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Theorem List for Intuitionistic Logic Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf2ndres 5701 Mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)
(2nd ↾ (A × B)):(A × B)⟶B
 
Theoremfo1stresm 5702* Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
(y y B → (1st ↾ (A × B)):(A × B)–ontoA)
 
Theoremfo2ndresm 5703* Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
(x x A → (2nd ↾ (A × B)):(A × B)–ontoB)
 
Theorem1stcof 5704 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
(𝐹:A⟶(B × 𝐶) → (1st𝐹):AB)
 
Theorem2ndcof 5705 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
(𝐹:A⟶(B × 𝐶) → (2nd𝐹):A𝐶)
 
Theoremxp1st 5706 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(A (B × 𝐶) → (1stA) B)
 
Theoremxp2nd 5707 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(A (B × 𝐶) → (2ndA) 𝐶)
 
Theorem1stexg 5708 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
(A 𝑉 → (1stA) V)
 
Theorem2ndexg 5709 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
(A 𝑉 → (2ndA) V)
 
Theoremelxp6 5710 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4726. (Contributed by NM, 9-Oct-2004.)
(A (B × 𝐶) ↔ (A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶)))
 
Theoremelxp7 5711 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4726. (Contributed by NM, 19-Aug-2006.)
(A (B × 𝐶) ↔ (A (V × V) ((1stA) B (2ndA) 𝐶)))
 
Theoremeqopi 5712 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
((A (𝑉 × 𝑊) ((1stA) = B (2ndA) = 𝐶)) → A = ⟨B, 𝐶⟩)
 
Theoremxp2 5713* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
(A × B) = {x (V × V) ∣ ((1stx) A (2ndx) B)}
 
Theoremunielxp 5714 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
(A (B × 𝐶) → A (B × 𝐶))
 
Theorem1st2nd2 5715 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 5716 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 5717 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 5718 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 5719* 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) = Bx A = ⟨B, x⟩))
 
Theorem2nd1st 5720 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
(A (B × 𝐶) → {A} = ⟨(2ndA), (1stA)⟩)
 
Theorem1st2nd 5721 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 = ⟨(1stA), (2ndA)⟩)
 
Theorem1stdm 5722 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) → (1stA) dom 𝑅)
 
Theorem2ndrn 5723 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) → (2ndA) ran 𝑅)
 
Theorem1st2ndbr 5724 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) → (1stA)B(2ndA))
 
Theoremreleldm2 5725* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel A → (B dom Ax A (1stx) = B))
 
Theoremreldm 5726* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel A → dom A = ran (x A ↦ (1stx)))
 
Theoremsbcopeq1a 5727 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2744 that avoids the existential quantifiers of copsexg 3947). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(A = ⟨x, y⟩ → ([(1stA) / x][(2ndA) / y]φφ))
 
Theoremcsbopeq1a 5728 Equality theorem for substitution of a class A for an ordered pair x, y in B (analog of csbeq1a 2831). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(A = ⟨x, y⟩ → (1stA) / x(2ndA) / yB = B)
 
Theoremdfopab2 5729* 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 5730* 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 5731* 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 5732* 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⟩ ∣ ((x A y B) ψ)}
 
Theoremdfoprab4f 5733* 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⟩ ∣ ((x A y B) ψ)}
 
Theoremdfxp3 5734* Define the cross product of three classes. Compare df-xp 4269. (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)}
 
Theoremelopabi 5735* 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 5736* 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 5737* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
(x A, y B𝐶) = (z x A ({x} × B) ↦ (1stz) / x(2ndz) / y𝐶)
 
Theoremmpt2mpts 5738* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
(x A, y B𝐶) = (z (A × B) ↦ (1stz) / x(2ndz) / y𝐶)
 
Theoremdmmpt2ssx 5739* The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝐹 = (x A, y B𝐶)       dom 𝐹 x A ({x} × B)
 
Theoremfmpt2x 5740* 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 B𝐶)       (x A y B 𝐶 𝐷𝐹: x A ({x} × B)⟶𝐷)
 
Theoremfmpt2 5741* Functionality, domain and range of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
𝐹 = (x A, y B𝐶)       (x A y B 𝐶 𝐷𝐹:(A × B)⟶𝐷)
 
Theoremfnmpt2 5742* Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
𝐹 = (x A, y B𝐶)       (x A y B 𝐶 𝑉𝐹 Fn (A × B))
 
Theoremmpt2fvex 5743* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐹 = (x A, y B𝐶)       ((xy 𝐶 𝑉 𝑅 𝑊 𝑆 𝑋) → (𝑅𝐹𝑆) V)
 
Theoremfnmpt2i 5744* Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
𝐹 = (x A, y B𝐶)    &   𝐶 V       𝐹 Fn (A × B)
 
Theoremdmmpt2 5745* Domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
𝐹 = (x A, y B𝐶)    &   𝐶 V       dom 𝐹 = (A × B)
 
Theoremmpt2fvexi 5746* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐹 = (x A, y B𝐶)    &   𝐶 V    &   𝑅 V    &   𝑆 V       (𝑅𝐹𝑆) V
 
Theoremmpt2exxg 5747* Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)
𝐹 = (x A, y B𝐶)       ((A 𝑅 x A B 𝑆) → 𝐹 V)
 
Theoremmpt2exg 5748* 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 B𝐶)       ((A 𝑅 B 𝑆) → 𝐹 V)
 
Theoremmpt2exga 5749* 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.)
((A 𝑉 B 𝑊) → (x A, y B𝐶) V)
 
Theoremmpt2ex 5750* 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       (x A, y B𝐶) V
 
Theoremfmpt2co 5751* Composition of two functions. Variation of fmptco 5246 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
((φ (x A y B)) → 𝑅 𝐶)    &   (φ𝐹 = (x A, y B𝑅))    &   (φ𝐺 = (z 𝐶𝑆))    &   (z = 𝑅𝑆 = 𝑇)       (φ → (𝐺𝐹) = (x A, y B𝑇))
 
Theoremoprabco 5752* 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 B𝐶)    &   𝐺 = (x A, y B ↦ (𝐻𝐶))       (𝐻 Fn 𝐷𝐺 = (𝐻𝐹))
 
Theoremoprab2co 5753* 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 = (𝑀𝐹))
 
Theoremdf1st2 5754* 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 5755* 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 5756 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-ontoA)
 
Theorem2ndconst 5757 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-ontoB)
 
Theoremdfmpt2 5758* Alternate definition for the "maps to" notation df-mpt2 5432 (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 B𝐶) = x A y B {⟨⟨x, y⟩, 𝐶⟩}
 
Theoremcnvf1olem 5759 Lemma for cnvf1o 5760. (Contributed by Mario Carneiro, 27-Apr-2014.)
((Rel A (B A 𝐶 = {B})) → (𝐶 A B = {𝐶}))
 
Theoremcnvf1o 5760* 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}):A1-1-ontoA)
 
Theoremf2ndf 5761 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 5762 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 5763 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 5764 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)
 
Theoremxporderlem 5765* 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⟩ ↔ (((𝑎 A 𝑐 A) (𝑏 B 𝑑 B)) (𝑎𝑅𝑐 (𝑎 = 𝑐 𝑏𝑆𝑑))))
 
Theorempoxp 5766* 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 A 𝑆 Po B) → 𝑇 Po (A × B))
 
2.6.15  Special "Maps to" operations

The following theorems are about maps-to operations (see df-mpt2 5432) 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 5496, ovmpt2x 5543 and fmpt2x 5740). 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 5767* 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)       ((𝑉 𝑋 𝑊 𝑌) → (𝑁 (⟨𝑉, 𝑊𝐹𝐾) → 𝐾 𝑉))
 
Theoremmpt2xopoveq 5768* 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})       (((𝑉 𝑋 𝑊 𝑌) 𝐾 𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛 𝑉[𝑉, 𝑊⟩ / x][𝐾 / y]φ})
 
Theoremmpt2xopovel 5769* 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})       ((𝑉 𝑋 𝑊 𝑌) → (𝑁 (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾 𝑉 𝑁 𝑉 [𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ)))
 
Theoremsprmpt2 5770* 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) → (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)𝑝 ψ)})
 
Theoremisprmpt2 5771* Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(φ𝑀 = {⟨f, 𝑝⟩ ∣ (f𝑊𝑝 ψ)})    &   ((f = 𝐹 𝑝 = 𝑃) → (ψχ))       (φ → ((𝐹 𝑋 𝑃 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 χ))))
 
2.6.16  Function transposition
 
Syntaxctpos 5772 The transposition of a function.
class tpos 𝐹
 
Definitiondf-tpos 5773* 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 5774 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
 
Theoremtposeq 5775 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
 
Theoremtposeqd 5776 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
(φ𝐹 = 𝐺)       (φ → tpos 𝐹 = tpos 𝐺)
 
Theoremtposssxp 5777 The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
 
Theoremreltpos 5778 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
Rel tpos 𝐹
 
Theorembrtpos2 5779 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)))
 
Theorembrtpos0 5780 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))
 
Theoremreldmtpos 5781 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 5782 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, A𝐹𝐶))
 
Theoremottposg 5783 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))
 
Theoremdmtpos 5784 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 5785 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 5786 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹 𝑉 → tpos 𝐹 V)
 
Theoremovtposg 5787 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))
 
Theoremtposfun 5788 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Fun 𝐹 → Fun tpos 𝐹)
 
Theoremdftpos2 5789* 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})))
 
Theoremdftpos3 5790* Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 4271. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom 𝐹 → tpos 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ⟨y, x𝐹z})
 
Theoremdftpos4 5791* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos 𝐹 = (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x}))
 
Theoremtpostpos 5792 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 5793 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 5794 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
(Rel A → (𝐹 Fn A → tpos 𝐹 Fn A))
 
Theoremtposfo2 5795 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel A → (𝐹:AontoB → tpos 𝐹:AontoB))
 
Theoremtposf2 5796 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(Rel A → (𝐹:AB → tpos 𝐹:AB))
 
Theoremtposf12 5797 Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel A → (𝐹:A1-1B → tpos 𝐹:A1-1B))
 
Theoremtposf1o2 5798 Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel A → (𝐹:A1-1-ontoB → tpos 𝐹:A1-1-ontoB))
 
Theoremtposfo 5799 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(𝐹:(A × B)–onto𝐶 → tpos 𝐹:(B × A)–onto𝐶)
 
Theoremtposf 5800 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(𝐹:(A × B)⟶𝐶 → tpos 𝐹:(B × A)⟶𝐶)
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