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Type | Label | Description |
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Statement | ||
Theorem | 1stval2 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) → (1^{st} ‘A) = ∩ ∩ A) | ||
Theorem | 2ndval2 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) → (2^{nd} ‘A) = ∩ ∩ ∩ ^{◡}{A}) | ||
Theorem | fo1st 5703 | The 1^{st} function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ 1^{st} :V–onto→V | ||
Theorem | fo2nd 5704 | The 2^{nd} function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ 2^{nd} :V–onto→V | ||
Theorem | f1stres 5705 | Mapping of a restriction of the 1^{st} (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (1^{st} ↾ (A × B)):(A × B)⟶A | ||
Theorem | f2ndres 5706 | 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 5707* | 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 5708* | 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 5709 | 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 5710 | 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 5711 | 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 5712 | 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 5713 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
⊢ (A ∈ 𝑉 → (1^{st} ‘A) ∈ V) | ||
Theorem | 2ndexg 5714 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
⊢ (A ∈ 𝑉 → (2^{nd} ‘A) ∈ V) | ||
Theorem | elxp6 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 = ⟨(1^{st} ‘A), (2^{nd} ‘A)⟩ ∧ ((1^{st} ‘A) ∈ B ∧ (2^{nd} ‘A) ∈ 𝐶))) | ||
Theorem | elxp7 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) ∧ ((1^{st} ‘A) ∈ B ∧ (2^{nd} ‘A) ∈ 𝐶))) | ||
Theorem | eqopi 5717 | 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 5718* | 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 5719 | 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 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 = ⟨(1^{st} ‘A), (2^{nd} ‘A)⟩) | ||
Theorem | xpopth 5721 | 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 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⟩ ↔ ((1^{st} ‘A) = B ∧ (2^{nd} ‘A) = 𝐶))) | ||
Theorem | eqop2 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) ∧ ((1^{st} ‘A) = B ∧ (2^{nd} ‘A) = 𝐶))) | ||
Theorem | op1steq 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) → ((1^{st} ‘A) = B ↔ ∃x A = ⟨B, x⟩)) | ||
Theorem | 2nd1st 5725 | 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 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 = ⟨(1^{st} ‘A), (2^{nd} ‘A)⟩) | ||
Theorem | 1stdm 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) → (1^{st} ‘A) ∈ dom 𝑅) | ||
Theorem | 2ndrn 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) → (2^{nd} ‘A) ∈ ran 𝑅) | ||
Theorem | 1st2ndbr 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) → (1^{st} ‘A)B(2^{nd} ‘A)) | ||
Theorem | releldm2 5730* | 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 5731* | 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 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⟩ → ([(1^{st} ‘A) / x][(2^{nd} ‘A) / y]φ ↔ φ)) | ||
Theorem | csbopeq1a 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⟩ → ⦋(1^{st} ‘A) / x⦌⦋(2^{nd} ‘A) / y⦌B = B) | ||
Theorem | dfopab2 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) ∣ [(1^{st} ‘z) / x][(2^{nd} ‘z) / y]φ} | ||
Theorem | dfoprab3s 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) ∧ [(1^{st} ‘w) / x][(2^{nd} ‘w) / y]φ)} | ||
Theorem | dfoprab3 5736* | 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 5737* | 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 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.) |
⊢ Ⅎxφ & ⊢ Ⅎyφ & ⊢ (w = ⟨x, y⟩ → (φ ↔ ψ)) ⇒ ⊢ {⟨w, z⟩ ∣ (w ∈ (A × B) ∧ φ)} = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ ψ)} | ||
Theorem | dfxp3 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)} | ||
Theorem | elopabi 5740* | 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 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 = (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 5742* | 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 5743* | 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 5744* | 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 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 ∈ B ↦ 𝐶) ⇒ ⊢ (∀x ∈ A ∀y ∈ B 𝐶 ∈ 𝐷 ↔ 𝐹:∪ x ∈ A ({x} × B)⟶𝐷) | ||
Theorem | fmpt2 5746* | 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 5747* | 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 5748* | 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 5749* | 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 5750* | 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 5751* | 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 5752* | 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 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 ∈ B ↦ 𝐶) ⇒ ⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆) → 𝐹 ∈ V) | ||
Theorem | mpt2exga 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 ∈ B ↦ 𝐶) ∈ V) | ||
Theorem | mpt2ex 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 ∈ B ↦ 𝐶) ∈ V | ||
Theorem | fmpt2co 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) & ⊢ (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 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 ∈ B ↦ 𝐶) & ⊢ 𝐺 = (x ∈ A, y ∈ B ↦ (𝐻‘𝐶)) ⇒ ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹)) | ||
Theorem | oprab2co 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 = (𝑀 ∘ 𝐹)) | ||
Theorem | df1st2 5759* | 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 5760* | 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 5761 | 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 5762 | 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 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 ∈ B ↦ 𝐶) = ∪ x ∈ A ∪ y ∈ B {⟨⟨x, y⟩, 𝐶⟩} | ||
Theorem | cnvf1olem 5764 | Lemma for cnvf1o 5765. (Contributed by Mario Carneiro, 27-Apr-2014.) |
⊢ ((Rel A ∧ (B ∈ A ∧ 𝐶 = ∪ ^{◡}{B})) → (𝐶 ∈ ^{◡}A ∧ B = ∪ ^{◡}{𝐶})) | ||
Theorem | cnvf1o 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}):A–1-1-onto→^{◡}A) | ||
Theorem | f2ndf 5766 | 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 5767 | 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 5768 | 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 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 ∘ 1^{st} )𝐶) = (𝐹‘B) | ||
Theorem | xporderlem 5770* | 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 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)) ∧ ((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 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. | ||
Theorem | mpt2xopn0yelv 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 ∈ (1^{st} ‘x) ↦ 𝐶) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉)) | ||
Theorem | mpt2xopoveq 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 ∈ (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 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 ∈ (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 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) → (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 5776* | 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 5777 | The transposition of a function. |
class tpos 𝐹 | ||
Definition | df-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})) | ||
Theorem | tposss 5779 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺) | ||
Theorem | tposeq 5780 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺) | ||
Theorem | tposeqd 5781 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ (φ → 𝐹 = 𝐺) ⇒ ⊢ (φ → tpos 𝐹 = tpos 𝐺) | ||
Theorem | tposssxp 5782 | The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ tpos 𝐹 ⊆ ((^{◡}dom 𝐹 ∪ {∅}) × ran 𝐹) | ||
Theorem | reltpos 5783 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ Rel tpos 𝐹 | ||
Theorem | brtpos2 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))) | ||
Theorem | brtpos0 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)) | ||
Theorem | reldmtpos 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) | ||
Theorem | brtposg 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, A⟩𝐹𝐶)) | ||
Theorem | ottposg 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)) | ||
Theorem | dmtpos 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) | ||
Theorem | rntpos 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) | ||
Theorem | tposexg 5791 | The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V) | ||
Theorem | ovtposg 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)) | ||
Theorem | tposfun 5793 | The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (Fun 𝐹 → Fun tpos 𝐹) | ||
Theorem | dftpos2 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}))) | ||
Theorem | dftpos3 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, x⟩𝐹z}) | ||
Theorem | dftpos4 5796* | Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ tpos 𝐹 = (𝐹 ∘ (x ∈ ((V × V) ∪ {∅}) ↦ ∪ ^{◡}{x})) | ||
Theorem | tpostpos 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)) | ||
Theorem | tpostpos2 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) | ||
Theorem | tposfn2 5799 | The domain of a transposition. (Contributed by NM, 10-Sep-2015.) |
⊢ (Rel A → (𝐹 Fn A → tpos 𝐹 Fn ^{◡}A)) | ||
Theorem | tposfo2 5800 | Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
⊢ (Rel A → (𝐹:A–onto→B → tpos 𝐹:^{◡}A–onto→B)) |
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