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Type | Label | Description |
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Statement | ||
Theorem | eusvobj1 7401* | Specify the same object in two ways when class π΅(π¦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
β’ π΅ β V β β’ (β!π₯βπ¦ β π΄ π₯ = π΅ β (β©π₯βπ¦ β π΄ π₯ = π΅) = (β©π₯βπ¦ β π΄ π₯ = π΅)) | ||
Theorem | f1ofveu 7402* | There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.) |
β’ ((πΉ:π΄β1-1-ontoβπ΅ β§ πΆ β π΅) β β!π₯ β π΄ (πΉβπ₯) = πΆ) | ||
Theorem | f1ocnvfv3 7403* | Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
β’ ((πΉ:π΄β1-1-ontoβπ΅ β§ πΆ β π΅) β (β‘πΉβπΆ) = (β©π₯ β π΄ (πΉβπ₯) = πΆ)) | ||
Theorem | riotaund 7404* | Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.) |
β’ (Β¬ β!π₯ β π΄ π β (β©π₯ β π΄ π) = β ) | ||
Theorem | riotassuni 7405* | The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.) |
β’ (β©π₯ β π΄ π) β (π« βͺ π΄ βͺ βͺ π΄) | ||
Theorem | riotaclb 7406* | Bidirectional closure of restricted iota when domain is not empty. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) (Revised by NM, 13-Sep-2018.) |
β’ (Β¬ β β π΄ β (β!π₯ β π΄ π β (β©π₯ β π΄ π) β π΄)) | ||
Theorem | riotarab 7407* | Restricted iota of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.) |
β’ (π₯ = π¦ β (π β π)) β β’ (β©π₯ β {π¦ β π΄ β£ π}π) = (β©π₯ β π΄ (π β§ π)) | ||
Syntax | co 7408 | Extend class notation to include the value of an operation πΉ (such as +) for two arguments π΄ and π΅. Note that the syntax is simply three class symbols in a row surrounded by parentheses. Since operation values are the only possible class expressions consisting of three class expressions in a row surrounded by parentheses, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 11444.) |
class (π΄πΉπ΅) | ||
Syntax | coprab 7409 | Extend class notation to include class abstraction (class builder) of nested ordered pairs. |
class {β¨β¨π₯, π¦β©, π§β© β£ π} | ||
Syntax | cmpo 7410 | Extend the definition of a class to include maps-to notation for defining an operation via a rule. |
class (π₯ β π΄, π¦ β π΅ β¦ πΆ) | ||
Definition | df-ov 7411 | Define the value of an operation. Definition of operation value in [Enderton] p. 79. Note that the syntax is simply three class expressions in a row bracketed by parentheses. There are no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation πΉ and its arguments π΄ and π΅- will be useful for proving meaningful theorems. For example, if class πΉ is the operation + and arguments π΄ and π΅ are 3 and 2, the expression (3 + 2) can be proved to equal 5 (see 3p2e5 12362). This definition is well-defined, although not very meaningful, when classes π΄ and/or π΅ are proper classes (i.e. are not sets); see ovprc1 7447 and ovprc2 7448. On the other hand, we often find uses for this definition when πΉ is a proper class, such as +o in oav 8510. πΉ is normally equal to a class of nested ordered pairs of the form defined by df-oprab 7412. (Contributed by NM, 28-Feb-1995.) |
β’ (π΄πΉπ΅) = (πΉββ¨π΄, π΅β©) | ||
Definition | df-oprab 7412* | Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally π₯, π¦, and π§ are distinct, although the definition doesn't strictly require it. See df-ov 7411 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of an operation given by a class abstraction is given by ovmpo 7567. (Contributed by NM, 12-Mar-1995.) |
β’ {β¨β¨π₯, π¦β©, π§β© β£ π} = {π€ β£ βπ₯βπ¦βπ§(π€ = β¨β¨π₯, π¦β©, π§β© β§ π)} | ||
Definition | df-mpo 7413* | Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from π₯, π¦ (in π΄ Γ π΅) to πΆ(π₯, π¦)". An extension of df-mpt 5232 for two arguments. (Contributed by NM, 17-Feb-2008.) |
β’ (π₯ β π΄, π¦ β π΅ β¦ πΆ) = {β¨β¨π₯, π¦β©, π§β© β£ ((π₯ β π΄ β§ π¦ β π΅) β§ π§ = πΆ)} | ||
Theorem | oveq 7414 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
β’ (πΉ = πΊ β (π΄πΉπ΅) = (π΄πΊπ΅)) | ||
Theorem | oveq1 7415 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
β’ (π΄ = π΅ β (π΄πΉπΆ) = (π΅πΉπΆ)) | ||
Theorem | oveq2 7416 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
β’ (π΄ = π΅ β (πΆπΉπ΄) = (πΆπΉπ΅)) | ||
Theorem | oveq12 7417 | Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.) |
β’ ((π΄ = π΅ β§ πΆ = π·) β (π΄πΉπΆ) = (π΅πΉπ·)) | ||
Theorem | oveq1i 7418 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
β’ π΄ = π΅ β β’ (π΄πΉπΆ) = (π΅πΉπΆ) | ||
Theorem | oveq2i 7419 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
β’ π΄ = π΅ β β’ (πΆπΉπ΄) = (πΆπΉπ΅) | ||
Theorem | oveq12i 7420 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ π΄ = π΅ & β’ πΆ = π· β β’ (π΄πΉπΆ) = (π΅πΉπ·) | ||
Theorem | oveqi 7421 | Equality inference for operation value. (Contributed by NM, 24-Nov-2007.) |
β’ π΄ = π΅ β β’ (πΆπ΄π·) = (πΆπ΅π·) | ||
Theorem | oveq123i 7422 | Equality inference for operation value. (Contributed by FL, 11-Jul-2010.) |
β’ π΄ = πΆ & β’ π΅ = π· & β’ πΉ = πΊ β β’ (π΄πΉπ΅) = (πΆπΊπ·) | ||
Theorem | oveq1d 7423 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
β’ (π β π΄ = π΅) β β’ (π β (π΄πΉπΆ) = (π΅πΉπΆ)) | ||
Theorem | oveq2d 7424 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
β’ (π β π΄ = π΅) β β’ (π β (πΆπΉπ΄) = (πΆπΉπ΅)) | ||
Theorem | oveqd 7425 | Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.) |
β’ (π β π΄ = π΅) β β’ (π β (πΆπ΄π·) = (πΆπ΅π·)) | ||
Theorem | oveq12d 7426 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ (π β π΄ = π΅) & β’ (π β πΆ = π·) β β’ (π β (π΄πΉπΆ) = (π΅πΉπ·)) | ||
Theorem | oveqan12d 7427 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
β’ (π β π΄ = π΅) & β’ (π β πΆ = π·) β β’ ((π β§ π) β (π΄πΉπΆ) = (π΅πΉπ·)) | ||
Theorem | oveqan12rd 7428 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
β’ (π β π΄ = π΅) & β’ (π β πΆ = π·) β β’ ((π β§ π) β (π΄πΉπΆ) = (π΅πΉπ·)) | ||
Theorem | oveq123d 7429 | Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.) |
β’ (π β πΉ = πΊ) & β’ (π β π΄ = π΅) & β’ (π β πΆ = π·) β β’ (π β (π΄πΉπΆ) = (π΅πΊπ·)) | ||
Theorem | fvoveq1d 7430 | Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.) |
β’ (π β π΄ = π΅) β β’ (π β (πΉβ(π΄ππΆ)) = (πΉβ(π΅ππΆ))) | ||
Theorem | fvoveq1 7431 | Equality theorem for nested function and operation value. Closed form of fvoveq1d 7430. (Contributed by AV, 23-Jul-2022.) |
β’ (π΄ = π΅ β (πΉβ(π΄ππΆ)) = (πΉβ(π΅ππΆ))) | ||
Theorem | ovanraleqv 7432* | Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.) |
β’ (π΅ = π β (π β π)) β β’ (π΅ = π β (βπ₯ β π (π β§ (π΄ Β· π΅) = πΆ) β βπ₯ β π (π β§ (π΄ Β· π) = πΆ))) | ||
Theorem | imbrov2fvoveq 7433 | Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.) |
β’ (π = π β (π β π)) β β’ (π = π β ((π β (πΉβ((πΊβπ) Β· π))π π΄) β (π β (πΉβ((πΊβπ) Β· π))π π΄))) | ||
Theorem | ovrspc2v 7434* | If an operation value is element of a class for all operands of two classes, then the operation value is an element of the class for specific operands of the two classes. (Contributed by Mario Carneiro, 6-Dec-2014.) |
β’ (((π β π΄ β§ π β π΅) β§ βπ₯ β π΄ βπ¦ β π΅ (π₯πΉπ¦) β πΆ) β (ππΉπ) β πΆ) | ||
Theorem | oveqrspc2v 7435* | Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.) |
β’ ((π β§ (π₯ β π΄ β§ π¦ β π΅)) β (π₯πΉπ¦) = (π₯πΊπ¦)) β β’ ((π β§ (π β π΄ β§ π β π΅)) β (ππΉπ) = (ππΊπ)) | ||
Theorem | oveqdr 7436 | Equality of two operations for any two operands. Useful in proofs using *propd theorems. (Contributed by Mario Carneiro, 29-Jun-2015.) |
β’ (π β πΉ = πΊ) β β’ ((π β§ π) β (π₯πΉπ¦) = (π₯πΊπ¦)) | ||
Theorem | nfovd 7437 | Deduction version of bound-variable hypothesis builder nfov 7438. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ (π β β²π₯π΄) & β’ (π β β²π₯πΉ) & β’ (π β β²π₯π΅) β β’ (π β β²π₯(π΄πΉπ΅)) | ||
Theorem | nfov 7438 | Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.) |
β’ β²π₯π΄ & β’ β²π₯πΉ & β’ β²π₯π΅ β β’ β²π₯(π΄πΉπ΅) | ||
Theorem | oprabidw 7439* | The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of oprabid 7440 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 20-Mar-2013.) Avoid ax-13 2371. (Revised by Gino Giotto, 26-Jan-2024.) |
β’ (β¨β¨π₯, π¦β©, π§β© β {β¨β¨π₯, π¦β©, π§β© β£ π} β π) | ||
Theorem | oprabid 7440 | The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker oprabidw 7439 when possible. (Contributed by Mario Carneiro, 20-Mar-2013.) (New usage is discouraged.) |
β’ (β¨β¨π₯, π¦β©, π§β© β {β¨β¨π₯, π¦β©, π§β© β£ π} β π) | ||
Theorem | ovex 7441 | The result of an operation is a set. (Contributed by NM, 13-Mar-1995.) |
β’ (π΄πΉπ΅) β V | ||
Theorem | ovexi 7442 | The result of an operation is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ π΄ = (π΅πΉπΆ) β β’ π΄ β V | ||
Theorem | ovexd 7443 | The result of an operation is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
β’ (π β (π΄πΉπ΅) β V) | ||
Theorem | ovssunirn 7444 | The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.) |
β’ (ππΉπ) β βͺ ran πΉ | ||
Theorem | 0ov 7445 | Operation value of the empty set. (Contributed by AV, 15-May-2021.) |
β’ (π΄β π΅) = β | ||
Theorem | ovprc 7446 | The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.) |
β’ Rel dom πΉ β β’ (Β¬ (π΄ β V β§ π΅ β V) β (π΄πΉπ΅) = β ) | ||
Theorem | ovprc1 7447 | The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.) |
β’ Rel dom πΉ β β’ (Β¬ π΄ β V β (π΄πΉπ΅) = β ) | ||
Theorem | ovprc2 7448 | The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.) |
β’ Rel dom πΉ β β’ (Β¬ π΅ β V β (π΄πΉπ΅) = β ) | ||
Theorem | ovrcl 7449 | Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.) |
β’ Rel dom πΉ β β’ (πΆ β (π΄πΉπ΅) β (π΄ β V β§ π΅ β V)) | ||
Theorem | csbov123 7450 | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Revised by NM, 23-Aug-2018.) |
β’ β¦π΄ / π₯β¦(π΅πΉπΆ) = (β¦π΄ / π₯β¦π΅β¦π΄ / π₯β¦πΉβ¦π΄ / π₯β¦πΆ) | ||
Theorem | csbov 7451* | Move class substitution in and out of an operation. (Contributed by NM, 23-Aug-2018.) |
β’ β¦π΄ / π₯β¦(π΅πΉπΆ) = (π΅β¦π΄ / π₯β¦πΉπΆ) | ||
Theorem | csbov12g 7452* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
β’ (π΄ β π β β¦π΄ / π₯β¦(π΅πΉπΆ) = (β¦π΄ / π₯β¦π΅πΉβ¦π΄ / π₯β¦πΆ)) | ||
Theorem | csbov1g 7453* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
β’ (π΄ β π β β¦π΄ / π₯β¦(π΅πΉπΆ) = (β¦π΄ / π₯β¦π΅πΉπΆ)) | ||
Theorem | csbov2g 7454* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
β’ (π΄ β π β β¦π΄ / π₯β¦(π΅πΉπΆ) = (π΅πΉβ¦π΄ / π₯β¦πΆ)) | ||
Theorem | rspceov 7455* | A frequently used special case of rspc2ev 3624 for operation values. (Contributed by NM, 21-Mar-2007.) |
β’ ((πΆ β π΄ β§ π· β π΅ β§ π = (πΆπΉπ·)) β βπ₯ β π΄ βπ¦ β π΅ π = (π₯πΉπ¦)) | ||
Theorem | elovimad 7456 | Elementhood of the image set of an operation value. (Contributed by Thierry Arnoux, 13-Mar-2017.) |
β’ (π β π΄ β πΆ) & β’ (π β π΅ β π·) & β’ (π β Fun πΉ) & β’ (π β (πΆ Γ π·) β dom πΉ) β β’ (π β (π΄πΉπ΅) β (πΉ β (πΆ Γ π·))) | ||
Theorem | fnbrovb 7457 | Value of a binary operation expressed as a binary relation. See also fnbrfvb 6944 for functions on Cartesian products. (Contributed by BJ, 15-Feb-2022.) |
β’ ((πΉ Fn (π Γ π) β§ (π΄ β π β§ π΅ β π)) β ((π΄πΉπ΅) = πΆ β β¨π΄, π΅β©πΉπΆ)) | ||
Theorem | fnotovb 7458 | Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6945. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 15-Feb-2022.) |
β’ ((πΉ Fn (π΄ Γ π΅) β§ πΆ β π΄ β§ π· β π΅) β ((πΆπΉπ·) = π β β¨πΆ, π·, π β© β πΉ)) | ||
Theorem | opabbrex 7459 | A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by BJ/AV, 20-Jun-2019.) (Proof shortened by OpenAI, 25-Mar-2020.) |
β’ ((βπ₯βπ¦(π₯π π¦ β π) β§ {β¨π₯, π¦β© β£ π} β π) β {β¨π₯, π¦β© β£ (π₯π π¦ β§ π)} β V) | ||
Theorem | opabresex2 7460* | Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable conditions betweem π, πΊ and π₯, π¦ to remove hypotheses. (Revised by SN, 13-Dec-2024.) |
β’ {β¨π₯, π¦β© β£ (π₯(πβπΊ)π¦ β§ π)} β V | ||
Theorem | opabresex2d 7461* | Obsolete version of opabresex2 7460 as of 13-Dec-2024. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((π β§ π₯(πβπΊ)π¦) β π) & β’ (π β {β¨π₯, π¦β© β£ π} β π) β β’ (π β {β¨π₯, π¦β© β£ (π₯(πβπΊ)π¦ β§ π)} β V) | ||
Theorem | fvmptopab 7462* | The function value of a mapping π to a restricted binary relation expressed as an ordered-pair class abstraction: The restricted binary relation is a binary relation given as value of a function πΉ restricted by the condition π. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.) Add disjoint variable condition on πΉ, π₯, π¦ to remove a sethood hypothesis. (Revised by SN, 13-Dec-2024.) |
β’ (π§ = π β (π β π)) & β’ π = (π§ β V β¦ {β¨π₯, π¦β© β£ (π₯(πΉβπ§)π¦ β§ π)}) β β’ (πβπ) = {β¨π₯, π¦β© β£ (π₯(πΉβπ)π¦ β§ π)} | ||
Theorem | fvmptopabOLD 7463* | Obsolete version of fvmptopab 7462 as of 13-Dec-2024. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((π β§ π§ = π) β (π β π)) & β’ (π β {β¨π₯, π¦β© β£ π₯(πΉβπ)π¦} β V) & β’ π = (π§ β V β¦ {β¨π₯, π¦β© β£ (π₯(πΉβπ§)π¦ β§ π)}) β β’ (π β (πβπ) = {β¨π₯, π¦β© β£ (π₯(πΉβπ)π¦ β§ π)}) | ||
Theorem | f1opr 7464* | Condition for an operation to be one-to-one. (Contributed by Jeff Madsen, 17-Jun-2010.) |
β’ (πΉ:(π΄ Γ π΅)β1-1βπΆ β (πΉ:(π΄ Γ π΅)βΆπΆ β§ βπ β π΄ βπ β π΅ βπ‘ β π΄ βπ’ β π΅ ((ππΉπ ) = (π‘πΉπ’) β (π = π‘ β§ π = π’)))) | ||
Theorem | brfvopab 7465 | The classes involved in a binary relation of a function value which is an ordered-pair class abstraction are sets. (Contributed by AV, 7-Jan-2021.) |
β’ (π β V β (πΉβπ) = {β¨π¦, π§β© β£ π}) β β’ (π΄(πΉβπ)π΅ β (π β V β§ π΄ β V β§ π΅ β V)) | ||
Theorem | dfoprab2 7466* | Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.) |
β’ {β¨β¨π₯, π¦β©, π§β© β£ π} = {β¨π€, π§β© β£ βπ₯βπ¦(π€ = β¨π₯, π¦β© β§ π)} | ||
Theorem | reloprab 7467* | An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.) |
β’ Rel {β¨β¨π₯, π¦β©, π§β© β£ π} | ||
Theorem | oprabv 7468* | If a pair and a class are in a relationship given by a class abstraction of a collection of nested ordered pairs, the involved classes are sets. (Contributed by Alexander van der Vekens, 8-Jul-2018.) |
β’ (β¨π, πβ©{β¨β¨π₯, π¦β©, π§β© β£ π}π β (π β V β§ π β V β§ π β V)) | ||
Theorem | nfoprab1 7469 | The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
β’ β²π₯{β¨β¨π₯, π¦β©, π§β© β£ π} | ||
Theorem | nfoprab2 7470 | The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.) |
β’ β²π¦{β¨β¨π₯, π¦β©, π§β© β£ π} | ||
Theorem | nfoprab3 7471 | The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.) |
β’ β²π§{β¨β¨π₯, π¦β©, π§β© β£ π} | ||
Theorem | nfoprab 7472* | Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.) |
β’ β²π€π β β’ β²π€{β¨β¨π₯, π¦β©, π§β© β£ π} | ||
Theorem | oprabbid 7473* | Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.) |
β’ β²π₯π & β’ β²π¦π & β’ β²π§π & β’ (π β (π β π)) β β’ (π β {β¨β¨π₯, π¦β©, π§β© β£ π} = {β¨β¨π₯, π¦β©, π§β© β£ π}) | ||
Theorem | oprabbidv 7474* | Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) |
β’ (π β (π β π)) β β’ (π β {β¨β¨π₯, π¦β©, π§β© β£ π} = {β¨β¨π₯, π¦β©, π§β© β£ π}) | ||
Theorem | oprabbii 7475* | Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
β’ (π β π) β β’ {β¨β¨π₯, π¦β©, π§β© β£ π} = {β¨β¨π₯, π¦β©, π§β© β£ π} | ||
Theorem | ssoprab2 7476 | Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 5546. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
β’ (βπ₯βπ¦βπ§(π β π) β {β¨β¨π₯, π¦β©, π§β© β£ π} β {β¨β¨π₯, π¦β©, π§β© β£ π}) | ||
Theorem | ssoprab2b 7477 | Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 5549. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) (New usage is discouraged.) |
β’ ({β¨β¨π₯, π¦β©, π§β© β£ π} β {β¨β¨π₯, π¦β©, π§β© β£ π} β βπ₯βπ¦βπ§(π β π)) | ||
Theorem | eqoprab2bw 7478* | Equivalence of ordered pair abstraction subclass and biconditional. Version of eqoprab2b 7479 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 4-Jan-2017.) Avoid ax-13 2371. (Revised by Gino Giotto, 26-Jan-2024.) |
β’ ({β¨β¨π₯, π¦β©, π§β© β£ π} = {β¨β¨π₯, π¦β©, π§β© β£ π} β βπ₯βπ¦βπ§(π β π)) | ||
Theorem | eqoprab2b 7479 | Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 5552. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker eqoprab2bw 7478 when possible. (Contributed by Mario Carneiro, 4-Jan-2017.) (New usage is discouraged.) |
β’ ({β¨β¨π₯, π¦β©, π§β© β£ π} = {β¨β¨π₯, π¦β©, π§β© β£ π} β βπ₯βπ¦βπ§(π β π)) | ||
Theorem | mpoeq123 7480* | An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.) |
β’ ((π΄ = π· β§ βπ₯ β π΄ (π΅ = πΈ β§ βπ¦ β π΅ πΆ = πΉ)) β (π₯ β π΄, π¦ β π΅ β¦ πΆ) = (π₯ β π·, π¦ β πΈ β¦ πΉ)) | ||
Theorem | mpoeq12 7481* | An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
β’ ((π΄ = πΆ β§ π΅ = π·) β (π₯ β π΄, π¦ β π΅ β¦ πΈ) = (π₯ β πΆ, π¦ β π· β¦ πΈ)) | ||
Theorem | mpoeq123dva 7482* | An equality deduction for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) |
β’ (π β π΄ = π·) & β’ ((π β§ π₯ β π΄) β π΅ = πΈ) & β’ ((π β§ (π₯ β π΄ β§ π¦ β π΅)) β πΆ = πΉ) β β’ (π β (π₯ β π΄, π¦ β π΅ β¦ πΆ) = (π₯ β π·, π¦ β πΈ β¦ πΉ)) | ||
Theorem | mpoeq123dv 7483* | An equality deduction for the maps-to notation. (Contributed by NM, 12-Sep-2011.) |
β’ (π β π΄ = π·) & β’ (π β π΅ = πΈ) & β’ (π β πΆ = πΉ) β β’ (π β (π₯ β π΄, π¦ β π΅ β¦ πΆ) = (π₯ β π·, π¦ β πΈ β¦ πΉ)) | ||
Theorem | mpoeq123i 7484 | An equality inference for the maps-to notation. (Contributed by NM, 15-Jul-2013.) |
β’ π΄ = π· & β’ π΅ = πΈ & β’ πΆ = πΉ β β’ (π₯ β π΄, π¦ β π΅ β¦ πΆ) = (π₯ β π·, π¦ β πΈ β¦ πΉ) | ||
Theorem | mpoeq3dva 7485* | Slightly more general equality inference for the maps-to notation. (Contributed by NM, 17-Oct-2013.) |
β’ ((π β§ π₯ β π΄ β§ π¦ β π΅) β πΆ = π·) β β’ (π β (π₯ β π΄, π¦ β π΅ β¦ πΆ) = (π₯ β π΄, π¦ β π΅ β¦ π·)) | ||
Theorem | mpoeq3ia 7486 | An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
β’ ((π₯ β π΄ β§ π¦ β π΅) β πΆ = π·) β β’ (π₯ β π΄, π¦ β π΅ β¦ πΆ) = (π₯ β π΄, π¦ β π΅ β¦ π·) | ||
Theorem | mpoeq3dv 7487* | An equality deduction for the maps-to notation restricted to the value of the operation. (Contributed by SO, 16-Jul-2018.) |
β’ (π β πΆ = π·) β β’ (π β (π₯ β π΄, π¦ β π΅ β¦ πΆ) = (π₯ β π΄, π¦ β π΅ β¦ π·)) | ||
Theorem | nfmpo1 7488 | Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.) |
β’ β²π₯(π₯ β π΄, π¦ β π΅ β¦ πΆ) | ||
Theorem | nfmpo2 7489 | Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.) |
β’ β²π¦(π₯ β π΄, π¦ β π΅ β¦ πΆ) | ||
Theorem | nfmpo 7490* | Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.) |
β’ β²π§π΄ & β’ β²π§π΅ & β’ β²π§πΆ β β’ β²π§(π₯ β π΄, π¦ β π΅ β¦ πΆ) | ||
Theorem | 0mpo0 7491* | A mapping operation with empty domain is empty. Generalization of mpo0 7493. (Contributed by AV, 27-Jan-2024.) |
β’ ((π΄ = β β¨ π΅ = β ) β (π₯ β π΄, π¦ β π΅ β¦ πΆ) = β ) | ||
Theorem | mpo0v 7492* | A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.) (Proof shortened by AV, 27-Jan-2024.) |
β’ (π₯ β β , π¦ β π΅ β¦ πΆ) = β | ||
Theorem | mpo0 7493 | A mapping operation with empty domain. In this version of mpo0v 7492, the class of the second operator may depend on the first operator. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.) |
β’ (π₯ β β , π¦ β π΅ β¦ πΆ) = β | ||
Theorem | oprab4 7494* | Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.) |
β’ {β¨β¨π₯, π¦β©, π§β© β£ (β¨π₯, π¦β© β (π΄ Γ π΅) β§ π)} = {β¨β¨π₯, π¦β©, π§β© β£ ((π₯ β π΄ β§ π¦ β π΅) β§ π)} | ||
Theorem | cbvoprab1 7495* | Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.) |
β’ β²π€π & β’ β²π₯π & β’ (π₯ = π€ β (π β π)) β β’ {β¨β¨π₯, π¦β©, π§β© β£ π} = {β¨β¨π€, π¦β©, π§β© β£ π} | ||
Theorem | cbvoprab2 7496* | Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.) |
β’ β²π€π & β’ β²π¦π & β’ (π¦ = π€ β (π β π)) β β’ {β¨β¨π₯, π¦β©, π§β© β£ π} = {β¨β¨π₯, π€β©, π§β© β£ π} | ||
Theorem | cbvoprab12 7497* | Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ β²π€π & β’ β²π£π & β’ β²π₯π & β’ β²π¦π & β’ ((π₯ = π€ β§ π¦ = π£) β (π β π)) β β’ {β¨β¨π₯, π¦β©, π§β© β£ π} = {β¨β¨π€, π£β©, π§β© β£ π} | ||
Theorem | cbvoprab12v 7498* | Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) |
β’ ((π₯ = π€ β§ π¦ = π£) β (π β π)) β β’ {β¨β¨π₯, π¦β©, π§β© β£ π} = {β¨β¨π€, π£β©, π§β© β£ π} | ||
Theorem | cbvoprab3 7499* | Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.) |
β’ β²π€π & β’ β²π§π & β’ (π§ = π€ β (π β π)) β β’ {β¨β¨π₯, π¦β©, π§β© β£ π} = {β¨β¨π₯, π¦β©, π€β© β£ π} | ||
Theorem | cbvoprab3v 7500* | Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.) |
β’ (π§ = π€ β (π β π)) β β’ {β¨β¨π₯, π¦β©, π§β© β£ π} = {β¨β¨π₯, π¦β©, π€β© β£ π} |
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