P. 61 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6001-6100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xp2 6001*	Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
⊢ (𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ 𝐵)}
Theorem	unielxp 6002	The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))
Theorem	1st2nd2 6003	Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
Theorem	xpopth 6004	An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)) ↔ 𝐴 = 𝐵))
Theorem	eqop 6005	Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))
Theorem	eqop2 6006	Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 = ⟨𝐵, 𝐶⟩ ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))
Theorem	op1steq 6007*	Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)
⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
Theorem	2nd1st 6008	Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩)
Theorem	1st2nd 6009	Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
Theorem	1stdm 6010	The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) ∈ dom 𝑅)
Theorem	2ndrn 6011	The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅)
Theorem	1st2ndbr 6012	Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴))
Theorem	releldm2 6013*	Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
⊢ (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵))
Theorem	reldm 6014*	An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))
Theorem	sbcopeq1a 6015	Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2871 that avoids the existential quantifiers of copsexg 4104). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑))
Theorem	csbopeq1a 6016	Equality theorem for substitution of a class 𝐴 for an ordered pair ⟨𝑥, 𝑦⟩ in 𝐵 (analog of csbeq1a 2963). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵)
Theorem	dfopab2 6017*	A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑}
Theorem	dfoprab3s 6018*	A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)}
Theorem	dfoprab3 6019*	Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Theorem	dfoprab4 6020*	Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}
Theorem	dfoprab4f 6021*	Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}
Theorem	dfxp3 6022*	Define the cross product of three classes. Compare df-xp 4483. (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 6023*	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 6024*	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 6025*	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 6026*	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 6027*	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 6028*	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 6029*	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 6030*	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	mpt2fvex 6031*	Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝑅𝐹𝑆) ∈ V)
Theorem	fnmpoi 6032*	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 6033*	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	mpt2fvexi 6034*	Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝑅 ∈ V    &   ⊢ 𝑆 ∈ V    ⇒   ⊢ (𝑅𝐹𝑆) ∈ V
Theorem	ovmpoelrn 6035*	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 6036*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6033. (Contributed by Alexander van der Vekens, 10-Feb-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵))
Theorem	dmmpog 6037*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6033. 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 6038*	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 6039*	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 6040*	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	mpoex 6041*	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	fnmpoovd 6042*	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	fmpoco 6043*	Composition of two functions. Variation of fmptco 5518 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 6044*	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 6045*	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 6046*	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 6047*	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 6048	The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)
Theorem	2ndconst 6049	The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵)
Theorem	dfmpo 6050*	Alternate definition for the maps-to notation df-mpo 5711 (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	cnvf1olem 6051	Lemma for cnvf1o 6052. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ◡{𝐵})) → (𝐶 ∈ ◡𝐴 ∧ 𝐵 = ∪ ◡{𝐶}))
Theorem	cnvf1o 6052*	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	f2ndf 6053	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 6054	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 6055	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	algrflem 6056	Lemma for algrf and related theorems. (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	algrflemg 6057	Lemma for algrf 11519 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Jim Kingdon, 22-Jul-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵))
Theorem	xporderlem 6058*	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 6059*	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	spc2ed 6060*	Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑦𝜒    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (𝜒 → ∃𝑥∃𝑦𝜓))
Theorem	cnvoprab 6061*	The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜑))    &   ⊢ (𝜓 → 𝑎 ∈ (V × V))    ⇒   ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
Theorem	f1od2 6062*	Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌))    &   ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽))))    ⇒   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)
Theorem	disjxp1 6063*	The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 (𝐵 × 𝐶))
Theorem	disjsnxp 6064*	The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
2.6.15  Special maps-to operations The following theorems are about maps-to operations (see df-mpo 5711) where the domain of the second argument depends on the domain of the first argument, especially when the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpox" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpox 5781, ovmpox 5831 and fmpox 6028). If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpoxop", and the maps-to operations are called "x-op maps-to operations" for short.
Theorem	opeliunxp2f 6065*	Membership in a union of Cartesian products, using bound-variable hypothesis for 𝐸 instead of distinct variable conditions as in opeliunxp2 4617. (Contributed by AV, 25-Oct-2020.)
⊢ Ⅎ𝑥𝐸    &   ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸)    ⇒   ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))
Theorem	mpoxopn0yelv 6066*	If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))
Theorem	mpoxopoveq 6067*	Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})    ⇒   ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
Theorem	mpoxopovel 6068*	Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
Theorem	sprmpt2 6069*	The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
⊢ 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣𝑊𝑒)𝑝 ∧ 𝜒)})    &   ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓))    &   ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝 → 𝜃))    &   ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)    ⇒   ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑀𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)})
Theorem	isprmpt2 6070*	Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)})    &   ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒))))
2.6.16  Function transposition
Syntax	ctpos 6071	The transposition of a function.
class tpos 𝐹
Definition	df-tpos 6072*	Define the transposition of a function, which is a function 𝐺 = tpos 𝐹 satisfying 𝐺(𝑥, 𝑦) = 𝐹(𝑦, 𝑥). (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
Theorem	tposss 6073	Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
Theorem	tposeq 6074	Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
Theorem	tposeqd 6075	Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → tpos 𝐹 = tpos 𝐺)
Theorem	tposssxp 6076	The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹)
Theorem	reltpos 6077	The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ Rel tpos 𝐹
Theorem	brtpos2 6078	Value of the transposition at a pair ⟨𝐴, 𝐵⟩. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))
Theorem	brtpos0 6079	The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))
Theorem	reldmtpos 6080	Necessary and sufficient condition for dom tpos 𝐹 to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
Theorem	brtposg 6081	The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶))
Theorem	ottposg 6082	The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ tpos 𝐹 ↔ ⟨𝐵, 𝐴, 𝐶⟩ ∈ 𝐹))
Theorem	dmtpos 6083	The domain of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
Theorem	rntpos 6084	The range of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
Theorem	tposexg 6085	The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V)
Theorem	ovtposg 6086	The transposition swaps the arguments in a two-argument function. When 𝐹 is a matrix, which is to say a function from ( 1 ... m ) × ( 1 ... n ) to the reals or some ring, tpos 𝐹 is the transposition of 𝐹, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴))
Theorem	tposfun 6087	The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (Fun 𝐹 → Fun tpos 𝐹)
Theorem	dftpos2 6088*	Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})))
Theorem	dftpos3 6089*	Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 4485. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧})
Theorem	dftpos4 6090*	Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
Theorem	tpostpos 6091	Value of the double transposition for a general class 𝐹. (Contributed by Mario Carneiro, 16-Sep-2015.)
⊢ tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))
Theorem	tpostpos2 6092	Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)
⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹)
Theorem	tposfn2 6093	The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴))
Theorem	tposfo2 6094	Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel 𝐴 → (𝐹:𝐴–onto→𝐵 → tpos 𝐹:◡𝐴–onto→𝐵))
Theorem	tposf2 6095	The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:◡𝐴⟶𝐵))
Theorem	tposf12 6096	Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel 𝐴 → (𝐹:𝐴–1-1→𝐵 → tpos 𝐹:◡𝐴–1-1→𝐵))
Theorem	tposf1o2 6097	Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel 𝐴 → (𝐹:𝐴–1-1-onto→𝐵 → tpos 𝐹:◡𝐴–1-1-onto→𝐵))
Theorem	tposfo 6098	The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 → tpos 𝐹:(𝐵 × 𝐴)–onto→𝐶)
Theorem	tposf 6099	The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → tpos 𝐹:(𝐵 × 𝐴)⟶𝐶)
Theorem	tposfn 6100	Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝐹 Fn (𝐴 × 𝐵) → tpos 𝐹 Fn (𝐵 × 𝐴))