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Type | Label | Description |
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Statement | ||
Theorem | ab2rexex2 5701* | Existence of an existentially restricted class abstraction. φ normally has free-variable parameters x, y, and z. Compare abrexex2 5693. (Contributed by NM, 20-Sep-2011.) |
⊢ A ∈ V & ⊢ B ∈ V & ⊢ {z ∣ φ} ∈ V ⇒ ⊢ {z ∣ ∃x ∈ A ∃y ∈ B φ} ∈ V | ||
Theorem | xpexgALT 5702 | The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4395 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (A × B) ∈ V) | ||
Theorem | offval3 5703* | General value of (𝐹 ∘_{𝑓} 𝑅𝐺) with no assumptions on functionality of 𝐹 and 𝐺. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘_{𝑓} 𝑅𝐺) = (x ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘x)𝑅(𝐺‘x)))) | ||
Theorem | offres 5704 | Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘_{𝑓} 𝑅𝐺) ↾ 𝐷) = ((𝐹 ↾ 𝐷) ∘_{𝑓} 𝑅(𝐺 ↾ 𝐷))) | ||
Theorem | ofmres 5705* | Equivalent expressions for a restriction of the function operation map. Unlike ∘_{𝑓} 𝑅 which is a proper class, ( ∘_{𝑓} 𝑅 ∣ ‘(A × B)) can be a set by ofmresex 5706, allowing it to be used as a function or structure argument. By ofmresval 5665, the restricted operation map values are the same as the original values, allowing theorems for ∘_{𝑓} 𝑅 to be reused. (Contributed by NM, 20-Oct-2014.) |
⊢ ( ∘_{𝑓} 𝑅 ↾ (A × B)) = (f ∈ A, g ∈ B ↦ (f ∘_{𝑓} 𝑅g)) | ||
Theorem | ofmresex 5706 | Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.) |
⊢ (φ → A ∈ 𝑉) & ⊢ (φ → B ∈ 𝑊) ⇒ ⊢ (φ → ( ∘_{𝑓} 𝑅 ↾ (A × B)) ∈ V) | ||
Syntax | c1st 5707 | Extend the definition of a class to include the first member an ordered pair function. |
class 1^{st} | ||
Syntax | c2nd 5708 | Extend the definition of a class to include the second member an ordered pair function. |
class 2^{nd} | ||
Definition | df-1st 5709 | Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 5715 proves that it does this. For example, (1^{st} ‘⟨ 3 , 4 ⟩) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 4745 and op1stb 4175). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.) |
⊢ 1^{st} = (x ∈ V ↦ ∪ dom {x}) | ||
Definition | df-2nd 5710 | Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 5716 proves that it does this. For example, (2^{nd} ‘⟨ 3 , 4 ⟩) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 4748 and op2ndb 4747). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.) |
⊢ 2^{nd} = (x ∈ V ↦ ∪ ran {x}) | ||
Theorem | 1stvalg 5711 | The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (A ∈ V → (1^{st} ‘A) = ∪ dom {A}) | ||
Theorem | 2ndvalg 5712 | The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (A ∈ V → (2^{nd} ‘A) = ∪ ran {A}) | ||
Theorem | 1st0 5713 | The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.) |
⊢ (1^{st} ‘∅) = ∅ | ||
Theorem | 2nd0 5714 | The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.) |
⊢ (2^{nd} ‘∅) = ∅ | ||
Theorem | op1st 5715 | Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.) |
⊢ A ∈ V & ⊢ B ∈ V ⇒ ⊢ (1^{st} ‘⟨A, B⟩) = A | ||
Theorem | op2nd 5716 | Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.) |
⊢ A ∈ V & ⊢ B ∈ V ⇒ ⊢ (2^{nd} ‘⟨A, B⟩) = B | ||
Theorem | op1std 5717 | Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ A ∈ V & ⊢ B ∈ V ⇒ ⊢ (𝐶 = ⟨A, B⟩ → (1^{st} ‘𝐶) = A) | ||
Theorem | op2ndd 5718 | Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ A ∈ V & ⊢ B ∈ V ⇒ ⊢ (𝐶 = ⟨A, B⟩ → (2^{nd} ‘𝐶) = B) | ||
Theorem | op1stg 5719 | Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (1^{st} ‘⟨A, B⟩) = A) | ||
Theorem | op2ndg 5720 | Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (2^{nd} ‘⟨A, B⟩) = B) | ||
Theorem | ot1stg 5721 | Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 5721, ot2ndg 5722, ot3rdgg 5723.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.) |
⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1^{st} ‘(1^{st} ‘⟨A, B, 𝐶⟩)) = A) | ||
Theorem | ot2ndg 5722 | Extract the second member of an ordered triple. (See ot1stg 5721 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.) |
⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2^{nd} ‘(1^{st} ‘⟨A, B, 𝐶⟩)) = B) | ||
Theorem | ot3rdgg 5723 | Extract the third member of an ordered triple. (See ot1stg 5721 comment.) (Contributed by NM, 3-Apr-2015.) |
⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2^{nd} ‘⟨A, B, 𝐶⟩) = 𝐶) | ||
Theorem | 1stval2 5724 | 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 5725 | 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 5726 | 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 5727 | 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 5728 | 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 5729 | 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 5730* | 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 5731* | 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 5732 | 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 5733 | 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 5734 | 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 5735 | 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 5736 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
⊢ (A ∈ 𝑉 → (1^{st} ‘A) ∈ V) | ||
Theorem | 2ndexg 5737 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
⊢ (A ∈ 𝑉 → (2^{nd} ‘A) ∈ V) | ||
Theorem | elxp6 5738 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4751. (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 5739 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4751. (Contributed by NM, 19-Aug-2006.) |
⊢ (A ∈ (B × 𝐶) ↔ (A ∈ (V × V) ∧ ((1^{st} ‘A) ∈ B ∧ (2^{nd} ‘A) ∈ 𝐶))) | ||
Theorem | eqopi 5740 | 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 5741* | 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 5742 | 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 5743 | 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 5744 | 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 5745 | 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 5746 | 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 5747* | 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 5748 | 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 5749 | 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 5750 | 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 5751 | 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 5752 | 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 5753* | 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 5754* | 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 5755 | Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2767 that avoids the existential quantifiers of copsexg 3972). (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 5756 | Equality theorem for substitution of a class A for an ordered pair ⟨x, y⟩ in B (analog of csbeq1a 2854). (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 5757* | 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 5758* | 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 5759* | 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 5760* | 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 5761* | 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 5762* | Define the cross product of three classes. Compare df-xp 4294. (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 5763* | 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 5764* | 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 5765* | 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 5766* | 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 5767* | 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 5768* | 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 5769* | 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 5770* | 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 5771* | 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 5772* | 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 5773* | 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 5774* | 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 5775* | 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 5776* | 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 5777* | 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 5778* | 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 5779* | Composition of two functions. Variation of fmptco 5273 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 5780* | 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 5781* | 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 5782* | 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 5783* | 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 5784 | 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 5785 | 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 5786* | Alternate definition for the "maps to" notation df-mpt2 5460 (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 5787 | Lemma for cnvf1o 5788. (Contributed by Mario Carneiro, 27-Apr-2014.) |
⊢ ((Rel A ∧ (B ∈ A ∧ 𝐶 = ∪ ^{◡}{B})) → (𝐶 ∈ ^{◡}A ∧ B = ∪ ^{◡}{𝐶})) | ||
Theorem | cnvf1o 5788* | 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 5789 | 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 5790 | 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 5791 | 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 5792 | 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 5793* | 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 5794* | 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 5460) 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 5524, ovmpt2x 5571 and fmpt2x 5768). 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 5795* | 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 5796* | 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 5797* | 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 5798* | 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 5799* | 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 5800 | The transposition of a function. |
class tpos 𝐹 |
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