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Theorem List for Metamath Proof Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeusvobj2 7401* Specify the same property in two ways when class ๐ต(๐‘ฆ) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
๐ต โˆˆ V    โ‡’   (โˆƒ!๐‘ฅโˆƒ๐‘ฆ โˆˆ ๐ด ๐‘ฅ = ๐ต โ†’ (โˆƒ๐‘ฆ โˆˆ ๐ด ๐‘ฅ = ๐ต โ†” โˆ€๐‘ฆ โˆˆ ๐ด ๐‘ฅ = ๐ต))
 
Theoremeusvobj1 7402* Specify the same object in two ways when class ๐ต(๐‘ฆ) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
๐ต โˆˆ V    โ‡’   (โˆƒ!๐‘ฅโˆƒ๐‘ฆ โˆˆ ๐ด ๐‘ฅ = ๐ต โ†’ (โ„ฉ๐‘ฅโˆƒ๐‘ฆ โˆˆ ๐ด ๐‘ฅ = ๐ต) = (โ„ฉ๐‘ฅโˆ€๐‘ฆ โˆˆ ๐ด ๐‘ฅ = ๐ต))
 
Theoremf1ofveu 7403* There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)
((๐น:๐ดโ€“1-1-ontoโ†’๐ต โˆง ๐ถ โˆˆ ๐ต) โ†’ โˆƒ!๐‘ฅ โˆˆ ๐ด (๐นโ€˜๐‘ฅ) = ๐ถ)
 
Theoremf1ocnvfv3 7404* 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.)
((๐น:๐ดโ€“1-1-ontoโ†’๐ต โˆง ๐ถ โˆˆ ๐ต) โ†’ (โ—ก๐นโ€˜๐ถ) = (โ„ฉ๐‘ฅ โˆˆ ๐ด (๐นโ€˜๐‘ฅ) = ๐ถ))
 
Theoremriotaund 7405* 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.)
(ยฌ โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ‘ โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) = โˆ…)
 
Theoremriotassuni 7406* The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)
(โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) โŠ† (๐’ซ โˆช ๐ด โˆช โˆช ๐ด)
 
Theoremriotaclb 7407* 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.)
(ยฌ โˆ… โˆˆ ๐ด โ†’ (โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ‘ โ†” (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) โˆˆ ๐ด))
 
Theoremriotarab 7408* Restricted iota of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.)
(๐‘ฅ = ๐‘ฆ โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   (โ„ฉ๐‘ฅ โˆˆ {๐‘ฆ โˆˆ ๐ด โˆฃ ๐œ“}๐œ’) = (โ„ฉ๐‘ฅ โˆˆ ๐ด (๐œ‘ โˆง ๐œ’))
 
2.3.19  Operations
 
Syntaxco 7409 Extend class notation to include the value of an operation ๐น (such as +) for two arguments ๐ด and ๐ต. 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 11445.)
class (๐ด๐น๐ต)
 
Syntaxcoprab 7410 Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘}
 
Syntaxcmpo 7411 Extend the definition of a class to include maps-to notation for defining an operation via a rule.
class (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ)
 
Definitiondf-ov 7412 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 ๐น and its arguments ๐ด and ๐ต- will be useful for proving meaningful theorems. For example, if class ๐น is the operation + and arguments ๐ด and ๐ต are 3 and 2, the expression (3 + 2) can be proved to equal 5 (see 3p2e5 12363). This definition is well-defined, although not very meaningful, when classes ๐ด and/or ๐ต are proper classes (i.e. are not sets); see ovprc1 7448 and ovprc2 7449. On the other hand, we often find uses for this definition when ๐น is a proper class, such as +o in oav 8511. ๐น is normally equal to a class of nested ordered pairs of the form defined by df-oprab 7413. (Contributed by NM, 28-Feb-1995.)
(๐ด๐น๐ต) = (๐นโ€˜โŸจ๐ด, ๐ตโŸฉ)
 
Definitiondf-oprab 7413* 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 ๐‘ฅ, ๐‘ฆ, and ๐‘ง are distinct, although the definition doesn't strictly require it. See df-ov 7412 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 7568. (Contributed by NM, 12-Mar-1995.)
{โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {๐‘ค โˆฃ โˆƒ๐‘ฅโˆƒ๐‘ฆโˆƒ๐‘ง(๐‘ค = โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆง ๐œ‘)}
 
Definitiondf-mpo 7414* Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from ๐‘ฅ, ๐‘ฆ (in ๐ด ร— ๐ต) to ๐ถ(๐‘ฅ, ๐‘ฆ)". An extension of df-mpt 5233 for two arguments. (Contributed by NM, 17-Feb-2008.)
(๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ((๐‘ฅ โˆˆ ๐ด โˆง ๐‘ฆ โˆˆ ๐ต) โˆง ๐‘ง = ๐ถ)}
 
Theoremoveq 7415 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(๐น = ๐บ โ†’ (๐ด๐น๐ต) = (๐ด๐บ๐ต))
 
Theoremoveq1 7416 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(๐ด = ๐ต โ†’ (๐ด๐น๐ถ) = (๐ต๐น๐ถ))
 
Theoremoveq2 7417 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(๐ด = ๐ต โ†’ (๐ถ๐น๐ด) = (๐ถ๐น๐ต))
 
Theoremoveq12 7418 Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
((๐ด = ๐ต โˆง ๐ถ = ๐ท) โ†’ (๐ด๐น๐ถ) = (๐ต๐น๐ท))
 
Theoremoveq1i 7419 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
๐ด = ๐ต    โ‡’   (๐ด๐น๐ถ) = (๐ต๐น๐ถ)
 
Theoremoveq2i 7420 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
๐ด = ๐ต    โ‡’   (๐ถ๐น๐ด) = (๐ถ๐น๐ต)
 
Theoremoveq12i 7421 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
๐ด = ๐ต    &   ๐ถ = ๐ท    โ‡’   (๐ด๐น๐ถ) = (๐ต๐น๐ท)
 
Theoremoveqi 7422 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
๐ด = ๐ต    โ‡’   (๐ถ๐ด๐ท) = (๐ถ๐ต๐ท)
 
Theoremoveq123i 7423 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
๐ด = ๐ถ    &   ๐ต = ๐ท    &   ๐น = ๐บ    โ‡’   (๐ด๐น๐ต) = (๐ถ๐บ๐ท)
 
Theoremoveq1d 7424 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(๐œ‘ โ†’ ๐ด = ๐ต)    โ‡’   (๐œ‘ โ†’ (๐ด๐น๐ถ) = (๐ต๐น๐ถ))
 
Theoremoveq2d 7425 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(๐œ‘ โ†’ ๐ด = ๐ต)    โ‡’   (๐œ‘ โ†’ (๐ถ๐น๐ด) = (๐ถ๐น๐ต))
 
Theoremoveqd 7426 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
(๐œ‘ โ†’ ๐ด = ๐ต)    โ‡’   (๐œ‘ โ†’ (๐ถ๐ด๐ท) = (๐ถ๐ต๐ท))
 
Theoremoveq12d 7427 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(๐œ‘ โ†’ ๐ด = ๐ต)    &   (๐œ‘ โ†’ ๐ถ = ๐ท)    โ‡’   (๐œ‘ โ†’ (๐ด๐น๐ถ) = (๐ต๐น๐ท))
 
Theoremoveqan12d 7428 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(๐œ‘ โ†’ ๐ด = ๐ต)    &   (๐œ“ โ†’ ๐ถ = ๐ท)    โ‡’   ((๐œ‘ โˆง ๐œ“) โ†’ (๐ด๐น๐ถ) = (๐ต๐น๐ท))
 
Theoremoveqan12rd 7429 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(๐œ‘ โ†’ ๐ด = ๐ต)    &   (๐œ“ โ†’ ๐ถ = ๐ท)    โ‡’   ((๐œ“ โˆง ๐œ‘) โ†’ (๐ด๐น๐ถ) = (๐ต๐น๐ท))
 
Theoremoveq123d 7430 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
(๐œ‘ โ†’ ๐น = ๐บ)    &   (๐œ‘ โ†’ ๐ด = ๐ต)    &   (๐œ‘ โ†’ ๐ถ = ๐ท)    โ‡’   (๐œ‘ โ†’ (๐ด๐น๐ถ) = (๐ต๐บ๐ท))
 
Theoremfvoveq1d 7431 Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.)
(๐œ‘ โ†’ ๐ด = ๐ต)    โ‡’   (๐œ‘ โ†’ (๐นโ€˜(๐ด๐‘‚๐ถ)) = (๐นโ€˜(๐ต๐‘‚๐ถ)))
 
Theoremfvoveq1 7432 Equality theorem for nested function and operation value. Closed form of fvoveq1d 7431. (Contributed by AV, 23-Jul-2022.)
(๐ด = ๐ต โ†’ (๐นโ€˜(๐ด๐‘‚๐ถ)) = (๐นโ€˜(๐ต๐‘‚๐ถ)))
 
Theoremovanraleqv 7433* 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.)
(๐ต = ๐‘‹ โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   (๐ต = ๐‘‹ โ†’ (โˆ€๐‘ฅ โˆˆ ๐‘‰ (๐œ‘ โˆง (๐ด ยท ๐ต) = ๐ถ) โ†” โˆ€๐‘ฅ โˆˆ ๐‘‰ (๐œ“ โˆง (๐ด ยท ๐‘‹) = ๐ถ)))
 
Theoremimbrov2fvoveq 7434 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.)
(๐‘‹ = ๐‘Œ โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   (๐‘‹ = ๐‘Œ โ†’ ((๐œ‘ โ†’ (๐นโ€˜((๐บโ€˜๐‘‹) ยท ๐‘‚))๐‘…๐ด) โ†” (๐œ“ โ†’ (๐นโ€˜((๐บโ€˜๐‘Œ) ยท ๐‘‚))๐‘…๐ด)))
 
Theoremovrspc2v 7435* 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.)
(((๐‘‹ โˆˆ ๐ด โˆง ๐‘Œ โˆˆ ๐ต) โˆง โˆ€๐‘ฅ โˆˆ ๐ด โˆ€๐‘ฆ โˆˆ ๐ต (๐‘ฅ๐น๐‘ฆ) โˆˆ ๐ถ) โ†’ (๐‘‹๐น๐‘Œ) โˆˆ ๐ถ)
 
Theoremoveqrspc2v 7436* Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.)
((๐œ‘ โˆง (๐‘ฅ โˆˆ ๐ด โˆง ๐‘ฆ โˆˆ ๐ต)) โ†’ (๐‘ฅ๐น๐‘ฆ) = (๐‘ฅ๐บ๐‘ฆ))    โ‡’   ((๐œ‘ โˆง (๐‘‹ โˆˆ ๐ด โˆง ๐‘Œ โˆˆ ๐ต)) โ†’ (๐‘‹๐น๐‘Œ) = (๐‘‹๐บ๐‘Œ))
 
Theoremoveqdr 7437 Equality of two operations for any two operands. Useful in proofs using *propd theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)
(๐œ‘ โ†’ ๐น = ๐บ)    โ‡’   ((๐œ‘ โˆง ๐œ“) โ†’ (๐‘ฅ๐น๐‘ฆ) = (๐‘ฅ๐บ๐‘ฆ))
 
Theoremnfovd 7438 Deduction version of bound-variable hypothesis builder nfov 7439. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(๐œ‘ โ†’ โ„ฒ๐‘ฅ๐ด)    &   (๐œ‘ โ†’ โ„ฒ๐‘ฅ๐น)    &   (๐œ‘ โ†’ โ„ฒ๐‘ฅ๐ต)    โ‡’   (๐œ‘ โ†’ โ„ฒ๐‘ฅ(๐ด๐น๐ต))
 
Theoremnfov 7439 Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
โ„ฒ๐‘ฅ๐ด    &   โ„ฒ๐‘ฅ๐น    &   โ„ฒ๐‘ฅ๐ต    โ‡’   โ„ฒ๐‘ฅ(๐ด๐น๐ต)
 
Theoremoprabidw 7440* The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of oprabid 7441 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 20-Mar-2013.) Avoid ax-13 2372. (Revised by Gino Giotto, 26-Jan-2024.)
(โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆˆ {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} โ†” ๐œ‘)
 
Theoremoprabid 7441 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 2372. Use the weaker oprabidw 7440 when possible. (Contributed by Mario Carneiro, 20-Mar-2013.) (New usage is discouraged.)
(โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆˆ {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} โ†” ๐œ‘)
 
Theoremovex 7442 The result of an operation is a set. (Contributed by NM, 13-Mar-1995.)
(๐ด๐น๐ต) โˆˆ V
 
Theoremovexi 7443 The result of an operation is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
๐ด = (๐ต๐น๐ถ)    โ‡’   ๐ด โˆˆ V
 
Theoremovexd 7444 The result of an operation is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(๐œ‘ โ†’ (๐ด๐น๐ต) โˆˆ V)
 
Theoremovssunirn 7445 The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.)
(๐‘‹๐น๐‘Œ) โŠ† โˆช ran ๐น
 
Theorem0ov 7446 Operation value of the empty set. (Contributed by AV, 15-May-2021.)
(๐ดโˆ…๐ต) = โˆ…
 
Theoremovprc 7447 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.)
Rel dom ๐น    โ‡’   (ยฌ (๐ด โˆˆ V โˆง ๐ต โˆˆ V) โ†’ (๐ด๐น๐ต) = โˆ…)
 
Theoremovprc1 7448 The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.)
Rel dom ๐น    โ‡’   (ยฌ ๐ด โˆˆ V โ†’ (๐ด๐น๐ต) = โˆ…)
 
Theoremovprc2 7449 The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
Rel dom ๐น    โ‡’   (ยฌ ๐ต โˆˆ V โ†’ (๐ด๐น๐ต) = โˆ…)
 
Theoremovrcl 7450 Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.)
Rel dom ๐น    โ‡’   (๐ถ โˆˆ (๐ด๐น๐ต) โ†’ (๐ด โˆˆ V โˆง ๐ต โˆˆ V))
 
Theoremcsbov123 7451 Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Revised by NM, 23-Aug-2018.)
โฆ‹๐ด / ๐‘ฅโฆŒ(๐ต๐น๐ถ) = (โฆ‹๐ด / ๐‘ฅโฆŒ๐ตโฆ‹๐ด / ๐‘ฅโฆŒ๐นโฆ‹๐ด / ๐‘ฅโฆŒ๐ถ)
 
Theoremcsbov 7452* Move class substitution in and out of an operation. (Contributed by NM, 23-Aug-2018.)
โฆ‹๐ด / ๐‘ฅโฆŒ(๐ต๐น๐ถ) = (๐ตโฆ‹๐ด / ๐‘ฅโฆŒ๐น๐ถ)
 
Theoremcsbov12g 7453* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(๐ด โˆˆ ๐‘‰ โ†’ โฆ‹๐ด / ๐‘ฅโฆŒ(๐ต๐น๐ถ) = (โฆ‹๐ด / ๐‘ฅโฆŒ๐ต๐นโฆ‹๐ด / ๐‘ฅโฆŒ๐ถ))
 
Theoremcsbov1g 7454* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(๐ด โˆˆ ๐‘‰ โ†’ โฆ‹๐ด / ๐‘ฅโฆŒ(๐ต๐น๐ถ) = (โฆ‹๐ด / ๐‘ฅโฆŒ๐ต๐น๐ถ))
 
Theoremcsbov2g 7455* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(๐ด โˆˆ ๐‘‰ โ†’ โฆ‹๐ด / ๐‘ฅโฆŒ(๐ต๐น๐ถ) = (๐ต๐นโฆ‹๐ด / ๐‘ฅโฆŒ๐ถ))
 
Theoremrspceov 7456* A frequently used special case of rspc2ev 3625 for operation values. (Contributed by NM, 21-Mar-2007.)
((๐ถ โˆˆ ๐ด โˆง ๐ท โˆˆ ๐ต โˆง ๐‘† = (๐ถ๐น๐ท)) โ†’ โˆƒ๐‘ฅ โˆˆ ๐ด โˆƒ๐‘ฆ โˆˆ ๐ต ๐‘† = (๐‘ฅ๐น๐‘ฆ))
 
Theoremelovimad 7457 Elementhood of the image set of an operation value. (Contributed by Thierry Arnoux, 13-Mar-2017.)
(๐œ‘ โ†’ ๐ด โˆˆ ๐ถ)    &   (๐œ‘ โ†’ ๐ต โˆˆ ๐ท)    &   (๐œ‘ โ†’ Fun ๐น)    &   (๐œ‘ โ†’ (๐ถ ร— ๐ท) โŠ† dom ๐น)    โ‡’   (๐œ‘ โ†’ (๐ด๐น๐ต) โˆˆ (๐น โ€œ (๐ถ ร— ๐ท)))
 
Theoremfnbrovb 7458 Value of a binary operation expressed as a binary relation. See also fnbrfvb 6945 for functions on Cartesian products. (Contributed by BJ, 15-Feb-2022.)
((๐น Fn (๐‘‰ ร— ๐‘Š) โˆง (๐ด โˆˆ ๐‘‰ โˆง ๐ต โˆˆ ๐‘Š)) โ†’ ((๐ด๐น๐ต) = ๐ถ โ†” โŸจ๐ด, ๐ตโŸฉ๐น๐ถ))
 
Theoremfnotovb 7459 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6946. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 15-Feb-2022.)
((๐น Fn (๐ด ร— ๐ต) โˆง ๐ถ โˆˆ ๐ด โˆง ๐ท โˆˆ ๐ต) โ†’ ((๐ถ๐น๐ท) = ๐‘… โ†” โŸจ๐ถ, ๐ท, ๐‘…โŸฉ โˆˆ ๐น))
 
Theoremopabbrex 7460 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.)
((โˆ€๐‘ฅโˆ€๐‘ฆ(๐‘ฅ๐‘…๐‘ฆ โ†’ ๐œ‘) โˆง {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ๐œ‘} โˆˆ ๐‘‰) โ†’ {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ (๐‘ฅ๐‘…๐‘ฆ โˆง ๐œ“)} โˆˆ V)
 
Theoremopabresex2 7461* 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 ๐‘Š, ๐บ and ๐‘ฅ, ๐‘ฆ to remove hypotheses. (Revised by SN, 13-Dec-2024.)
{โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ (๐‘ฅ(๐‘Šโ€˜๐บ)๐‘ฆ โˆง ๐œƒ)} โˆˆ V
 
Theoremopabresex2d 7462* Obsolete version of opabresex2 7461 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.)
((๐œ‘ โˆง ๐‘ฅ(๐‘Šโ€˜๐บ)๐‘ฆ) โ†’ ๐œ“)    &   (๐œ‘ โ†’ {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ๐œ“} โˆˆ ๐‘‰)    โ‡’   (๐œ‘ โ†’ {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ (๐‘ฅ(๐‘Šโ€˜๐บ)๐‘ฆ โˆง ๐œƒ)} โˆˆ V)
 
Theoremfvmptopab 7463* The function value of a mapping ๐‘€ 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 ๐น restricted by the condition ๐œ“. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.) Add disjoint variable condition on ๐น, ๐‘ฅ, ๐‘ฆ to remove a sethood hypothesis. (Revised by SN, 13-Dec-2024.)
(๐‘ง = ๐‘ โ†’ (๐œ‘ โ†” ๐œ“))    &   ๐‘€ = (๐‘ง โˆˆ V โ†ฆ {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ (๐‘ฅ(๐นโ€˜๐‘ง)๐‘ฆ โˆง ๐œ‘)})    โ‡’   (๐‘€โ€˜๐‘) = {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ (๐‘ฅ(๐นโ€˜๐‘)๐‘ฆ โˆง ๐œ“)}
 
TheoremfvmptopabOLD 7464* Obsolete version of fvmptopab 7463 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.)
((๐œ‘ โˆง ๐‘ง = ๐‘) โ†’ (๐œ’ โ†” ๐œ“))    &   (๐œ‘ โ†’ {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ๐‘ฅ(๐นโ€˜๐‘)๐‘ฆ} โˆˆ V)    &   ๐‘€ = (๐‘ง โˆˆ V โ†ฆ {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ (๐‘ฅ(๐นโ€˜๐‘ง)๐‘ฆ โˆง ๐œ’)})    โ‡’   (๐œ‘ โ†’ (๐‘€โ€˜๐‘) = {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ (๐‘ฅ(๐นโ€˜๐‘)๐‘ฆ โˆง ๐œ“)})
 
Theoremf1opr 7465* Condition for an operation to be one-to-one. (Contributed by Jeff Madsen, 17-Jun-2010.)
(๐น:(๐ด ร— ๐ต)โ€“1-1โ†’๐ถ โ†” (๐น:(๐ด ร— ๐ต)โŸถ๐ถ โˆง โˆ€๐‘Ÿ โˆˆ ๐ด โˆ€๐‘  โˆˆ ๐ต โˆ€๐‘ก โˆˆ ๐ด โˆ€๐‘ข โˆˆ ๐ต ((๐‘Ÿ๐น๐‘ ) = (๐‘ก๐น๐‘ข) โ†’ (๐‘Ÿ = ๐‘ก โˆง ๐‘  = ๐‘ข))))
 
Theorembrfvopab 7466 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.)
(๐‘‹ โˆˆ V โ†’ (๐นโ€˜๐‘‹) = {โŸจ๐‘ฆ, ๐‘งโŸฉ โˆฃ ๐œ‘})    โ‡’   (๐ด(๐นโ€˜๐‘‹)๐ต โ†’ (๐‘‹ โˆˆ V โˆง ๐ด โˆˆ V โˆง ๐ต โˆˆ V))
 
Theoremdfoprab2 7467* Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)
{โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {โŸจ๐‘ค, ๐‘งโŸฉ โˆฃ โˆƒ๐‘ฅโˆƒ๐‘ฆ(๐‘ค = โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆง ๐œ‘)}
 
Theoremreloprab 7468* An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
Rel {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘}
 
Theoremoprabv 7469* 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.)
(โŸจ๐‘‹, ๐‘ŒโŸฉ{โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘}๐‘ โ†’ (๐‘‹ โˆˆ V โˆง ๐‘Œ โˆˆ V โˆง ๐‘ โˆˆ V))
 
Theoremnfoprab1 7470 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
โ„ฒ๐‘ฅ{โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘}
 
Theoremnfoprab2 7471 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.)
โ„ฒ๐‘ฆ{โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘}
 
Theoremnfoprab3 7472 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
โ„ฒ๐‘ง{โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘}
 
Theoremnfoprab 7473* Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)
โ„ฒ๐‘ค๐œ‘    โ‡’   โ„ฒ๐‘ค{โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘}
 
Theoremoprabbid 7474* Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)
โ„ฒ๐‘ฅ๐œ‘    &   โ„ฒ๐‘ฆ๐œ‘    &   โ„ฒ๐‘ง๐œ‘    &   (๐œ‘ โ†’ (๐œ“ โ†” ๐œ’))    โ‡’   (๐œ‘ โ†’ {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“} = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ’})
 
Theoremoprabbidv 7475* Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.)
(๐œ‘ โ†’ (๐œ“ โ†” ๐œ’))    โ‡’   (๐œ‘ โ†’ {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“} = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ’})
 
Theoremoprabbii 7476* Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
(๐œ‘ โ†” ๐œ“)    โ‡’   {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“}
 
Theoremssoprab2 7477 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 5547. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
(โˆ€๐‘ฅโˆ€๐‘ฆโˆ€๐‘ง(๐œ‘ โ†’ ๐œ“) โ†’ {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} โŠ† {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“})
 
Theoremssoprab2b 7478 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 5550. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) (New usage is discouraged.)
({โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} โŠ† {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“} โ†” โˆ€๐‘ฅโˆ€๐‘ฆโˆ€๐‘ง(๐œ‘ โ†’ ๐œ“))
 
Theoremeqoprab2bw 7479* Equivalence of ordered pair abstraction subclass and biconditional. Version of eqoprab2b 7480 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 4-Jan-2017.) Avoid ax-13 2372. (Revised by Gino Giotto, 26-Jan-2024.)
({โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“} โ†” โˆ€๐‘ฅโˆ€๐‘ฆโˆ€๐‘ง(๐œ‘ โ†” ๐œ“))
 
Theoremeqoprab2b 7480 Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 5553. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker eqoprab2bw 7479 when possible. (Contributed by Mario Carneiro, 4-Jan-2017.) (New usage is discouraged.)
({โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“} โ†” โˆ€๐‘ฅโˆ€๐‘ฆโˆ€๐‘ง(๐œ‘ โ†” ๐œ“))
 
Theoremmpoeq123 7481* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.)
((๐ด = ๐ท โˆง โˆ€๐‘ฅ โˆˆ ๐ด (๐ต = ๐ธ โˆง โˆ€๐‘ฆ โˆˆ ๐ต ๐ถ = ๐น)) โ†’ (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = (๐‘ฅ โˆˆ ๐ท, ๐‘ฆ โˆˆ ๐ธ โ†ฆ ๐น))
 
Theoremmpoeq12 7482* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((๐ด = ๐ถ โˆง ๐ต = ๐ท) โ†’ (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ธ) = (๐‘ฅ โˆˆ ๐ถ, ๐‘ฆ โˆˆ ๐ท โ†ฆ ๐ธ))
 
Theoremmpoeq123dva 7483* An equality deduction for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
(๐œ‘ โ†’ ๐ด = ๐ท)    &   ((๐œ‘ โˆง ๐‘ฅ โˆˆ ๐ด) โ†’ ๐ต = ๐ธ)    &   ((๐œ‘ โˆง (๐‘ฅ โˆˆ ๐ด โˆง ๐‘ฆ โˆˆ ๐ต)) โ†’ ๐ถ = ๐น)    โ‡’   (๐œ‘ โ†’ (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = (๐‘ฅ โˆˆ ๐ท, ๐‘ฆ โˆˆ ๐ธ โ†ฆ ๐น))
 
Theoremmpoeq123dv 7484* An equality deduction for the maps-to notation. (Contributed by NM, 12-Sep-2011.)
(๐œ‘ โ†’ ๐ด = ๐ท)    &   (๐œ‘ โ†’ ๐ต = ๐ธ)    &   (๐œ‘ โ†’ ๐ถ = ๐น)    โ‡’   (๐œ‘ โ†’ (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = (๐‘ฅ โˆˆ ๐ท, ๐‘ฆ โˆˆ ๐ธ โ†ฆ ๐น))
 
Theoremmpoeq123i 7485 An equality inference for the maps-to notation. (Contributed by NM, 15-Jul-2013.)
๐ด = ๐ท    &   ๐ต = ๐ธ    &   ๐ถ = ๐น    โ‡’   (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = (๐‘ฅ โˆˆ ๐ท, ๐‘ฆ โˆˆ ๐ธ โ†ฆ ๐น)
 
Theoremmpoeq3dva 7486* Slightly more general equality inference for the maps-to notation. (Contributed by NM, 17-Oct-2013.)
((๐œ‘ โˆง ๐‘ฅ โˆˆ ๐ด โˆง ๐‘ฆ โˆˆ ๐ต) โ†’ ๐ถ = ๐ท)    โ‡’   (๐œ‘ โ†’ (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ท))
 
Theoremmpoeq3ia 7487 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((๐‘ฅ โˆˆ ๐ด โˆง ๐‘ฆ โˆˆ ๐ต) โ†’ ๐ถ = ๐ท)    โ‡’   (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ท)
 
Theoremmpoeq3dv 7488* An equality deduction for the maps-to notation restricted to the value of the operation. (Contributed by SO, 16-Jul-2018.)
(๐œ‘ โ†’ ๐ถ = ๐ท)    โ‡’   (๐œ‘ โ†’ (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ท))
 
Theoremnfmpo1 7489 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
โ„ฒ๐‘ฅ(๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ)
 
Theoremnfmpo2 7490 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
โ„ฒ๐‘ฆ(๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ)
 
Theoremnfmpo 7491* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
โ„ฒ๐‘ง๐ด    &   โ„ฒ๐‘ง๐ต    &   โ„ฒ๐‘ง๐ถ    โ‡’   โ„ฒ๐‘ง(๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ)
 
Theorem0mpo0 7492* A mapping operation with empty domain is empty. Generalization of mpo0 7494. (Contributed by AV, 27-Jan-2024.)
((๐ด = โˆ… โˆจ ๐ต = โˆ…) โ†’ (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = โˆ…)
 
Theoremmpo0v 7493* 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.)
(๐‘ฅ โˆˆ โˆ…, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = โˆ…
 
Theoremmpo0 7494 A mapping operation with empty domain. In this version of mpo0v 7493, 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.)
(๐‘ฅ โˆˆ โˆ…, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = โˆ…
 
Theoremoprab4 7495* Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.)
{โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ (โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆˆ (๐ด ร— ๐ต) โˆง ๐œ‘)} = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ((๐‘ฅ โˆˆ ๐ด โˆง ๐‘ฆ โˆˆ ๐ต) โˆง ๐œ‘)}
 
Theoremcbvoprab1 7496* 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.)
โ„ฒ๐‘ค๐œ‘    &   โ„ฒ๐‘ฅ๐œ“    &   (๐‘ฅ = ๐‘ค โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {โŸจโŸจ๐‘ค, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“}
 
Theoremcbvoprab2 7497* Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
โ„ฒ๐‘ค๐œ‘    &   โ„ฒ๐‘ฆ๐œ“    &   (๐‘ฆ = ๐‘ค โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {โŸจโŸจ๐‘ฅ, ๐‘คโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“}
 
Theoremcbvoprab12 7498* 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.)
โ„ฒ๐‘ค๐œ‘    &   โ„ฒ๐‘ฃ๐œ‘    &   โ„ฒ๐‘ฅ๐œ“    &   โ„ฒ๐‘ฆ๐œ“    &   ((๐‘ฅ = ๐‘ค โˆง ๐‘ฆ = ๐‘ฃ) โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {โŸจโŸจ๐‘ค, ๐‘ฃโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“}
 
Theoremcbvoprab12v 7499* Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)
((๐‘ฅ = ๐‘ค โˆง ๐‘ฆ = ๐‘ฃ) โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {โŸจโŸจ๐‘ค, ๐‘ฃโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“}
 
Theoremcbvoprab3 7500* Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)
โ„ฒ๐‘ค๐œ‘    &   โ„ฒ๐‘ง๐œ“    &   (๐‘ง = ๐‘ค โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘คโŸฉ โˆฃ ๐œ“}
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