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Theorem List for Metamath Proof Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem1st2val 7701* 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) = (1st𝐴)

Theorem2nd2val 7702* 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) = (2nd𝐴)

Theorem1stcof 7703 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
(𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹):𝐴𝐵)

Theorem2ndcof 7704 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
(𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹):𝐴𝐶)

Theoremxp1st 7705 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 ∈ (𝐵 × 𝐶) → (1st𝐴) ∈ 𝐵)

Theoremxp2nd 7706 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 ∈ (𝐵 × 𝐶) → (2nd𝐴) ∈ 𝐶)

Theoremelxp6 7707 Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7611. (Contributed by NM, 9-Oct-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))

Theoremelxp7 7708 Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7611. (Contributed by NM, 19-Aug-2006.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))

Theoremeqopi 7709 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)

Theoremxp2 7710* Representation of Cartesian product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
(𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ 𝐵)}

Theoremunielxp 7711 The membership relation for a Cartesian product is inherited by union. (Contributed by NM, 16-Sep-2006.)
(𝐴 ∈ (𝐵 × 𝐶) → 𝐴 (𝐵 × 𝐶))

Theorem1st2nd2 7712 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 = ⟨(1st𝐴), (2nd𝐴)⟩)

Theorem1st2ndb 7713 Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.)
(𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)

Theoremxpopth 7714 An ordered pair theorem for members of Cartesian products. (Contributed by NM, 20-Jun-2007.)
((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))

Theoremeqop 7715 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
(𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)))

Theoremeqop2 7716 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐴 = ⟨𝐵, 𝐶⟩ ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)))

Theoremop1steq 7717* Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)
(𝐴 ∈ (𝑉 × 𝑊) → ((1st𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))

Theoremopreuopreu 7718* There is a unique ordered pair fulfilling a wff iff its components fulfil a corresponding wff. (Contributed by AV, 2-Jul-2023.)
((𝑎 = (1st𝑝) ∧ 𝑏 = (2nd𝑝)) → (𝜓𝜑))       (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))

Theoremel2xptp 7719* A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
(𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)

Theoremel2xptp0 7720 A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
((𝑋𝑈𝑌𝑉𝑍𝑊) → ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍)) ↔ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩))

Theorem2nd1st 7721 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
(𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = ⟨(2nd𝐴), (1st𝐴)⟩)

Theorem1st2nd 7722 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
((Rel 𝐵𝐴𝐵) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)

Theorem1stdm 7723 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 𝑅𝐴𝑅) → (1st𝐴) ∈ dom 𝑅)

Theorem2ndrn 7724 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 𝑅𝐴𝑅) → (2nd𝐴) ∈ ran 𝑅)

Theorem1st2ndbr 7725 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
((Rel 𝐵𝐴𝐵) → (1st𝐴)𝐵(2nd𝐴))

Theoremreleldm2 7726* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))

Theoremreldm 7727* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel 𝐴 → dom 𝐴 = ran (𝑥𝐴 ↦ (1st𝑥)))

Theoremreleldmdifi 7728* One way of expressing membership in the difference of domains of two nested relations. (Contributed by AV, 26-Oct-2023.)
((Rel 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))

Theoremfunfv1st2nd 7729 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 𝐹𝑋𝐹) → (𝐹‘(1st𝑋)) = (2nd𝑋))

Theoremfunelss 7730 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 𝐴𝐵𝐴𝑋𝐴) → ((1st𝑋) ∈ dom 𝐵𝑋𝐵))

Theoremfuneldmdif 7731* Two ways of expressing membership in the difference of domains of two nested functions. (Contributed by AV, 27-Oct-2023.)
((Fun 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))

Theoremsbcopeq1a 7732 Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a 3731 that avoids the existential quantifiers of copsexg 5347). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st𝐴) / 𝑥][(2nd𝐴) / 𝑦]𝜑𝜑))

Theoremcsbopeq1a 7733 Equality theorem for substitution of a class 𝐴 for an ordered pair 𝑥, 𝑦 in 𝐵 (analogue of csbeq1a 3842). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵 = 𝐵)

Theoremdfopab2 7734* A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑}

Theoremdfoprab3s 7735* A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑)}

Theoremdfoprab3 7736* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
(𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Theoremdfoprab4 7737* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}

Theoremdfoprab4f 7738* 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.)
𝑥𝜑    &   𝑦𝜑    &   (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}

Theoremopabex2 7739* Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑𝜓) → 𝑥𝐴)    &   ((𝜑𝜓) → 𝑦𝐵)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)

Theoremopabn1stprc 7740* 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)

Theoremopiota 7741* The property of a uniquely specified ordered pair. The proof uses properties of the description binder. (Contributed by Mario Carneiro, 21-May-2015.)
𝐼 = (℩𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))    &   𝑋 = (1st𝐼)    &   𝑌 = (2nd𝐼)    &   (𝑥 = 𝐶 → (𝜑𝜓))    &   (𝑦 = 𝐷 → (𝜓𝜒))       (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ((𝐶𝐴𝐷𝐵𝜒) ↔ (𝐶 = 𝑋𝐷 = 𝑌)))

Theoremcnvoprab 7742* 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⟩ → (𝜓𝜑))    &   (𝜓𝑎 ∈ (V × V))       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Theoremdfxp3 7743* Define the Cartesian product of three classes. Compare df-xp 5525. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
((𝐴 × 𝐵) × 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥𝐴𝑦𝐵𝑧𝐶)}

Theoremelopabi 7744* A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
(𝑥 = (1st𝐴) → (𝜑𝜓))    &   (𝑦 = (2nd𝐴) → (𝜓𝜒))       (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)

Theoremeloprabi 7745* A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
(𝑥 = (1st ‘(1st𝐴)) → (𝜑𝜓))    &   (𝑦 = (2nd ‘(1st𝐴)) → (𝜓𝜒))    &   (𝑧 = (2nd𝐴) → (𝜒𝜃))       (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)

Theoremmpomptsx 7746* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)

Theoremmpompts 7747* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
(𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)

Theoremdmmpossx 7748* The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       dom 𝐹 𝑥𝐴 ({𝑥} × 𝐵)

Theoremfmpox 7749* 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 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷)

Theoremfmpo 7750* Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹:(𝐴 × 𝐵)⟶𝐷)

Theoremfnmpo 7751* Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝑉𝐹 Fn (𝐴 × 𝐵))

Theoremfnmpoi 7752* Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐶 ∈ V       𝐹 Fn (𝐴 × 𝐵)

Theoremdmmpo 7753* Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐶 ∈ V       dom 𝐹 = (𝐴 × 𝐵)

Theoremovmpoelrn 7754* An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.)
𝑂 = (𝑥𝐴, 𝑦𝐵𝐶)       ((∀𝑥𝐴𝑦𝐵 𝐶𝑀𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) ∈ 𝑀)

Theoremdmmpoga 7755* Domain of an operation given by the maps-to notation, closed form of dmmpo 7753. (Contributed by Alexander van der Vekens, 10-Feb-2019.) (Proof shortened by Lammen, 29-May-2024.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))

TheoremdmmpogaOLD 7756* Obsolete version of dmmpoga 7755 as of 29-May-2024. (Contributed by Alexander van der Vekens, 10-Feb-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))

Theoremdmmpog 7757* Domain of an operation given by the maps-to notation, closed form of dmmpo 7753. 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 = (𝑥𝐴, 𝑦𝐵𝐶)       (𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))

Theoremmpoexxg 7758* Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝐴𝑅 ∧ ∀𝑥𝐴 𝐵𝑆) → 𝐹 ∈ V)

Theoremmpoexg 7759* Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝐴𝑅𝐵𝑆) → 𝐹 ∈ V)

Theoremmpoexga 7760* 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.)
((𝐴𝑉𝐵𝑊) → (𝑥𝐴, 𝑦𝐵𝐶) ∈ V)

Theoremmpoexw 7761* Weak version of mpoex 7762 that holds without ax-rep 5154. 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    &   𝑥𝐴𝑦𝐵 𝐶𝐷       (𝑥𝐴, 𝑦𝐵𝐶) ∈ V

Theoremmpoex 7762* 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       (𝑥𝐴, 𝑦𝐵𝐶) ∈ V

Theoremmptmpoopabbrd 7763* 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.)
(𝜑𝐺𝑊)    &   (𝜑𝑋 ∈ (𝐴𝐺))    &   (𝜑𝑌 ∈ (𝐵𝐺))    &   (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)    &   ((𝜑𝑓(𝐷𝐺)) → 𝜓)    &   ((𝑎 = 𝑋𝑏 = 𝑌) → (𝜏𝜃))    &   (𝑔 = 𝐺 → (𝜒𝜏))    &   𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}))       (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})

Theoremmptmpoopabovd 7764* 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.)
(𝜑𝐺𝑊)    &   (𝜑𝑋 ∈ (𝐴𝐺))    &   (𝜑𝑌 ∈ (𝐵𝐺))    &   (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)    &   ((𝜑𝑓(𝐷𝐺)) → 𝜓)    &   𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝐷𝑔))}))       (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝑓(𝑋(𝐶𝐺)𝑌)𝑓(𝐷𝐺))})

Theoremel2mpocsbcl 7765* 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 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸))       (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))

Theoremel2mpocl 7766* 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 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸))    &   ((𝑥 = 𝑋𝑦 = 𝑌) → (𝐶 = 𝐹𝐷 = 𝐺))       (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐹𝑇𝐺))))

Theoremfnmpoovd 7767* 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.)
(𝜑𝑀 Fn (𝐴 × 𝐵))    &   ((𝑖 = 𝑎𝑗 = 𝑏) → 𝐷 = 𝐶)    &   ((𝜑𝑖𝐴𝑗𝐵) → 𝐷𝑈)    &   ((𝜑𝑎𝐴𝑏𝐵) → 𝐶𝑉)       (𝜑 → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = 𝐷))

Theoremoffval22 7768* The function operation expressed as a mapping, variation of offval2 7408. (Contributed by SO, 15-Jul-2018.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑥𝐴𝑦𝐵) → 𝐶𝑋)    &   ((𝜑𝑥𝐴𝑦𝐵) → 𝐷𝑌)    &   (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝐶))    &   (𝜑𝐺 = (𝑥𝐴, 𝑦𝐵𝐷))       (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑅𝐷)))

Theorembrovpreldm 7769 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)

Theorembropopvvv 7770* 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 ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ 𝜑}))    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜑𝜓))    &   (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉𝑂𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ 𝜃})       (𝐹(𝐴(𝑉𝑂𝐸)𝐵)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))

Theorembropfvvvvlem 7771* Lemma for bropfvvvv 7772. (Contributed by AV, 31-Dec-2020.) (Revised by AV, 16-Jan-2021.)
𝑂 = (𝑎𝑈 ↦ (𝑏𝑉, 𝑐𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}))    &   ((𝐴𝑈𝐵𝑆𝐶𝑇) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})       ((⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) ∧ 𝐷(𝐵(𝑂𝐴)𝐶)𝐸) → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))

Theorembropfvvvv 7772* 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 = (𝑎𝑈 ↦ (𝑏𝑉, 𝑐𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}))    &   ((𝐴𝑈𝐵𝑆𝐶𝑇) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})    &   (𝑎 = 𝐴𝑉 = 𝑆)    &   (𝑎 = 𝐴𝑊 = 𝑇)    &   (𝑎 = 𝐴 → (𝜑𝜓))       ((𝑆𝑋𝑇𝑌) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))

Theoremovmptss 7773* 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 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)

Theoremrelmpoopab 7774* Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨𝑧, 𝑤⟩ ∣ 𝜑})       Rel (𝐶𝐹𝐷)

Theoremfmpoco 7775* Composition of two functions. Variation of fmptco 6868 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑅𝐶)    &   (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝑅))    &   (𝜑𝐺 = (𝑧𝐶𝑆))    &   (𝑧 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴, 𝑦𝐵𝑇))

Theoremoprabco 7776* Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
((𝑥𝐴𝑦𝐵) → 𝐶𝐷)    &   𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐻𝐶))       (𝐻 Fn 𝐷𝐺 = (𝐻𝐹))

Theoremoprab2co 7777* Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
((𝑥𝐴𝑦𝐵) → 𝐶𝑅)    &   ((𝑥𝐴𝑦𝐵) → 𝐷𝑆)    &   𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨𝐶, 𝐷⟩)    &   𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑀𝐷))       (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))

Theoremdf1st2 7778* An alternate possible definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))

Theoremdf2nd2 7779* An alternate possible definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))

Theorem1stconst 7780 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.)
(𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)

Theorem2ndconst 7781 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.)
(𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)

Theoremdfmpo 7782* Alternate definition for the maps-to notation df-mpo 7140 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐶 ∈ V       (𝑥𝐴, 𝑦𝐵𝐶) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}

Theoremmposn 7783* An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.)
𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶)    &   (𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑦 = 𝐵𝐷 = 𝐸)       ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})

Theoremcurry1 7784* 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) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))

Theoremcurry1val 7785 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) ∧ 𝐶𝐴) → (𝐺𝐷) = (𝐶𝐹𝐷))

Theoremcurry1f 7786 Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)
𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))       ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → 𝐺:𝐵𝐷)

Theoremcurry2 7787* 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) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))

Theoremcurry2f 7788 Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
𝐺 = (𝐹(1st ↾ (V × {𝐶})))       ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → 𝐺:𝐴𝐷)

Theoremcurry2val 7789 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) ∧ 𝐶𝐵) → (𝐺𝐷) = (𝐷𝐹𝐶))

Theoremcnvf1olem 7790 Lemma for cnvf1o 7791. (Contributed by Mario Carneiro, 27-Apr-2014.)
((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → (𝐶𝐴𝐵 = {𝐶}))

Theoremcnvf1o 7791* Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
(Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)

Theoremfparlem1 7792 Lemma for fpar 7796. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
((1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)

Theoremfparlem2 7793 Lemma for fpar 7796. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})

Theoremfparlem3 7794* Lemma for fpar 7796. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐹 Fn 𝐴 → ((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)))

Theoremfparlem4 7795* Lemma for fpar 7796. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐺 Fn 𝐵 → ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))

Theoremfpar 7796* Merge two functions in parallel. Use as the second argument of a composition with a binary operation to build compound functions such as (𝑥 ∈ (0[,)+∞), 𝑦 ∈ ℝ ↦ ((√‘𝑥) + (sin‘𝑦))), see also ex-fpar 28254. (Contributed by NM, 17-Sep-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))       ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))

Theoremfsplit 7797 A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 7796 in order to build compound functions such as (𝑥 ∈ (0[,)+∞) ↦ ((√‘𝑥) + (sin‘𝑥))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5428 with df-id 5425. (Revised by BJ, 31-Dec-2023.)
(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

TheoremfsplitOLD 7798 Obsolete proof of fsplit 7797 as of 31-Dec-2023 . (Contributed by NM, 17-Sep-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

Theoremfsplitfpar 7799* Merge two functions with a common argument in parallel. Combination of fsplit 7797 and fpar 7796. (Contributed by AV, 3-Jan-2024.)
𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   𝑆 = ((1st ↾ I ) ↾ 𝐴)       ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))

Theoremoffsplitfpar 7800 Express the function operation map f by the functions defined in fsplit 7797 and fpar 7796. (Contributed by AV, 4-Jan-2024.)
𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   𝑆 = ((1st ↾ I ) ↾ 𝐴)       (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻𝑆)) = (𝐹f + 𝐺))

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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