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Theorem List for Intuitionistic Logic Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoveqi 5901 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 5902 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 5903 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 5904 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 5905 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 5906 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 5907 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 5908 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 5909 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 5910 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 5911 Equality theorem for nested function and operation value. Closed form of fvoveq1d 5910. (Contributed by AV, 23-Jul-2022.)
(๐ด = ๐ต โ†’ (๐นโ€˜(๐ด๐‘‚๐ถ)) = (๐นโ€˜(๐ต๐‘‚๐ถ)))
 
Theoremovanraleqv 5912* 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 5913 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 5914* 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 5915* 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 5916 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 5917 Deduction version of bound-variable hypothesis builder nfov 5918. (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 5918 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)
 
Theoremoprabidlem 5919* Slight elaboration of exdistrfor 1810. A lemma for oprabid 5920. (Contributed by Jim Kingdon, 15-Jan-2019.)
(โˆƒ๐‘ฅโˆƒ๐‘ฆ(๐‘ฅ = ๐‘ง โˆง ๐œ“) โ†’ โˆƒ๐‘ฅ(๐‘ฅ = ๐‘ง โˆง โˆƒ๐‘ฆ๐œ“))
 
Theoremoprabid 5920 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable condition between ๐‘ฅ, ๐‘ฆ, and ๐‘ง, we use ax-bndl 1519 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)
(โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆˆ {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} โ†” ๐œ‘)
 
Theoremfnovex 5921 The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.)
((๐น Fn (๐ถ ร— ๐ท) โˆง ๐ด โˆˆ ๐ถ โˆง ๐ต โˆˆ ๐ท) โ†’ (๐ด๐น๐ต) โˆˆ V)
 
Theoremovexg 5922 Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.)
((๐ด โˆˆ ๐‘‰ โˆง ๐น โˆˆ ๐‘Š โˆง ๐ต โˆˆ ๐‘‹) โ†’ (๐ด๐น๐ต) โˆˆ V)
 
Theoremovprc 5923 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 5924 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 5925 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…)
 
Theoremcsbov123g 5926 Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
(๐ด โˆˆ ๐ท โ†’ 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))
 
Theoremcsbov12g 5927* 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 5928* 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 5929* 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 5930* A frequently used special case of rspc2ev 2868 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))
 
Theoremfnotovb 5931 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5570. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
((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 5932* A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
((๐‘‰ โˆˆ V โˆง ๐ธ โˆˆ V) โ†’ (๐‘“(๐‘‰๐‘Š๐ธ)๐‘ โ†’ ๐œƒ))    &   ((๐‘‰ โˆˆ V โˆง ๐ธ โˆˆ V) โ†’ {โŸจ๐‘“, ๐‘โŸฉ โˆฃ ๐œƒ} โˆˆ V)    โ‡’   ((๐‘‰ โˆˆ V โˆง ๐ธ โˆˆ V) โ†’ {โŸจ๐‘“, ๐‘โŸฉ โˆฃ (๐‘“(๐‘‰๐‘Š๐ธ)๐‘ โˆง ๐œ“)} โˆˆ V)
 
Theorem0neqopab 5933 The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
ยฌ โˆ… โˆˆ {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ๐œ‘}
 
Theorembrabvv 5934* If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Jim Kingdon, 16-Jan-2019.)
(๐‘‹{โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ๐œ‘}๐‘Œ โ†’ (๐‘‹ โˆˆ V โˆง ๐‘Œ โˆˆ V))
 
Theoremdfoprab2 5935* 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 5936* An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
Rel {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘}
 
Theoremnfoprab1 5937 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 5938 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 5939 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 5940* 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 5941* 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 5942* 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 5943* 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 5944 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4287. (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‘} โІ {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“})
 
Theoremssoprab2b 5945 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4288. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
({โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} โІ {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“} โ†” โˆ€๐‘ฅโˆ€๐‘ฆโˆ€๐‘ง(๐œ‘ โ†’ ๐œ“))
 
Theoremeqoprab2b 5946 Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4291. (Contributed by Mario Carneiro, 4-Jan-2017.)
({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 5947* 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 5948* 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 5949* 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 5950* 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 5951 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 5952* 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 5953 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 5954* 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 5955 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 5956 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 5957* 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)
 
Theoremmpo0 5958 A mapping operation with empty domain. (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 5959* 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 5960* 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 5961* 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 5962* 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 5963* 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 5964* 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“}
 
Theoremcbvoprab3v 5965* 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.)
(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“}
 
Theoremcbvmpox 5966* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 5967 allows ๐ต to be a function of ๐‘ฅ. (Contributed by NM, 29-Dec-2014.)
โ„ฒ๐‘ง๐ต    &   โ„ฒ๐‘ฅ๐ท    &   โ„ฒ๐‘ง๐ถ    &   โ„ฒ๐‘ค๐ถ    &   โ„ฒ๐‘ฅ๐ธ    &   โ„ฒ๐‘ฆ๐ธ    &   (๐‘ฅ = ๐‘ง โ†’ ๐ต = ๐ท)    &   ((๐‘ฅ = ๐‘ง โˆง ๐‘ฆ = ๐‘ค) โ†’ ๐ถ = ๐ธ)    โ‡’   (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = (๐‘ง โˆˆ ๐ด, ๐‘ค โˆˆ ๐ท โ†ฆ ๐ธ)
 
Theoremcbvmpo 5967* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)
โ„ฒ๐‘ง๐ถ    &   โ„ฒ๐‘ค๐ถ    &   โ„ฒ๐‘ฅ๐ท    &   โ„ฒ๐‘ฆ๐ท    &   ((๐‘ฅ = ๐‘ง โˆง ๐‘ฆ = ๐‘ค) โ†’ ๐ถ = ๐ท)    โ‡’   (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = (๐‘ง โˆˆ ๐ด, ๐‘ค โˆˆ ๐ต โ†ฆ ๐ท)
 
Theoremcbvmpov 5968* Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4110, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
(๐‘ฅ = ๐‘ง โ†’ ๐ถ = ๐ธ)    &   (๐‘ฆ = ๐‘ค โ†’ ๐ธ = ๐ท)    โ‡’   (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ) = (๐‘ง โˆˆ ๐ด, ๐‘ค โˆˆ ๐ต โ†ฆ ๐ท)
 
Theoremdmoprab 5969* The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
dom {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ โˆƒ๐‘ง๐œ‘}
 
Theoremdmoprabss 5970* The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)
dom {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ((๐‘ฅ โˆˆ ๐ด โˆง ๐‘ฆ โˆˆ ๐ต) โˆง ๐œ‘)} โІ (๐ด ร— ๐ต)
 
Theoremrnoprab 5971* The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)
ran {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {๐‘ง โˆฃ โˆƒ๐‘ฅโˆƒ๐‘ฆ๐œ‘}
 
Theoremrnoprab2 5972* The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)
ran {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ((๐‘ฅ โˆˆ ๐ด โˆง ๐‘ฆ โˆˆ ๐ต) โˆง ๐œ‘)} = {๐‘ง โˆฃ โˆƒ๐‘ฅ โˆˆ ๐ด โˆƒ๐‘ฆ โˆˆ ๐ต ๐œ‘}
 
Theoremreldmoprab 5973* The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)
Rel dom {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘}
 
Theoremoprabss 5974* Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)
{โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} โІ ((V ร— V) ร— V)
 
Theoremeloprabga 5975* The law of concretion for operation class abstraction. Compare elopab 4270. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
((๐‘ฅ = ๐ด โˆง ๐‘ฆ = ๐ต โˆง ๐‘ง = ๐ถ) โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   ((๐ด โˆˆ ๐‘‰ โˆง ๐ต โˆˆ ๐‘Š โˆง ๐ถ โˆˆ ๐‘‹) โ†’ (โŸจโŸจ๐ด, ๐ตโŸฉ, ๐ถโŸฉ โˆˆ {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} โ†” ๐œ“))
 
Theoremeloprabg 5976* The law of concretion for operation class abstraction. Compare elopab 4270. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
(๐‘ฅ = ๐ด โ†’ (๐œ‘ โ†” ๐œ“))    &   (๐‘ฆ = ๐ต โ†’ (๐œ“ โ†” ๐œ’))    &   (๐‘ง = ๐ถ โ†’ (๐œ’ โ†” ๐œƒ))    โ‡’   ((๐ด โˆˆ ๐‘‰ โˆง ๐ต โˆˆ ๐‘Š โˆง ๐ถ โˆˆ ๐‘‹) โ†’ (โŸจโŸจ๐ด, ๐ตโŸฉ, ๐ถโŸฉ โˆˆ {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} โ†” ๐œƒ))
 
Theoremssoprab2i 5977* Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)
(๐œ‘ โ†’ ๐œ“)    โ‡’   {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} โІ {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ“}
 
Theoremmpov 5978* Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
(๐‘ฅ โˆˆ V, ๐‘ฆ โˆˆ V โ†ฆ ๐ถ) = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐‘ง = ๐ถ}
 
Theoremmpomptx 5979* Express a two-argument function as a one-argument function, or vice-versa. In this version ๐ต(๐‘ฅ) is not assumed to be constant w.r.t ๐‘ฅ. (Contributed by Mario Carneiro, 29-Dec-2014.)
(๐‘ง = โŸจ๐‘ฅ, ๐‘ฆโŸฉ โ†’ ๐ถ = ๐ท)    โ‡’   (๐‘ง โˆˆ โˆช ๐‘ฅ โˆˆ ๐ด ({๐‘ฅ} ร— ๐ต) โ†ฆ ๐ถ) = (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ท)
 
Theoremmpompt 5980* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.)
(๐‘ง = โŸจ๐‘ฅ, ๐‘ฆโŸฉ โ†’ ๐ถ = ๐ท)    โ‡’   (๐‘ง โˆˆ (๐ด ร— ๐ต) โ†ฆ ๐ถ) = (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ท)
 
Theoremmpodifsnif 5981 A mapping with two arguments with the first argument from a difference set with a singleton and a conditional as result. (Contributed by AV, 13-Feb-2019.)
(๐‘– โˆˆ (๐ด โˆ– {๐‘‹}), ๐‘— โˆˆ ๐ต โ†ฆ if(๐‘– = ๐‘‹, ๐ถ, ๐ท)) = (๐‘– โˆˆ (๐ด โˆ– {๐‘‹}), ๐‘— โˆˆ ๐ต โ†ฆ ๐ท)
 
Theoremmposnif 5982 A mapping with two arguments with the first argument from a singleton and a conditional as result. (Contributed by AV, 14-Feb-2019.)
(๐‘– โˆˆ {๐‘‹}, ๐‘— โˆˆ ๐ต โ†ฆ if(๐‘– = ๐‘‹, ๐ถ, ๐ท)) = (๐‘– โˆˆ {๐‘‹}, ๐‘— โˆˆ ๐ต โ†ฆ ๐ถ)
 
Theoremfconstmpo 5983* Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.)
((๐ด ร— ๐ต) ร— {๐ถ}) = (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ)
 
Theoremresoprab 5984* Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)
({โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘} โ†พ (๐ด ร— ๐ต)) = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ((๐‘ฅ โˆˆ ๐ด โˆง ๐‘ฆ โˆˆ ๐ต) โˆง ๐œ‘)}
 
Theoremresoprab2 5985* Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
((๐ถ โІ ๐ด โˆง ๐ท โІ ๐ต) โ†’ ({โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ((๐‘ฅ โˆˆ ๐ด โˆง ๐‘ฆ โˆˆ ๐ต) โˆง ๐œ‘)} โ†พ (๐ถ ร— ๐ท)) = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ((๐‘ฅ โˆˆ ๐ถ โˆง ๐‘ฆ โˆˆ ๐ท) โˆง ๐œ‘)})
 
Theoremresmpo 5986* Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)
((๐ถ โІ ๐ด โˆง ๐ท โІ ๐ต) โ†’ ((๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ธ) โ†พ (๐ถ ร— ๐ท)) = (๐‘ฅ โˆˆ ๐ถ, ๐‘ฆ โˆˆ ๐ท โ†ฆ ๐ธ))
 
Theoremfunoprabg 5987* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.)
(โˆ€๐‘ฅโˆ€๐‘ฆโˆƒ*๐‘ง๐œ‘ โ†’ Fun {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘})
 
Theoremfunoprab 5988* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.)
โˆƒ*๐‘ง๐œ‘    โ‡’   Fun {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘}
 
Theoremfnoprabg 5989* Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)
(โˆ€๐‘ฅโˆ€๐‘ฆ(๐œ‘ โ†’ โˆƒ!๐‘ง๐œ“) โ†’ {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ (๐œ‘ โˆง ๐œ“)} Fn {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ๐œ‘})
 
Theoremmpofun 5990* The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)
๐น = (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ)    โ‡’   Fun ๐น
 
Theoremfnoprab 5991* Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.)
(๐œ‘ โ†’ โˆƒ!๐‘ง๐œ“)    โ‡’   {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ (๐œ‘ โˆง ๐œ“)} Fn {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ๐œ‘}
 
Theoremffnov 5992* An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.)
(๐น:(๐ด ร— ๐ต)โŸถ๐ถ โ†” (๐น Fn (๐ด ร— ๐ต) โˆง โˆ€๐‘ฅ โˆˆ ๐ด โˆ€๐‘ฆ โˆˆ ๐ต (๐‘ฅ๐น๐‘ฆ) โˆˆ ๐ถ))
 
Theoremfovcl 5993 Closure law for an operation. (Contributed by NM, 19-Apr-2007.)
๐น:(๐‘… ร— ๐‘†)โŸถ๐ถ    โ‡’   ((๐ด โˆˆ ๐‘… โˆง ๐ต โˆˆ ๐‘†) โ†’ (๐ด๐น๐ต) โˆˆ ๐ถ)
 
Theoremeqfnov 5994* Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)
((๐น Fn (๐ด ร— ๐ต) โˆง ๐บ Fn (๐ถ ร— ๐ท)) โ†’ (๐น = ๐บ โ†” ((๐ด ร— ๐ต) = (๐ถ ร— ๐ท) โˆง โˆ€๐‘ฅ โˆˆ ๐ด โˆ€๐‘ฆ โˆˆ ๐ต (๐‘ฅ๐น๐‘ฆ) = (๐‘ฅ๐บ๐‘ฆ))))
 
Theoremeqfnov2 5995* Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.)
((๐น Fn (๐ด ร— ๐ต) โˆง ๐บ Fn (๐ด ร— ๐ต)) โ†’ (๐น = ๐บ โ†” โˆ€๐‘ฅ โˆˆ ๐ด โˆ€๐‘ฆ โˆˆ ๐ต (๐‘ฅ๐น๐‘ฆ) = (๐‘ฅ๐บ๐‘ฆ)))
 
Theoremfnovim 5996* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 16-Jan-2019.)
(๐น Fn (๐ด ร— ๐ต) โ†’ ๐น = (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ (๐‘ฅ๐น๐‘ฆ)))
 
Theoremmpo2eqb 5997* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5995. (Contributed by Mario Carneiro, 4-Jan-2017.)
(โˆ€๐‘ฅ โˆˆ ๐ด โˆ€๐‘ฆ โˆˆ ๐ต ๐ถ โˆˆ ๐‘‰ 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) โ†” โˆ€๐‘ฅ โˆˆ ๐ด โˆ€๐‘ฆ โˆˆ ๐ต ๐ถ = ๐ท))
 
Theoremrnmpo 5998* The range of an operation given by the maps-to notation. (Contributed by FL, 20-Jun-2011.)
๐น = (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ)    โ‡’   ran ๐น = {๐‘ง โˆฃ โˆƒ๐‘ฅ โˆˆ ๐ด โˆƒ๐‘ฆ โˆˆ ๐ต ๐‘ง = ๐ถ}
 
Theoremreldmmpo 5999* The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)
๐น = (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ)    โ‡’   Rel dom ๐น
 
Theoremelrnmpog 6000* Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
๐น = (๐‘ฅ โˆˆ ๐ด, ๐‘ฆ โˆˆ ๐ต โ†ฆ ๐ถ)    โ‡’   (๐ท โˆˆ ๐‘‰ โ†’ (๐ท โˆˆ ran ๐น โ†” โˆƒ๐‘ฅ โˆˆ ๐ด โˆƒ๐‘ฆ โˆˆ ๐ต ๐ท = ๐ถ))
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