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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | f2ndres 5701 | Mapping of a restriction of the 2^{nd} (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (2^{nd} ↾ (A × B)):(A × B)⟶B | ||
Theorem | fo1stresm 5702* | Onto mapping of a restriction of the 1^{st} (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) |
⊢ (∃y y ∈ B → (1^{st} ↾ (A × B)):(A × B)–onto→A) | ||
Theorem | fo2ndresm 5703* | Onto mapping of a restriction of the 2^{nd} (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) |
⊢ (∃x x ∈ A → (2^{nd} ↾ (A × B)):(A × B)–onto→B) | ||
Theorem | 1stcof 5704 | Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.) |
⊢ (𝐹:A⟶(B × 𝐶) → (1^{st} ∘ 𝐹):A⟶B) | ||
Theorem | 2ndcof 5705 | Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.) |
⊢ (𝐹:A⟶(B × 𝐶) → (2^{nd} ∘ 𝐹):A⟶𝐶) | ||
Theorem | xp1st 5706 | Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (A ∈ (B × 𝐶) → (1^{st} ‘A) ∈ B) | ||
Theorem | xp2nd 5707 | Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (A ∈ (B × 𝐶) → (2^{nd} ‘A) ∈ 𝐶) | ||
Theorem | 1stexg 5708 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
⊢ (A ∈ 𝑉 → (1^{st} ‘A) ∈ V) | ||
Theorem | 2ndexg 5709 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
⊢ (A ∈ 𝑉 → (2^{nd} ‘A) ∈ V) | ||
Theorem | elxp6 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 = ⟨(1^{st} ‘A), (2^{nd} ‘A)⟩ ∧ ((1^{st} ‘A) ∈ B ∧ (2^{nd} ‘A) ∈ 𝐶))) | ||
Theorem | elxp7 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) ∧ ((1^{st} ‘A) ∈ B ∧ (2^{nd} ‘A) ∈ 𝐶))) | ||
Theorem | eqopi 5712 | Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.) |
⊢ ((A ∈ (𝑉 × 𝑊) ∧ ((1^{st} ‘A) = B ∧ (2^{nd} ‘A) = 𝐶)) → A = ⟨B, 𝐶⟩) | ||
Theorem | xp2 5713* | Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.) |
⊢ (A × B) = {x ∈ (V × V) ∣ ((1^{st} ‘x) ∈ A ∧ (2^{nd} ‘x) ∈ B)} | ||
Theorem | unielxp 5714 | 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 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 = ⟨(1^{st} ‘A), (2^{nd} ‘A)⟩) | ||
Theorem | xpopth 5716 | An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.) |
⊢ ((A ∈ (𝐶 × 𝐷) ∧ B ∈ (𝑅 × 𝑆)) → (((1^{st} ‘A) = (1^{st} ‘B) ∧ (2^{nd} ‘A) = (2^{nd} ‘B)) ↔ A = B)) | ||
Theorem | eqop 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⟩ ↔ ((1^{st} ‘A) = B ∧ (2^{nd} ‘A) = 𝐶))) | ||
Theorem | eqop2 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) ∧ ((1^{st} ‘A) = B ∧ (2^{nd} ‘A) = 𝐶))) | ||
Theorem | op1steq 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) → ((1^{st} ‘A) = B ↔ ∃x A = ⟨B, x⟩)) | ||
Theorem | 2nd1st 5720 | Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.) |
⊢ (A ∈ (B × 𝐶) → ∪ ^{◡}{A} = ⟨(2^{nd} ‘A), (1^{st} ‘A)⟩) | ||
Theorem | 1st2nd 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 = ⟨(1^{st} ‘A), (2^{nd} ‘A)⟩) | ||
Theorem | 1stdm 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) → (1^{st} ‘A) ∈ dom 𝑅) | ||
Theorem | 2ndrn 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) → (2^{nd} ‘A) ∈ ran 𝑅) | ||
Theorem | 1st2ndbr 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) → (1^{st} ‘A)B(2^{nd} ‘A)) | ||
Theorem | releldm2 5725* | Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
⊢ (Rel A → (B ∈ dom A ↔ ∃x ∈ A (1^{st} ‘x) = B)) | ||
Theorem | reldm 5726* | An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
⊢ (Rel A → dom A = ran (x ∈ A ↦ (1^{st} ‘x))) | ||
Theorem | sbcopeq1a 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⟩ → ([(1^{st} ‘A) / x][(2^{nd} ‘A) / y]φ ↔ φ)) | ||
Theorem | csbopeq1a 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⟩ → ⦋(1^{st} ‘A) / x⦌⦋(2^{nd} ‘A) / y⦌B = B) | ||
Theorem | dfopab2 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) ∣ [(1^{st} ‘z) / x][(2^{nd} ‘z) / y]φ} | ||
Theorem | dfoprab3s 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) ∧ [(1^{st} ‘w) / x][(2^{nd} ‘w) / y]φ)} | ||
Theorem | dfoprab3 5731* | 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 5732* | 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 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.) |
⊢ Ⅎxφ & ⊢ Ⅎyφ & ⊢ (w = ⟨x, y⟩ → (φ ↔ ψ)) ⇒ ⊢ {⟨w, z⟩ ∣ (w ∈ (A × B) ∧ φ)} = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ ψ)} | ||
Theorem | dfxp3 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)} | ||
Theorem | elopabi 5735* | A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.) |
⊢ (x = (1^{st} ‘A) → (φ ↔ ψ)) & ⊢ (y = (2^{nd} ‘A) → (ψ ↔ χ)) ⇒ ⊢ (A ∈ {⟨x, y⟩ ∣ φ} → χ) | ||
Theorem | eloprabi 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 = (1^{st} ‘(1^{st} ‘A)) → (φ ↔ ψ)) & ⊢ (y = (2^{nd} ‘(1^{st} ‘A)) → (ψ ↔ χ)) & ⊢ (z = (2^{nd} ‘A) → (χ ↔ θ)) ⇒ ⊢ (A ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} → θ) | ||
Theorem | mpt2mptsx 5737* | 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) ↦ ⦋(1^{st} ‘z) / x⦌⦋(2^{nd} ‘z) / y⦌𝐶) | ||
Theorem | mpt2mpts 5738* | 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) ↦ ⦋(1^{st} ‘z) / x⦌⦋(2^{nd} ‘z) / y⦌𝐶) | ||
Theorem | dmmpt2ssx 5739* | 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 | fmpt2x 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 ∈ B ↦ 𝐶) ⇒ ⊢ (∀x ∈ A ∀y ∈ B 𝐶 ∈ 𝐷 ↔ 𝐹:∪ x ∈ A ({x} × B)⟶𝐷) | ||
Theorem | fmpt2 5741* | 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 | fnmpt2 5742* | 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 5743* | 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 | fnmpt2i 5744* | 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 | dmmpt2 5745* | 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 5746* | 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 | mpt2exxg 5747* | 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 | mpt2exg 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 ∈ B ↦ 𝐶) ⇒ ⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆) → 𝐹 ∈ V) | ||
Theorem | mpt2exga 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 ∈ B ↦ 𝐶) ∈ V) | ||
Theorem | mpt2ex 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 ∈ B ↦ 𝐶) ∈ V | ||
Theorem | fmpt2co 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) & ⊢ (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 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 ∈ B ↦ 𝐶) & ⊢ 𝐺 = (x ∈ A, y ∈ B ↦ (𝐻‘𝐶)) ⇒ ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹)) | ||
Theorem | oprab2co 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 = (𝑀 ∘ 𝐹)) | ||
Theorem | df1st2 5754* | An alternate possible definition of the 1^{st} function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ {⟨⟨x, y⟩, z⟩ ∣ z = x} = (1^{st} ↾ (V × V)) | ||
Theorem | df2nd2 5755* | An alternate possible definition of the 2^{nd} function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ {⟨⟨x, y⟩, z⟩ ∣ z = y} = (2^{nd} ↾ (V × V)) | ||
Theorem | 1stconst 5756 | The mapping of a restriction of the 1^{st} function to a constant function. (Contributed by NM, 14-Dec-2008.) |
⊢ (B ∈ 𝑉 → (1^{st} ↾ (A × {B})):(A × {B})–1-1-onto→A) | ||
Theorem | 2ndconst 5757 | The mapping of a restriction of the 2^{nd} function to a converse constant function. (Contributed by NM, 27-Mar-2008.) |
⊢ (A ∈ 𝑉 → (2^{nd} ↾ ({A} × B)):({A} × B)–1-1-onto→B) | ||
Theorem | dfmpt2 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 ∈ B ↦ 𝐶) = ∪ x ∈ A ∪ y ∈ B {⟨⟨x, y⟩, 𝐶⟩} | ||
Theorem | cnvf1olem 5759 | Lemma for cnvf1o 5760. (Contributed by Mario Carneiro, 27-Apr-2014.) |
⊢ ((Rel A ∧ (B ∈ A ∧ 𝐶 = ∪ ^{◡}{B})) → (𝐶 ∈ ^{◡}A ∧ B = ∪ ^{◡}{𝐶})) | ||
Theorem | cnvf1o 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}):A–1-1-onto→^{◡}A) | ||
Theorem | f2ndf 5761 | The 2^{nd} (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:A⟶B → (2^{nd} ↾ 𝐹):𝐹⟶B) | ||
Theorem | fo2ndf 5762 | The 2^{nd} (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:A⟶B → (2^{nd} ↾ 𝐹):𝐹–onto→ran 𝐹) | ||
Theorem | f1o2ndf1 5763 | The 2^{nd} (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:A–1-1→B → (2^{nd} ↾ 𝐹):𝐹–1-1-onto→ran 𝐹) | ||
Theorem | algrflem 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 ∘ 1^{st} )𝐶) = (𝐹‘B) | ||
Theorem | xporderlem 5765* | Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.) |
⊢ 𝑇 = {⟨x, y⟩ ∣ ((x ∈ (A × B) ∧ y ∈ (A × B)) ∧ ((1^{st} ‘x)𝑅(1^{st} ‘y) ∨ ((1^{st} ‘x) = (1^{st} ‘y) ∧ (2^{nd} ‘x)𝑆(2^{nd} ‘y))))} ⇒ ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ (((𝑎 ∈ A ∧ 𝑐 ∈ A) ∧ (𝑏 ∈ B ∧ 𝑑 ∈ B)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑)))) | ||
Theorem | poxp 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)) ∧ ((1^{st} ‘x)𝑅(1^{st} ‘y) ∨ ((1^{st} ‘x) = (1^{st} ‘y) ∧ (2^{nd} ‘x)𝑆(2^{nd} ‘y))))} ⇒ ⊢ ((𝑅 Po A ∧ 𝑆 Po B) → 𝑇 Po (A × B)) | ||
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. | ||
Theorem | mpt2xopn0yelv 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 ∈ (1^{st} ‘x) ↦ 𝐶) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉)) | ||
Theorem | mpt2xopoveq 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 ∈ (1^{st} ‘x) ↦ {𝑛 ∈ (1^{st} ‘x) ∣ φ}) ⇒ ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / x][𝐾 / y]φ}) | ||
Theorem | mpt2xopovel 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 ∈ (1^{st} ‘x) ↦ {𝑛 ∈ (1^{st} ‘x) ∣ φ}) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ))) | ||
Theorem | sprmpt2 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) → (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 5771* | 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)))) | ||
Syntax | ctpos 5772 | The transposition of a function. |
class tpos 𝐹 | ||
Definition | df-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})) | ||
Theorem | tposss 5774 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺) | ||
Theorem | tposeq 5775 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺) | ||
Theorem | tposeqd 5776 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ (φ → 𝐹 = 𝐺) ⇒ ⊢ (φ → tpos 𝐹 = tpos 𝐺) | ||
Theorem | tposssxp 5777 | The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ tpos 𝐹 ⊆ ((^{◡}dom 𝐹 ∪ {∅}) × ran 𝐹) | ||
Theorem | reltpos 5778 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ Rel tpos 𝐹 | ||
Theorem | brtpos2 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))) | ||
Theorem | brtpos0 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)) | ||
Theorem | reldmtpos 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) | ||
Theorem | brtposg 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, A⟩𝐹𝐶)) | ||
Theorem | ottposg 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)) | ||
Theorem | dmtpos 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) | ||
Theorem | rntpos 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) | ||
Theorem | tposexg 5786 | The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V) | ||
Theorem | ovtposg 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)) | ||
Theorem | tposfun 5788 | The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (Fun 𝐹 → Fun tpos 𝐹) | ||
Theorem | dftpos2 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}))) | ||
Theorem | dftpos3 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, x⟩𝐹z}) | ||
Theorem | dftpos4 5791* | Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ tpos 𝐹 = (𝐹 ∘ (x ∈ ((V × V) ∪ {∅}) ↦ ∪ ^{◡}{x})) | ||
Theorem | tpostpos 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)) | ||
Theorem | tpostpos2 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) | ||
Theorem | tposfn2 5794 | The domain of a transposition. (Contributed by NM, 10-Sep-2015.) |
⊢ (Rel A → (𝐹 Fn A → tpos 𝐹 Fn ^{◡}A)) | ||
Theorem | tposfo2 5795 | Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
⊢ (Rel A → (𝐹:A–onto→B → tpos 𝐹:^{◡}A–onto→B)) | ||
Theorem | tposf2 5796 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
⊢ (Rel A → (𝐹:A⟶B → tpos 𝐹:^{◡}A⟶B)) | ||
Theorem | tposf12 5797 | 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 5798 | 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 5799 | 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 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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