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Theorem List for Intuitionistic Logic Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremunielxp 5901 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
(𝐴 ∈ (𝐵 × 𝐶) → 𝐴 (𝐵 × 𝐶))
 
Theorem1st2nd2 5902 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 = ⟨(1st𝐴), (2nd𝐴)⟩)
 
Theoremxpopth 5903 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
 
Theoremeqop 5904 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⟩ ↔ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)))
 
Theoremeqop2 5905 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) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)))
 
Theoremop1steq 5906* 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) → ((1st𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
 
Theorem2nd1st 5907 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
(𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = ⟨(2nd𝐴), (1st𝐴)⟩)
 
Theorem1st2nd 5908 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
((Rel 𝐵𝐴𝐵) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
 
Theorem1stdm 5909 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 𝑅𝐴𝑅) → (1st𝐴) ∈ dom 𝑅)
 
Theorem2ndrn 5910 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 𝑅𝐴𝑅) → (2nd𝐴) ∈ ran 𝑅)
 
Theorem1st2ndbr 5911 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
((Rel 𝐵𝐴𝐵) → (1st𝐴)𝐵(2nd𝐴))
 
Theoremreleldm2 5912* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))
 
Theoremreldm 5913* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel 𝐴 → dom 𝐴 = ran (𝑥𝐴 ↦ (1st𝑥)))
 
Theoremsbcopeq1a 5914 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2838 that avoids the existential quantifiers of copsexg 4045). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st𝐴) / 𝑥][(2nd𝐴) / 𝑦]𝜑𝜑))
 
Theoremcsbopeq1a 5915 Equality theorem for substitution of a class 𝐴 for an ordered pair 𝑥, 𝑦 in 𝐵 (analog of csbeq1a 2930). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵 = 𝐵)
 
Theoremdfopab2 5916* 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) ∣ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑}
 
Theoremdfoprab3s 5917* 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) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑)}
 
Theoremdfoprab3 5918* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
(𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
 
Theoremdfoprab4 5919* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}
 
Theoremdfoprab4f 5920* 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.)
𝑥𝜑    &   𝑦𝜑    &   (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}
 
Theoremdfxp3 5921* Define the cross product of three classes. Compare df-xp 4417. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
((𝐴 × 𝐵) × 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥𝐴𝑦𝐵𝑧𝐶)}
 
Theoremelopabi 5922* A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
(𝑥 = (1st𝐴) → (𝜑𝜓))    &   (𝑦 = (2nd𝐴) → (𝜓𝜒))       (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)
 
Theoremeloprabi 5923* 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 ‘(1st𝐴)) → (𝜑𝜓))    &   (𝑦 = (2nd ‘(1st𝐴)) → (𝜓𝜒))    &   (𝑧 = (2nd𝐴) → (𝜒𝜃))       (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)
 
Theoremmpt2mptsx 5924* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
 
Theoremmpt2mpts 5925* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
(𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
 
Theoremdmmpt2ssx 5926* The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       dom 𝐹 𝑥𝐴 ({𝑥} × 𝐵)
 
Theoremfmpt2x 5927* 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 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷)
 
Theoremfmpt2 5928* Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹:(𝐴 × 𝐵)⟶𝐷)
 
Theoremfnmpt2 5929* Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝑉𝐹 Fn (𝐴 × 𝐵))
 
Theoremmpt2fvex 5930* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → (𝑅𝐹𝑆) ∈ V)
 
Theoremfnmpt2i 5931* Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐶 ∈ V       𝐹 Fn (𝐴 × 𝐵)
 
Theoremdmmpt2 5932* Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐶 ∈ V       dom 𝐹 = (𝐴 × 𝐵)
 
Theoremmpt2fvexi 5933* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐶 ∈ V    &   𝑅 ∈ V    &   𝑆 ∈ V       (𝑅𝐹𝑆) ∈ V
 
Theoremmpt2exxg 5934* Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝐴𝑅 ∧ ∀𝑥𝐴 𝐵𝑆) → 𝐹 ∈ V)
 
Theoremmpt2exg 5935* Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝐴𝑅𝐵𝑆) → 𝐹 ∈ V)
 
Theoremmpt2exga 5936* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 12-Sep-2011.)
((𝐴𝑉𝐵𝑊) → (𝑥𝐴, 𝑦𝐵𝐶) ∈ V)
 
Theoremmpt2ex 5937* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by Mario Carneiro, 20-Dec-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
 
Theoremfmpt2co 5938* Composition of two functions. Variation of fmptco 5427 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑅𝐶)    &   (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝑅))    &   (𝜑𝐺 = (𝑧𝐶𝑆))    &   (𝑧 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴, 𝑦𝐵𝑇))
 
Theoremoprabco 5939* Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
((𝑥𝐴𝑦𝐵) → 𝐶𝐷)    &   𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐻𝐶))       (𝐻 Fn 𝐷𝐺 = (𝐻𝐹))
 
Theoremoprab2co 5940* Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
((𝑥𝐴𝑦𝐵) → 𝐶𝑅)    &   ((𝑥𝐴𝑦𝐵) → 𝐷𝑆)    &   𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨𝐶, 𝐷⟩)    &   𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑀𝐷))       (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
 
Theoremdf1st2 5941* 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))
 
Theoremdf2nd2 5942* 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))
 
Theorem1stconst 5943 The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
(𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)
 
Theorem2ndconst 5944 The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
(𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)
 
Theoremdfmpt2 5945* Alternate definition for the maps-to notation df-mpt2 5618 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐶 ∈ V       (𝑥𝐴, 𝑦𝐵𝐶) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
 
Theoremcnvf1olem 5946 Lemma for cnvf1o 5947. (Contributed by Mario Carneiro, 27-Apr-2014.)
((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → (𝐶𝐴𝐵 = {𝐶}))
 
Theoremcnvf1o 5947* Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
(Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
 
Theoremf2ndf 5948 The 2nd (second member of an ordered pair) function restricted to a function 𝐹 is a function of 𝐹 into the codomain of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
(𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)
 
Theoremfo2ndf 5949 The 2nd (second member of an ordered pair) function restricted to a function 𝐹 is a function of 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
(𝐹:𝐴𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)
 
Theoremf1o2ndf1 5950 The 2nd (second member of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function of 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
(𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹1-1-onto→ran 𝐹)
 
Theoremalgrflem 5951 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) = (𝐹𝐵)
 
Theoremalgrflemg 5952 Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.)
((𝐵𝑉𝐶𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵))
 
Theoremxporderlem 5953* Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))}       (⟨𝑎, 𝑏𝑇𝑐, 𝑑⟩ ↔ (((𝑎𝐴𝑐𝐴) ∧ (𝑏𝐵𝑑𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐𝑏𝑆𝑑))))
 
Theorempoxp 5954* 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)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))}       ((𝑅 Po 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐴 × 𝐵))
 
Theoremspc2ed 5955* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
𝑥𝜒    &   𝑦𝜒    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (𝜒 → ∃𝑥𝑦𝜓))
 
Theoremcnvoprab 5956* The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)
𝑥𝜓    &   𝑦𝜓    &   (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))    &   (𝜓𝑎 ∈ (V × V))       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
 
Theoremf1od2 5957* Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝑊)    &   ((𝜑𝑧𝐷) → (𝐼𝑋𝐽𝑌))    &   (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽))))       (𝜑𝐹:(𝐴 × 𝐵)–1-1-onto𝐷)
 
2.6.15  Special maps-to operations

The following theorems are about maps-to operations (see df-mpt2 5618) where 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 "mpt2x" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpt2x 5683, ovmpt2x 5730 and fmpt2x 5927). However, there is a proposal by Norman Megill to use the abbreviation "mpo" or "mpto" instead of "mpt2" (see beginning of set.mm). If this proposal will be realized, the labels in the following should also be adapted. If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpt2xop", and the maps-to operations are called "x-op maps-to operations" for short.

 
Theoremmpt2xopn0yelv 5958* 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 ∈ (1st𝑥) ↦ 𝐶)       ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → 𝐾𝑉))
 
Theoremmpt2xopoveq 5959* 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 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})       (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
 
Theoremmpt2xopovel 5960* 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 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})       ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾𝑉𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
 
Theoremsprmpt2 5961* 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)𝑝𝜒)})    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜒𝜓))    &   ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝𝜃))    &   ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)       ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑀𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)})
 
Theoremisprmpt2 5962* Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)})    &   ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))       (𝜑 → ((𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒))))
 
2.6.16  Function transposition
 
Syntaxctpos 5963 The transposition of a function.
class tpos 𝐹
 
Definitiondf-tpos 5964* 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}))
 
Theoremtposss 5965 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
 
Theoremtposeq 5966 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
 
Theoremtposeqd 5967 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐹 = 𝐺)       (𝜑 → tpos 𝐹 = tpos 𝐺)
 
Theoremtposssxp 5968 The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
 
Theoremreltpos 5969 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
Rel tpos 𝐹
 
Theorembrtpos2 5970 Value of the transposition at a pair 𝐴, 𝐵. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐵𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
 
Theorembrtpos0 5971 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.)
(𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))
 
Theoremreldmtpos 5972 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)
 
Theorembrtposg 5973 The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))
 
Theoremottposg 5974 The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ tpos 𝐹 ↔ ⟨𝐵, 𝐴, 𝐶⟩ ∈ 𝐹))
 
Theoremdmtpos 5975 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)
 
Theoremrntpos 5976 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)
 
Theoremtposexg 5977 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹𝑉 → tpos 𝐹 ∈ V)
 
Theoremovtposg 5978 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.)
((𝐴𝑉𝐵𝑊) → (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴))
 
Theoremtposfun 5979 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Fun 𝐹 → Fun tpos 𝐹)
 
Theoremdftpos2 5980* Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})))
 
Theoremdftpos3 5981* Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 4419. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧})
 
Theoremdftpos4 5982* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
 
Theoremtpostpos 5983 Value of the double transposition for a general class 𝐹. (Contributed by Mario Carneiro, 16-Sep-2015.)
tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))
 
Theoremtpostpos2 5984 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)
 
Theoremtposfn2 5985 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴))
 
Theoremtposfo2 5986 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹:𝐴onto𝐵 → tpos 𝐹:𝐴onto𝐵))
 
Theoremtposf2 5987 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹:𝐴𝐵 → tpos 𝐹:𝐴𝐵))
 
Theoremtposf12 5988 Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹:𝐴1-1𝐵 → tpos 𝐹:𝐴1-1𝐵))
 
Theoremtposf1o2 5989 Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹:𝐴1-1-onto𝐵 → tpos 𝐹:𝐴1-1-onto𝐵))
 
Theoremtposfo 5990 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(𝐹:(𝐴 × 𝐵)–onto𝐶 → tpos 𝐹:(𝐵 × 𝐴)–onto𝐶)
 
Theoremtposf 5991 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(𝐹:(𝐴 × 𝐵)⟶𝐶 → tpos 𝐹:(𝐵 × 𝐴)⟶𝐶)
 
Theoremtposfn 5992 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝐹 Fn (𝐴 × 𝐵) → tpos 𝐹 Fn (𝐵 × 𝐴))
 
Theoremtpos0 5993 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
tpos ∅ = ∅
 
Theoremtposco 5994 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos (𝐹𝐺) = (𝐹 ∘ tpos 𝐺)
 
Theoremtpossym 5995* Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)))
 
Theoremtposeqi 5996 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 = 𝐺       tpos 𝐹 = tpos 𝐺
 
Theoremtposex 5997 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 ∈ V       tpos 𝐹 ∈ V
 
Theoremnftpos 5998 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝑥𝐹       𝑥tpos 𝐹
 
Theoremtposoprab 5999* Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}       tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremtposmpt2 6000* Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       tpos 𝐹 = (𝑦𝐵, 𝑥𝐴𝐶)
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