𝑆 Fr 𝐵)    ⇒   ⊢ (𝜑 → 𝑇 Fr (𝐴 × 𝐵))
Theorem	xpord2pred 8087*	Calculate the predecessor class in frxp2 8086. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑌⟩) = (((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) ∖ {⟨𝑋, 𝑌⟩}))
Theorem	sexp2 8088*	Condition for the relation in frxp2 8086 to be set-like. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑆 Se 𝐵)    ⇒   ⊢ (𝜑 → 𝑇 Se (𝐴 × 𝐵))
Theorem	xpord2indlem 8089*	Induction over the Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ 𝑅 Fr 𝐴    &   ⊢ 𝑅 Po 𝐴    &   ⊢ 𝑅 Se 𝐴    &   ⊢ 𝑆 Fr 𝐵    &   ⊢ 𝑆 Po 𝐵    &   ⊢ 𝑆 Se 𝐵    &   ⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))    &   ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑))    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)
Theorem	xpord2ind 8090*	Induction over the Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ 𝑅 Fr 𝐴    &   ⊢ 𝑅 Po 𝐴    &   ⊢ 𝑅 Se 𝐴    &   ⊢ 𝑆 Fr 𝐵    &   ⊢ 𝑆 Po 𝐵    &   ⊢ 𝑆 Se 𝐵    &   ⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))    &   ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑))    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)
Theorem	xpord3lem 8091*	Lemma for triple ordering. Calculate the value of the relation. (Contributed by Scott Fenton, 21-Aug-2024.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    ⇒   ⊢ (⟨𝑎, 𝑏, 𝑐⟩𝑈⟨𝑑, 𝑒, 𝑓⟩ ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶) ∧ (((𝑎𝑅𝑑 ∨ 𝑎 = 𝑑) ∧ (𝑏𝑆𝑒 ∨ 𝑏 = 𝑒) ∧ (𝑐𝑇𝑓 ∨ 𝑐 = 𝑓)) ∧ (𝑎 ≠ 𝑑 ∨ 𝑏 ≠ 𝑒 ∨ 𝑐 ≠ 𝑓))))
Theorem	poxp3 8092*	Triple Cartesian product partial ordering. (Contributed by Scott Fenton, 21-Aug-2024.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜑 → 𝑅 Po 𝐴)    &   ⊢ (𝜑 → 𝑆 Po 𝐵)    &   ⊢ (𝜑 → 𝑇 Po 𝐶)    ⇒   ⊢ (𝜑 → 𝑈 Po ((𝐴 × 𝐵) × 𝐶))
Theorem	frxp3 8093*	Give well-foundedness over a triple Cartesian product. (Contributed by Scott Fenton, 21-Aug-2024.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜑 → 𝑅 Fr 𝐴)    &   ⊢ (𝜑 → 𝑆 Fr 𝐵)    &   ⊢ (𝜑 → 𝑇 Fr 𝐶)    ⇒   ⊢ (𝜑 → 𝑈 Fr ((𝐴 × 𝐵) × 𝐶))
Theorem	xpord3pred 8094*	Calculate the predecsessor class for the triple order. (Contributed by Scott Fenton, 31-Jan-2025.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑋, 𝑌, 𝑍⟩) = ((((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) × (Pred(𝑇, 𝐶, 𝑍) ∪ {𝑍})) ∖ {⟨𝑋, 𝑌, 𝑍⟩}))
Theorem	sexp3 8095*	Show that the triple order is set-like. (Contributed by Scott Fenton, 21-Aug-2024.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑆 Se 𝐵)    &   ⊢ (𝜑 → 𝑇 Se 𝐶)    ⇒   ⊢ (𝜑 → 𝑈 Se ((𝐴 × 𝐵) × 𝐶))
Theorem	xpord3inddlem 8096*	Induction over the triple Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜅 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜅 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜅 → 𝑍 ∈ 𝐶)    &   ⊢ (𝜅 → 𝑅 Fr 𝐴)    &   ⊢ (𝜅 → 𝑅 Po 𝐴)    &   ⊢ (𝜅 → 𝑅 Se 𝐴)    &   ⊢ (𝜅 → 𝑆 Fr 𝐵)    &   ⊢ (𝜅 → 𝑆 Po 𝐵)    &   ⊢ (𝜅 → 𝑆 Se 𝐵)    &   ⊢ (𝜅 → 𝑇 Fr 𝐶)    &   ⊢ (𝜅 → 𝑇 Po 𝐶)    &   ⊢ (𝜅 → 𝑇 Se 𝐶)    &   ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ⊢ ((𝜅 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑))    ⇒   ⊢ (𝜅 → 𝜆)
Theorem	xpord3indd 8097*	Induction over the triple Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝜅 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜅 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜅 → 𝑍 ∈ 𝐶)    &   ⊢ (𝜅 → 𝑅 Fr 𝐴)    &   ⊢ (𝜅 → 𝑅 Po 𝐴)    &   ⊢ (𝜅 → 𝑅 Se 𝐴)    &   ⊢ (𝜅 → 𝑆 Fr 𝐵)    &   ⊢ (𝜅 → 𝑆 Po 𝐵)    &   ⊢ (𝜅 → 𝑆 Se 𝐵)    &   ⊢ (𝜅 → 𝑇 Fr 𝐶)    &   ⊢ (𝜅 → 𝑇 Po 𝐶)    &   ⊢ (𝜅 → 𝑇 Se 𝐶)    &   ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ⊢ ((𝜅 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑))    ⇒   ⊢ (𝜅 → 𝜆)
Theorem	xpord3ind 8098*	Induction over the triple Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 4-Sep-2024.)
⊢ 𝑅 Fr 𝐴    &   ⊢ 𝑅 Po 𝐴    &   ⊢ 𝑅 Se 𝐴    &   ⊢ 𝑆 Fr 𝐵    &   ⊢ 𝑆 Po 𝐵    &   ⊢ 𝑆 Se 𝐵    &   ⊢ 𝑇 Fr 𝐶    &   ⊢ 𝑇 Po 𝐶    &   ⊢ 𝑇 Se 𝐶    &   ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑))    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝜆)
2.4.11  Ordering Ordinal Sequences
Theorem	orderseqlem 8099*	Lemma for poseq 8100 and soseq 8101. The function value of a sequence is either in 𝐴 or null. (Contributed by Scott Fenton, 8-Jun-2011.)
⊢ 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶𝐴}    ⇒   ⊢ (𝐺 ∈ 𝐹 → (𝐺‘𝑋) ∈ (𝐴 ∪ {∅}))
Theorem	poseq 8100*	A partial ordering of ordinal sequences. (Contributed by Scott Fenton, 8-Jun-2011.)
⊢ 𝑅 Po (𝐴 ∪ {∅})    &   ⊢ 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶𝐴}    &   ⊢ 𝑆 = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ∧ (𝑓‘𝑥)𝑅(𝑔‘𝑥)))}    ⇒   ⊢ 𝑆 Po 𝐹