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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 2ndval2 7701 | 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) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) | ||
Theorem | oteqimp 7702 | The components of an ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.) |
⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((1st ‘(1st ‘𝑇)) = 𝐴 ∧ (2nd ‘(1st ‘𝑇)) = 𝐵 ∧ (2nd ‘𝑇) = 𝐶))) | ||
Theorem | fo1st 7703 | 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 | ||
Theorem | fo2nd 7704 | 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 | ||
Theorem | br1steqg 7705 | Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉1st 𝐶 ↔ 𝐶 = 𝐴)) | ||
Theorem | br2ndeqg 7706 | Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉2nd 𝐶 ↔ 𝐶 = 𝐵)) | ||
Theorem | f1stres 7707 | 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 | ||
Theorem | f2ndres 7708 | 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 | ||
Theorem | fo1stres 7709 | Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.) |
⊢ (𝐵 ≠ ∅ → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴) | ||
Theorem | fo2ndres 7710 | Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.) |
⊢ (𝐴 ≠ ∅ → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵) | ||
Theorem | 1st2val 7711* | Value of an alternate definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴) | ||
Theorem | 2nd2val 7712* | Value of an alternate definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴) | ||
Theorem | 1stcof 7713 | Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.) |
⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹):𝐴⟶𝐵) | ||
Theorem | 2ndcof 7714 | Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.) |
⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹):𝐴⟶𝐶) | ||
Theorem | xp1st 7715 | Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵) | ||
Theorem | xp2nd 7716 | Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ 𝐶) | ||
Theorem | elxp6 7717 | Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7621. (Contributed by NM, 9-Oct-2004.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) | ||
Theorem | elxp7 7718 | Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7621. (Contributed by NM, 19-Aug-2006.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) | ||
Theorem | eqopi 7719 | Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.) |
⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = 〈𝐵, 𝐶〉) | ||
Theorem | xp2 7720* | Representation of Cartesian product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.) |
⊢ (𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ 𝐵)} | ||
Theorem | unielxp 7721 | The membership relation for a Cartesian product is inherited by union. (Contributed by NM, 16-Sep-2006.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶)) | ||
Theorem | 1st2nd2 7722 | Reconstruction of a member of a Cartesian product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | ||
Theorem | 1st2ndb 7723 | Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.) |
⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | ||
Theorem | xpopth 7724 | An ordered pair theorem for members of Cartesian products. (Contributed by NM, 20-Jun-2007.) |
⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)) ↔ 𝐴 = 𝐵)) | ||
Theorem | eqop 7725 | 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〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) | ||
Theorem | eqop2 7726 | 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) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) | ||
Theorem | op1steq 7727* | 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) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) | ||
Theorem | opreuopreu 7728* | There is a unique ordered pair fulfilling a wff iff its components fulfil a corresponding wff. (Contributed by AV, 2-Jul-2023.) |
⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → (𝜓 ↔ 𝜑)) ⇒ ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)∃𝑎∃𝑏(𝑝 = 〈𝑎, 𝑏〉 ∧ 𝜓)) | ||
Theorem | el2xptp 7729* | A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.) |
⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | ||
Theorem | el2xptp0 7730 | A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.) |
⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st ‘𝐴)) = 𝑋 ∧ (2nd ‘(1st ‘𝐴)) = 𝑌 ∧ (2nd ‘𝐴) = 𝑍)) ↔ 𝐴 = 〈𝑋, 𝑌, 𝑍〉)) | ||
Theorem | 2nd1st 7731 | Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) | ||
Theorem | 1st2nd 7732 | 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 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | ||
Theorem | 1stdm 7733 | 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) → (1st ‘𝐴) ∈ dom 𝑅) | ||
Theorem | 2ndrn 7734 | 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) → (2nd ‘𝐴) ∈ ran 𝑅) | ||
Theorem | 1st2ndbr 7735 | 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) → (1st ‘𝐴)𝐵(2nd ‘𝐴)) | ||
Theorem | releldm2 7736* | Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
⊢ (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵)) | ||
Theorem | reldm 7737* | An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))) | ||
Theorem | releldmdifi 7738* | One way of expressing membership in the difference of domains of two nested relations. (Contributed by AV, 26-Oct-2023.) |
⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶)) | ||
Theorem | funfv1st2nd 7739 | The function value for the first component of an ordered pair is the second component of the ordered pair. (Contributed by AV, 17-Oct-2023.) |
⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋)) | ||
Theorem | funelss 7740 | If the first component of an element of a function is in the domain of a subset of the function, the element is a member of this subset. (Contributed by AV, 27-Oct-2023.) |
⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐴) → ((1st ‘𝑋) ∈ dom 𝐵 → 𝑋 ∈ 𝐵)) | ||
Theorem | funeldmdif 7741* | Two ways of expressing membership in the difference of domains of two nested functions. (Contributed by AV, 27-Oct-2023.) |
⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶)) | ||
Theorem | sbcopeq1a 7742 | Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a 3783 that avoids the existential quantifiers of copsexg 5375). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 = 〈𝑥, 𝑦〉 → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑)) | ||
Theorem | csbopeq1a 7743 | Equality theorem for substitution of a class 𝐴 for an ordered pair 〈𝑥, 𝑦〉 in 𝐵 (analogue of csbeq1a 3897). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵) | ||
Theorem | dfopab2 7744* | 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) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑} | ||
Theorem | dfoprab3s 7745* | 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) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)} | ||
Theorem | dfoprab3 7746* | 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 7747* | 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 7748* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 20-Dec-2008.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (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 | opabex2 7749* | Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ V) | ||
Theorem | opabn1stprc 7750* | An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wff. (Contributed by AV, 27-Dec-2020.) |
⊢ (∃𝑦𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜑} ∉ V) | ||
Theorem | opiota 7751* | The property of a uniquely specified ordered pair. The proof uses properties of the ℩ description binder. (Contributed by Mario Carneiro, 21-May-2015.) |
⊢ 𝐼 = (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) & ⊢ 𝑋 = (1st ‘𝐼) & ⊢ 𝑌 = (2nd ‘𝐼) & ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐷 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝜒) ↔ (𝐶 = 𝑋 ∧ 𝐷 = 𝑌))) | ||
Theorem | cnvoprab 7752* | The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) (Proof shortened by Thierry Arnoux, 20-Feb-2022.) |
⊢ (𝑎 = 〈𝑥, 𝑦〉 → (𝜓 ↔ 𝜑)) & ⊢ (𝜓 → 𝑎 ∈ (V × V)) ⇒ ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑧, 𝑎〉 ∣ 𝜓} | ||
Theorem | dfxp3 7753* | Define the Cartesian product of three classes. Compare df-xp 5556. (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 7754* | A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.) |
⊢ (𝑥 = (1st ‘𝐴) → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = (2nd ‘𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝜒) | ||
Theorem | eloprabi 7755* | 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 ‘(1st ‘𝐴)) → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = (2nd ‘(1st ‘𝐴)) → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = (2nd ‘𝐴) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝐴 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → 𝜃) | ||
Theorem | mpomptsx 7756* | 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) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶) | ||
Theorem | mpompts 7757* | 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) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶) | ||
Theorem | dmmpossx 7758* | 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 | fmpox 7759* | 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 | fmpo 7760* | 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 | fnmpo 7761* | 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 | fnmpoi 7762* | 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 | dmmpo 7763* | 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 | ovmpoelrn 7764* | An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.) |
⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) ∈ 𝑀) | ||
Theorem | dmmpoga 7765* | Domain of an operation given by the maps-to notation, closed form of dmmpo 7763. (Contributed by Alexander van der Vekens, 10-Feb-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵)) | ||
Theorem | dmmpog 7766* | Domain of an operation given by the maps-to notation, closed form of dmmpo 7763. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵)) | ||
Theorem | mpoexxg 7767* | 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 | mpoexg 7768* | 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 | mpoexga 7769* | If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by NM, 12-Sep-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) | ||
Theorem | mpoexw 7770* | Weak version of mpoex 7771 that holds without ax-rep 5183. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐷 ∈ V & ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ⇒ ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V | ||
Theorem | mpoex 7771* | If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by Mario Carneiro, 20-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V | ||
Theorem | mptmpoopabbrd 7772* | The operation value of a function value of a collection of ordered pairs of elements related in two ways. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) |
⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺)) & ⊢ (𝜑 → {〈𝑓, ℎ〉 ∣ 𝜓} ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓) & ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝜏 ↔ 𝜃)) & ⊢ (𝑔 = 𝐺 → (𝜒 ↔ 𝜏)) & ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {〈𝑓, ℎ〉 ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)})) ⇒ ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {〈𝑓, ℎ〉 ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)}) | ||
Theorem | mptmpoopabovd 7773* | The operation value of a function value of a collection of ordered pairs of related elements (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) |
⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺)) & ⊢ (𝜑 → {〈𝑓, ℎ〉 ∣ 𝜓} ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓) & ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {〈𝑓, ℎ〉 ∣ (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ∧ 𝑓(𝐷‘𝑔)ℎ)})) ⇒ ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {〈𝑓, ℎ〉 ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)}) | ||
Theorem | el2mpocsbcl 7774* | If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. (Contributed by AV, 21-May-2021.) |
⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷)))) | ||
Theorem | el2mpocl 7775* | If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. Using implicit substitution. (Contributed by AV, 21-May-2021.) |
⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸)) & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐶 = 𝐹 ∧ 𝐷 = 𝐺)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺)))) | ||
Theorem | fnmpoovd 7776* | A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.) (Revised by AV, 3-Jul-2022.) |
⊢ (𝜑 → 𝑀 Fn (𝐴 × 𝐵)) & ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷)) | ||
Theorem | offval22 7777* | The function operation expressed as a mapping, variation of offval2 7420. (Contributed by SO, 15-Jul-2018.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑅𝐷))) | ||
Theorem | brovpreldm 7778 | If a binary relation holds for the result of an operation, the operands are in the domain of the operation. (Contributed by AV, 31-Dec-2020.) |
⊢ (𝐷(𝐵𝐴𝐶)𝐸 → 〈𝐵, 𝐶〉 ∈ dom 𝐴) | ||
Theorem | bropopvvv 7779* | If a binary relation holds for the result of an operation which is a result of an operation, the involved classes are sets. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Proof shortened by AV, 3-Jan-2021.) |
⊢ 𝑂 = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {〈𝑓, 𝑝〉 ∣ 𝜑})) & ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜑 ↔ 𝜓)) & ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉𝑂𝐸)𝐵) = {〈𝑓, 𝑝〉 ∣ 𝜃}) ⇒ ⊢ (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) | ||
Theorem | bropfvvvvlem 7780* | Lemma for bropfvvvv 7781. (Contributed by AV, 31-Dec-2020.) (Revised by AV, 16-Jan-2021.) |
⊢ 𝑂 = (𝑎 ∈ 𝑈 ↦ (𝑏 ∈ 𝑉, 𝑐 ∈ 𝑊 ↦ {〈𝑑, 𝑒〉 ∣ 𝜑})) & ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (𝐵(𝑂‘𝐴)𝐶) = {〈𝑑, 𝑒〉 ∣ 𝜃}) ⇒ ⊢ ((〈𝐵, 𝐶〉 ∈ (𝑆 × 𝑇) ∧ 𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸) → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))) | ||
Theorem | bropfvvvv 7781* | If a binary relation holds for the result of an operation which is a function value, the involved classes are sets. (Contributed by AV, 31-Dec-2020.) (Revised by AV, 16-Jan-2021.) |
⊢ 𝑂 = (𝑎 ∈ 𝑈 ↦ (𝑏 ∈ 𝑉, 𝑐 ∈ 𝑊 ↦ {〈𝑑, 𝑒〉 ∣ 𝜑})) & ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (𝐵(𝑂‘𝐴)𝐶) = {〈𝑑, 𝑒〉 ∣ 𝜃}) & ⊢ (𝑎 = 𝐴 → 𝑉 = 𝑆) & ⊢ (𝑎 = 𝐴 → 𝑊 = 𝑇) & ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝑆 ∈ 𝑋 ∧ 𝑇 ∈ 𝑌) → (𝐷(𝐵(𝑂‘𝐴)𝐶)𝐸 → (𝐴 ∈ 𝑈 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))) | ||
Theorem | ovmptss 7782* | If all the values of the mapping are subsets of a class 𝑋, then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋) | ||
Theorem | relmpoopab 7783* | Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈𝑧, 𝑤〉 ∣ 𝜑}) ⇒ ⊢ Rel (𝐶𝐹𝐷) | ||
Theorem | fmpoco 7784* | Composition of two functions. Variation of fmptco 6886 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 7785* | 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 7786* | 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 7787* | 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)) | ||
Theorem | df2nd2 7788* | 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)) | ||
Theorem | 1stconst 7789 | The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴) | ||
Theorem | 2ndconst 7790 | The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵) | ||
Theorem | dfmpo 7791* | Alternate definition for the maps-to notation df-mpo 7155 (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 | mposn 7792* | An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.) |
⊢ 𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶) & ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) & ⊢ (𝑦 = 𝐵 → 𝐷 = 𝐸) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → 𝐹 = {〈〈𝐴, 𝐵〉, 𝐸〉}) | ||
Theorem | curry1 7793* | Composition with ◡(2nd ↾ ({𝐶} × V)) turns any binary operation 𝐹 with a constant first operand into a function 𝐺 of the second operand only. This transformation is called "currying." (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.) |
⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ⇒ ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))) | ||
Theorem | curry1val 7794 | The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ⇒ ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷)) | ||
Theorem | curry1f 7795 | Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.) |
⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ⇒ ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺:𝐵⟶𝐷) | ||
Theorem | curry2 7796* | Composition with ◡(1st ↾ (V × {𝐶})) turns any binary operation 𝐹 with a constant second operand into a function 𝐺 of the first operand only. This transformation is called "currying." (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.) |
⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) ⇒ ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))) | ||
Theorem | curry2f 7797 | Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.) |
⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) ⇒ ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → 𝐺:𝐴⟶𝐷) | ||
Theorem | curry2val 7798 | The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.) |
⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) ⇒ ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = (𝐷𝐹𝐶)) | ||
Theorem | cnvf1olem 7799 | Lemma for cnvf1o 7800. (Contributed by Mario Carneiro, 27-Apr-2014.) |
⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ◡{𝐵})) → (𝐶 ∈ ◡𝐴 ∧ 𝐵 = ∪ ◡{𝐶})) | ||
Theorem | cnvf1o 7800* | 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) |
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