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Theorem List for Intuitionistic Logic Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1stval2 5701 Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
(A (V × V) → (1stA) = A)
 
Theorem2ndval2 5702 Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
(A (V × V) → (2ndA) = {A})
 
Theoremfo1st 5703 The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
1st :V–onto→V
 
Theoremfo2nd 5704 The 2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
2nd :V–onto→V
 
Theoremf1stres 5705 Mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(1st ↾ (A × B)):(A × B)⟶A
 
Theoremf2ndres 5706 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 5707* 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 5708* 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 5709 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
(𝐹:A⟶(B × 𝐶) → (1st𝐹):AB)
 
Theorem2ndcof 5710 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
(𝐹:A⟶(B × 𝐶) → (2nd𝐹):A𝐶)
 
Theoremxp1st 5711 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(A (B × 𝐶) → (1stA) B)
 
Theoremxp2nd 5712 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(A (B × 𝐶) → (2ndA) 𝐶)
 
Theorem1stexg 5713 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
(A 𝑉 → (1stA) V)
 
Theorem2ndexg 5714 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
(A 𝑉 → (2ndA) V)
 
Theoremelxp6 5715 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4731. (Contributed by NM, 9-Oct-2004.)
(A (B × 𝐶) ↔ (A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶)))
 
Theoremelxp7 5716 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4731. (Contributed by NM, 19-Aug-2006.)
(A (B × 𝐶) ↔ (A (V × V) ((1stA) B (2ndA) 𝐶)))
 
Theoremeqopi 5717 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 5718* 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 5719 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
(A (B × 𝐶) → A (B × 𝐶))
 
Theorem1st2nd2 5720 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 5721 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 5722 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 5723 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 5724* 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 5725 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
(A (B × 𝐶) → {A} = ⟨(2ndA), (1stA)⟩)
 
Theorem1st2nd 5726 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 5727 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 5728 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 5729 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 5730* 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 5731* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel A → dom A = ran (x A ↦ (1stx)))
 
Theoremsbcopeq1a 5732 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2746 that avoids the existential quantifiers of copsexg 3951). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(A = ⟨x, y⟩ → ([(1stA) / x][(2ndA) / y]φφ))
 
Theoremcsbopeq1a 5733 Equality theorem for substitution of a class A for an ordered pair x, y in B (analog of csbeq1a 2833). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(A = ⟨x, y⟩ → (1stA) / x(2ndA) / yB = B)
 
Theoremdfopab2 5734* 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 5735* 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 5736* 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 5737* 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 5738* 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 5739* Define the cross product of three classes. Compare df-xp 4274. (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 5740* 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 5741* 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 5742* 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 5743* 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 5744* 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 5745* 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 5746* 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 5747* 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 5748* 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 5749* 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 5750* 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 5751* 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 5752* 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 5753* 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 5754* 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 5755* 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 5756* Composition of two functions. Variation of fmptco 5251 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 5757* 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 5758* 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 5759* 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 5760* 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 5761 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 5762 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 5763* Alternate definition for the "maps to" notation df-mpt2 5437 (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 5764 Lemma for cnvf1o 5765. (Contributed by Mario Carneiro, 27-Apr-2014.)
((Rel A (B A 𝐶 = {B})) → (𝐶 A B = {𝐶}))
 
Theoremcnvf1o 5765* 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 5766 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 5767 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 5768 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 5769 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 5770* 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 5771* 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 5437) 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 5501, ovmpt2x 5548 and fmpt2x 5745). 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 5772* 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 5773* 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 5774* 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 5775* 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 5776* 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 5777 The transposition of a function.
class tpos 𝐹
 
Definitiondf-tpos 5778* 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 5779 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
 
Theoremtposeq 5780 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
 
Theoremtposeqd 5781 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
(φ𝐹 = 𝐺)       (φ → tpos 𝐹 = tpos 𝐺)
 
Theoremtposssxp 5782 The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
 
Theoremreltpos 5783 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
Rel tpos 𝐹
 
Theorembrtpos2 5784 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 5785 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 5786 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 5787 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 5788 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 5789 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 5790 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 5791 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹 𝑉 → tpos 𝐹 V)
 
Theoremovtposg 5792 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 5793 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Fun 𝐹 → Fun tpos 𝐹)
 
Theoremdftpos2 5794* 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 5795* Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 4276. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom 𝐹 → tpos 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ⟨y, x𝐹z})
 
Theoremdftpos4 5796* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos 𝐹 = (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x}))
 
Theoremtpostpos 5797 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 5798 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 5799 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
(Rel A → (𝐹 Fn A → tpos 𝐹 Fn A))
 
Theoremtposfo2 5800 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel A → (𝐹:AontoB → tpos 𝐹:AontoB))
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