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Theorem List for Metamath Proof Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeqfunressuc 7301 Law for equality of restriction to successors. This is primarily useful when š‘‹ is an ordinal, but it does not require that. (Contributed by Scott Fenton, 6-Dec-2021.)
(((Fun š¹ āˆ§ Fun šŗ) āˆ§ (š¹ ā†¾ š‘‹) = (šŗ ā†¾ š‘‹) āˆ§ (š‘‹ āˆˆ dom š¹ āˆ§ š‘‹ āˆˆ dom šŗ āˆ§ (š¹ā€˜š‘‹) = (šŗā€˜š‘‹))) ā†’ (š¹ ā†¾ suc š‘‹) = (šŗ ā†¾ suc š‘‹))
 
Theoremfnssintima 7302* Condition for subset of an intersection of an image. (Contributed by Scott Fenton, 16-Aug-2024.)
((š¹ Fn š“ āˆ§ šµ āŠ† š“) ā†’ (š¶ āŠ† āˆ© (š¹ ā€œ šµ) ā†” āˆ€š‘„ āˆˆ šµ š¶ āŠ† (š¹ā€˜š‘„)))
 
2.3.17  Cantor's Theorem
 
Theoremcanth 7303 No set š“ is equinumerous to its power set (Cantor's theorem), i.e., no function can map š“ onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 9008. Note that š“ must be a set: this theorem does not hold when š“ is too large to be a set; see ncanth 7304 for a counterexample. (Use nex 1803 if you want the form Ā¬ āˆƒš‘“š‘“:š“ā€“ontoā†’š’« š“.) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
š“ āˆˆ V    ā‡’    Ā¬ š¹:š“ā€“ontoā†’š’« š“
 
Theoremncanth 7304 Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5271). Specifically, the identity function maps the universe onto its power class. Compare canth 7303 that works for sets.

This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3737): š’« V, being a class, cannot contain proper classes, so it is no larger than V, which is why the identity function "succeeds" in being surjective onto š’« V (see pwv 4861). See also the remark in ru 3737 about NF, in which Cantor's theorem fails for sets that are "too large". This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set. (Contributed by NM, 29-Jun-2004.) (Proof shortened by BJ, 29-Dec-2023.)

I :Vā€“ontoā†’š’« V
 
2.3.18  Restricted iota (description binder)
 
Syntaxcrio 7305 Extend class notation with restricted description binder.
class (ā„©š‘„ āˆˆ š“ šœ‘)
 
Definitiondf-riota 7306 Define restricted description binder. In case there is no unique š‘„ such that (š‘„ āˆˆ š“ āˆ§ šœ‘) holds, it evaluates to the empty set. See also comments for df-iota 6444. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)
(ā„©š‘„ āˆˆ š“ šœ‘) = (ā„©š‘„(š‘„ āˆˆ š“ āˆ§ šœ‘))
 
Theoremriotaeqdv 7307* Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.)
(šœ‘ ā†’ š“ = šµ)    ā‡’   (šœ‘ ā†’ (ā„©š‘„ āˆˆ š“ šœ“) = (ā„©š‘„ āˆˆ šµ šœ“))
 
Theoremriotabidv 7308* Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)
(šœ‘ ā†’ (šœ“ ā†” šœ’))    ā‡’   (šœ‘ ā†’ (ā„©š‘„ āˆˆ š“ šœ“) = (ā„©š‘„ āˆˆ š“ šœ’))
 
Theoremriotaeqbidv 7309* Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
(šœ‘ ā†’ š“ = šµ)    &   (šœ‘ ā†’ (šœ“ ā†” šœ’))    ā‡’   (šœ‘ ā†’ (ā„©š‘„ āˆˆ š“ šœ“) = (ā„©š‘„ āˆˆ šµ šœ’))
 
Theoremriotaex 7310 Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)
(ā„©š‘„ āˆˆ š“ šœ“) āˆˆ V
 
Theoremriotav 7311 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
(ā„©š‘„ āˆˆ V šœ‘) = (ā„©š‘„šœ‘)
 
Theoremriotauni 7312 Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
(āˆƒ!š‘„ āˆˆ š“ šœ‘ ā†’ (ā„©š‘„ āˆˆ š“ šœ‘) = āˆŖ {š‘„ āˆˆ š“ āˆ£ šœ‘})
 
Theoremnfriota1 7313* The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
ā„²š‘„(ā„©š‘„ āˆˆ š“ šœ‘)
 
Theoremnfriotadw 7314* Deduction version of nfriota 7319 with a disjoint variable condition, which contrary to nfriotad 7318 does not require ax-13 2372. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2372. (Revised by Gino Giotto, 26-Jan-2024.)
ā„²š‘¦šœ‘    &   (šœ‘ ā†’ ā„²š‘„šœ“)    &   (šœ‘ ā†’ ā„²š‘„š“)    ā‡’   (šœ‘ ā†’ ā„²š‘„(ā„©š‘¦ āˆˆ š“ šœ“))
 
Theoremcbvriotaw 7315* Change bound variable in a restricted description binder. Version of cbvriota 7320 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 18-Mar-2013.) Avoid ax-13 2372. (Revised by Gino Giotto, 26-Jan-2024.)
ā„²š‘¦šœ‘    &   ā„²š‘„šœ“    &   (š‘„ = š‘¦ ā†’ (šœ‘ ā†” šœ“))    ā‡’   (ā„©š‘„ āˆˆ š“ šœ‘) = (ā„©š‘¦ āˆˆ š“ šœ“)
 
Theoremcbvriotavw 7316* Change bound variable in a restricted description binder. Version of cbvriotav 7321 with a disjoint variable condition, which requires fewer axioms . (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 30-Sep-2024.)
(š‘„ = š‘¦ ā†’ (šœ‘ ā†” šœ“))    ā‡’   (ā„©š‘„ āˆˆ š“ šœ‘) = (ā„©š‘¦ āˆˆ š“ šœ“)
 
TheoremcbvriotavwOLD 7317* Obsolete version of cbvriotavw 7316 as of 30-Sep-2024. (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 26-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(š‘„ = š‘¦ ā†’ (šœ‘ ā†” šœ“))    ā‡’   (ā„©š‘„ āˆˆ š“ šœ‘) = (ā„©š‘¦ āˆˆ š“ šœ“)
 
Theoremnfriotad 7318 Deduction version of nfriota 7319. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfriotadw 7314 when possible. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
ā„²š‘¦šœ‘    &   (šœ‘ ā†’ ā„²š‘„šœ“)    &   (šœ‘ ā†’ ā„²š‘„š“)    ā‡’   (šœ‘ ā†’ ā„²š‘„(ā„©š‘¦ āˆˆ š“ šœ“))
 
Theoremnfriota 7319* A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)
ā„²š‘„šœ‘    &   ā„²š‘„š“    ā‡’   ā„²š‘„(ā„©š‘¦ āˆˆ š“ šœ‘)
 
Theoremcbvriota 7320* Change bound variable in a restricted description binder. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvriotaw 7315 when possible. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
ā„²š‘¦šœ‘    &   ā„²š‘„šœ“    &   (š‘„ = š‘¦ ā†’ (šœ‘ ā†” šœ“))    ā‡’   (ā„©š‘„ āˆˆ š“ šœ‘) = (ā„©š‘¦ āˆˆ š“ šœ“)
 
Theoremcbvriotav 7321* Change bound variable in a restricted description binder. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvriotavw 7316 when possible. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
(š‘„ = š‘¦ ā†’ (šœ‘ ā†” šœ“))    ā‡’   (ā„©š‘„ āˆˆ š“ šœ‘) = (ā„©š‘¦ āˆˆ š“ šœ“)
 
Theoremcsbriota 7322* Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 2-Sep-2018.)
ā¦‹š“ / š‘„ā¦Œ(ā„©š‘¦ āˆˆ šµ šœ‘) = (ā„©š‘¦ āˆˆ šµ [š“ / š‘„]šœ‘)
 
Theoremriotacl2 7323 Membership law for "the unique element in š“ such that šœ‘". (Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(āˆƒ!š‘„ āˆˆ š“ šœ‘ ā†’ (ā„©š‘„ āˆˆ š“ šœ‘) āˆˆ {š‘„ āˆˆ š“ āˆ£ šœ‘})
 
Theoremriotacl 7324* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
(āˆƒ!š‘„ āˆˆ š“ šœ‘ ā†’ (ā„©š‘„ āˆˆ š“ šœ‘) āˆˆ š“)
 
Theoremriotasbc 7325 Substitution law for descriptions. Compare iotasbc 42432. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
(āˆƒ!š‘„ āˆˆ š“ šœ‘ ā†’ [(ā„©š‘„ āˆˆ š“ šœ‘) / š‘„]šœ‘)
 
Theoremriotabidva 7326* Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3413 analog.) (Contributed by NM, 17-Jan-2012.)
((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (šœ“ ā†” šœ’))    ā‡’   (šœ‘ ā†’ (ā„©š‘„ āˆˆ š“ šœ“) = (ā„©š‘„ āˆˆ š“ šœ’))
 
Theoremriotabiia 7327 Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3410 analog.) (Contributed by NM, 16-Jan-2012.)
(š‘„ āˆˆ š“ ā†’ (šœ‘ ā†” šœ“))    ā‡’   (ā„©š‘„ āˆˆ š“ šœ‘) = (ā„©š‘„ āˆˆ š“ šœ“)
 
Theoremriota1 7328* Property of restricted iota. Compare iota1 6469. (Contributed by Mario Carneiro, 15-Oct-2016.)
(āˆƒ!š‘„ āˆˆ š“ šœ‘ ā†’ ((š‘„ āˆˆ š“ āˆ§ šœ‘) ā†” (ā„©š‘„ āˆˆ š“ šœ‘) = š‘„))
 
Theoremriota1a 7329 Property of iota. (Contributed by NM, 23-Aug-2011.)
((š‘„ āˆˆ š“ āˆ§ āˆƒ!š‘„ āˆˆ š“ šœ‘) ā†’ (šœ‘ ā†” (ā„©š‘„(š‘„ āˆˆ š“ āˆ§ šœ‘)) = š‘„))
 
Theoremriota2df 7330* A deduction version of riota2f 7331. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
ā„²š‘„šœ‘    &   (šœ‘ ā†’ ā„²š‘„šµ)    &   (šœ‘ ā†’ ā„²š‘„šœ’)    &   (šœ‘ ā†’ šµ āˆˆ š“)    &   ((šœ‘ āˆ§ š‘„ = šµ) ā†’ (šœ“ ā†” šœ’))    ā‡’   ((šœ‘ āˆ§ āˆƒ!š‘„ āˆˆ š“ šœ“) ā†’ (šœ’ ā†” (ā„©š‘„ āˆˆ š“ šœ“) = šµ))
 
Theoremriota2f 7331* This theorem shows a condition that allows to represent a descriptor with a class expression šµ. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
ā„²š‘„šµ    &   ā„²š‘„šœ“    &   (š‘„ = šµ ā†’ (šœ‘ ā†” šœ“))    ā‡’   ((šµ āˆˆ š“ āˆ§ āˆƒ!š‘„ āˆˆ š“ šœ‘) ā†’ (šœ“ ā†” (ā„©š‘„ āˆˆ š“ šœ‘) = šµ))
 
Theoremriota2 7332* This theorem shows a condition that allows to represent a descriptor with a class expression šµ. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.)
(š‘„ = šµ ā†’ (šœ‘ ā†” šœ“))    ā‡’   ((šµ āˆˆ š“ āˆ§ āˆƒ!š‘„ āˆˆ š“ šœ‘) ā†’ (šœ“ ā†” (ā„©š‘„ āˆˆ š“ šœ‘) = šµ))
 
Theoremriotaeqimp 7333* If two restricted iota descriptors for an equality are equal, then the terms of the equality are equal. (Contributed by AV, 6-Dec-2020.)
š¼ = (ā„©š‘Ž āˆˆ š‘‰ š‘‹ = š“)    &   š½ = (ā„©š‘Ž āˆˆ š‘‰ š‘Œ = š“)    &   (šœ‘ ā†’ āˆƒ!š‘Ž āˆˆ š‘‰ š‘‹ = š“)    &   (šœ‘ ā†’ āˆƒ!š‘Ž āˆˆ š‘‰ š‘Œ = š“)    ā‡’   ((šœ‘ āˆ§ š¼ = š½) ā†’ š‘‹ = š‘Œ)
 
Theoremriotaprop 7334* Properties of a restricted definite description operator. (Contributed by NM, 23-Nov-2013.)
ā„²š‘„šœ“    &   šµ = (ā„©š‘„ āˆˆ š“ šœ‘)    &   (š‘„ = šµ ā†’ (šœ‘ ā†” šœ“))    ā‡’   (āˆƒ!š‘„ āˆˆ š“ šœ‘ ā†’ (šµ āˆˆ š“ āˆ§ šœ“))
 
Theoremriota5f 7335* A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(šœ‘ ā†’ ā„²š‘„šµ)    &   (šœ‘ ā†’ šµ āˆˆ š“)    &   ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (šœ“ ā†” š‘„ = šµ))    ā‡’   (šœ‘ ā†’ (ā„©š‘„ āˆˆ š“ šœ“) = šµ)
 
Theoremriota5 7336* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)
(šœ‘ ā†’ šµ āˆˆ š“)    &   ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (šœ“ ā†” š‘„ = šµ))    ā‡’   (šœ‘ ā†’ (ā„©š‘„ āˆˆ š“ šœ“) = šµ)
 
Theoremriotass2 7337* Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
(((š“ āŠ† šµ āˆ§ āˆ€š‘„ āˆˆ š“ (šœ‘ ā†’ šœ“)) āˆ§ (āˆƒš‘„ āˆˆ š“ šœ‘ āˆ§ āˆƒ!š‘„ āˆˆ šµ šœ“)) ā†’ (ā„©š‘„ āˆˆ š“ šœ‘) = (ā„©š‘„ āˆˆ šµ šœ“))
 
Theoremriotass 7338* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
((š“ āŠ† šµ āˆ§ āˆƒš‘„ āˆˆ š“ šœ‘ āˆ§ āˆƒ!š‘„ āˆˆ šµ šœ‘) ā†’ (ā„©š‘„ āˆˆ š“ šœ‘) = (ā„©š‘„ āˆˆ šµ šœ‘))
 
Theoremmoriotass 7339* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
((š“ āŠ† šµ āˆ§ āˆƒš‘„ āˆˆ š“ šœ‘ āˆ§ āˆƒ*š‘„ āˆˆ šµ šœ‘) ā†’ (ā„©š‘„ āˆˆ š“ šœ‘) = (ā„©š‘„ āˆˆ šµ šœ‘))
 
Theoremsnriota 7340 A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
(āˆƒ!š‘„ āˆˆ š“ šœ‘ ā†’ {š‘„ āˆˆ š“ āˆ£ šœ‘} = {(ā„©š‘„ āˆˆ š“ šœ‘)})
 
Theoremriotaxfrd 7341* Change the variable š‘„ in the expression for "the unique š‘„ such that šœ“ " to another variable š‘¦ contained in expression šµ. Use reuhypd 5373 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
ā„²š‘¦š¶    &   ((šœ‘ āˆ§ š‘¦ āˆˆ š“) ā†’ šµ āˆˆ š“)    &   ((šœ‘ āˆ§ (ā„©š‘¦ āˆˆ š“ šœ’) āˆˆ š“) ā†’ š¶ āˆˆ š“)    &   (š‘„ = šµ ā†’ (šœ“ ā†” šœ’))    &   (š‘¦ = (ā„©š‘¦ āˆˆ š“ šœ’) ā†’ šµ = š¶)    &   ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ āˆƒ!š‘¦ āˆˆ š“ š‘„ = šµ)    ā‡’   ((šœ‘ āˆ§ āˆƒ!š‘„ āˆˆ š“ šœ“) ā†’ (ā„©š‘„ āˆˆ š“ šœ“) = š¶)
 
Theoremeusvobj2 7342* Specify the same property in two ways when class šµ(š‘¦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
šµ āˆˆ V    ā‡’   (āˆƒ!š‘„āˆƒš‘¦ āˆˆ š“ š‘„ = šµ ā†’ (āˆƒš‘¦ āˆˆ š“ š‘„ = šµ ā†” āˆ€š‘¦ āˆˆ š“ š‘„ = šµ))
 
Theoremeusvobj1 7343* Specify the same object in two ways when class šµ(š‘¦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
šµ āˆˆ V    ā‡’   (āˆƒ!š‘„āˆƒš‘¦ āˆˆ š“ š‘„ = šµ ā†’ (ā„©š‘„āˆƒš‘¦ āˆˆ š“ š‘„ = šµ) = (ā„©š‘„āˆ€š‘¦ āˆˆ š“ š‘„ = šµ))
 
Theoremf1ofveu 7344* There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)
((š¹:š“ā€“1-1-ontoā†’šµ āˆ§ š¶ āˆˆ šµ) ā†’ āˆƒ!š‘„ āˆˆ š“ (š¹ā€˜š‘„) = š¶)
 
Theoremf1ocnvfv3 7345* 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.)
((š¹:š“ā€“1-1-ontoā†’šµ āˆ§ š¶ āˆˆ šµ) ā†’ (ā—”š¹ā€˜š¶) = (ā„©š‘„ āˆˆ š“ (š¹ā€˜š‘„) = š¶))
 
Theoremriotaund 7346* 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.)
(Ā¬ āˆƒ!š‘„ āˆˆ š“ šœ‘ ā†’ (ā„©š‘„ āˆˆ š“ šœ‘) = āˆ…)
 
Theoremriotassuni 7347* The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)
(ā„©š‘„ āˆˆ š“ šœ‘) āŠ† (š’« āˆŖ š“ āˆŖ āˆŖ š“)
 
Theoremriotaclb 7348* 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.)
(Ā¬ āˆ… āˆˆ š“ ā†’ (āˆƒ!š‘„ āˆˆ š“ šœ‘ ā†” (ā„©š‘„ āˆˆ š“ šœ‘) āˆˆ š“))
 
Theoremriotarab 7349* Restricted iota of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.)
(š‘„ = š‘¦ ā†’ (šœ‘ ā†” šœ“))    ā‡’   (ā„©š‘„ āˆˆ {š‘¦ āˆˆ š“ āˆ£ šœ“}šœ’) = (ā„©š‘„ āˆˆ š“ (šœ‘ āˆ§ šœ’))
 
2.3.19  Operations
 
Syntaxco 7350 Extend class notation to include the value of an operation š¹ (such as +) for two arguments š“ and šµ. 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 11320.)
class (š“š¹šµ)
 
Syntaxcoprab 7351 Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ šœ‘}
 
Syntaxcmpo 7352 Extend the definition of a class to include maps-to notation for defining an operation via a rule.
class (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)
 
Definitiondf-ov 7353 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 š¹ and its arguments š“ and šµ- will be useful for proving meaningful theorems. For example, if class š¹ is the operation + and arguments š“ and šµ are 3 and 2, the expression (3 + 2) can be proved to equal 5 (see 3p2e5 12238). This definition is well-defined, although not very meaningful, when classes š“ and/or šµ are proper classes (i.e. are not sets); see ovprc1 7389 and ovprc2 7390. On the other hand, we often find uses for this definition when š¹ is a proper class, such as +o in oav 8425. š¹ is normally equal to a class of nested ordered pairs of the form defined by df-oprab 7354. (Contributed by NM, 28-Feb-1995.)
(š“š¹šµ) = (š¹ā€˜āŸØš“, šµāŸ©)
 
Definitiondf-oprab 7354* 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 š‘„, š‘¦, and š‘§ are distinct, although the definition doesn't strictly require it. See df-ov 7353 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 7508. (Contributed by NM, 12-Mar-1995.)
{āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ šœ‘} = {š‘¤ āˆ£ āˆƒš‘„āˆƒš‘¦āˆƒš‘§(š‘¤ = āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ§ šœ‘)}
 
Definitiondf-mpo 7355* Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from š‘„, š‘¦ (in š“ Ɨ šµ) to š¶(š‘„, š‘¦)". An extension of df-mpt 5188 for two arguments. (Contributed by NM, 17-Feb-2008.)
(š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶) = {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) āˆ§ š‘§ = š¶)}
 
Theoremoveq 7356 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(š¹ = šŗ ā†’ (š“š¹šµ) = (š“šŗšµ))
 
Theoremoveq1 7357 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(š“ = šµ ā†’ (š“š¹š¶) = (šµš¹š¶))
 
Theoremoveq2 7358 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(š“ = šµ ā†’ (š¶š¹š“) = (š¶š¹šµ))
 
Theoremoveq12 7359 Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
((š“ = šµ āˆ§ š¶ = š·) ā†’ (š“š¹š¶) = (šµš¹š·))
 
Theoremoveq1i 7360 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
š“ = šµ    ā‡’   (š“š¹š¶) = (šµš¹š¶)
 
Theoremoveq2i 7361 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
š“ = šµ    ā‡’   (š¶š¹š“) = (š¶š¹šµ)
 
Theoremoveq12i 7362 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
š“ = šµ    &   š¶ = š·    ā‡’   (š“š¹š¶) = (šµš¹š·)
 
Theoremoveqi 7363 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
š“ = šµ    ā‡’   (š¶š“š·) = (š¶šµš·)
 
Theoremoveq123i 7364 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
š“ = š¶    &   šµ = š·    &   š¹ = šŗ    ā‡’   (š“š¹šµ) = (š¶šŗš·)
 
Theoremoveq1d 7365 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(šœ‘ ā†’ š“ = šµ)    ā‡’   (šœ‘ ā†’ (š“š¹š¶) = (šµš¹š¶))
 
Theoremoveq2d 7366 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(šœ‘ ā†’ š“ = šµ)    ā‡’   (šœ‘ ā†’ (š¶š¹š“) = (š¶š¹šµ))
 
Theoremoveqd 7367 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
(šœ‘ ā†’ š“ = šµ)    ā‡’   (šœ‘ ā†’ (š¶š“š·) = (š¶šµš·))
 
Theoremoveq12d 7368 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(šœ‘ ā†’ š“ = šµ)    &   (šœ‘ ā†’ š¶ = š·)    ā‡’   (šœ‘ ā†’ (š“š¹š¶) = (šµš¹š·))
 
Theoremoveqan12d 7369 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(šœ‘ ā†’ š“ = šµ)    &   (šœ“ ā†’ š¶ = š·)    ā‡’   ((šœ‘ āˆ§ šœ“) ā†’ (š“š¹š¶) = (šµš¹š·))
 
Theoremoveqan12rd 7370 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(šœ‘ ā†’ š“ = šµ)    &   (šœ“ ā†’ š¶ = š·)    ā‡’   ((šœ“ āˆ§ šœ‘) ā†’ (š“š¹š¶) = (šµš¹š·))
 
Theoremoveq123d 7371 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
(šœ‘ ā†’ š¹ = šŗ)    &   (šœ‘ ā†’ š“ = šµ)    &   (šœ‘ ā†’ š¶ = š·)    ā‡’   (šœ‘ ā†’ (š“š¹š¶) = (šµšŗš·))
 
Theoremfvoveq1d 7372 Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.)
(šœ‘ ā†’ š“ = šµ)    ā‡’   (šœ‘ ā†’ (š¹ā€˜(š“š‘‚š¶)) = (š¹ā€˜(šµš‘‚š¶)))
 
Theoremfvoveq1 7373 Equality theorem for nested function and operation value. Closed form of fvoveq1d 7372. (Contributed by AV, 23-Jul-2022.)
(š“ = šµ ā†’ (š¹ā€˜(š“š‘‚š¶)) = (š¹ā€˜(šµš‘‚š¶)))
 
Theoremovanraleqv 7374* 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.)
(šµ = š‘‹ ā†’ (šœ‘ ā†” šœ“))    ā‡’   (šµ = š‘‹ ā†’ (āˆ€š‘„ āˆˆ š‘‰ (šœ‘ āˆ§ (š“ Ā· šµ) = š¶) ā†” āˆ€š‘„ āˆˆ š‘‰ (šœ“ āˆ§ (š“ Ā· š‘‹) = š¶)))
 
Theoremimbrov2fvoveq 7375 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.)
(š‘‹ = š‘Œ ā†’ (šœ‘ ā†” šœ“))    ā‡’   (š‘‹ = š‘Œ ā†’ ((šœ‘ ā†’ (š¹ā€˜((šŗā€˜š‘‹) Ā· š‘‚))š‘…š“) ā†” (šœ“ ā†’ (š¹ā€˜((šŗā€˜š‘Œ) Ā· š‘‚))š‘…š“)))
 
Theoremovrspc2v 7376* 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.)
(((š‘‹ āˆˆ š“ āˆ§ š‘Œ āˆˆ šµ) āˆ§ āˆ€š‘„ āˆˆ š“ āˆ€š‘¦ āˆˆ šµ (š‘„š¹š‘¦) āˆˆ š¶) ā†’ (š‘‹š¹š‘Œ) āˆˆ š¶)
 
Theoremoveqrspc2v 7377* Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.)
((šœ‘ āˆ§ (š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ)) ā†’ (š‘„š¹š‘¦) = (š‘„šŗš‘¦))    ā‡’   ((šœ‘ āˆ§ (š‘‹ āˆˆ š“ āˆ§ š‘Œ āˆˆ šµ)) ā†’ (š‘‹š¹š‘Œ) = (š‘‹šŗš‘Œ))
 
Theoremoveqdr 7378 Equality of two operations for any two operands. Useful in proofs using *propd theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)
(šœ‘ ā†’ š¹ = šŗ)    ā‡’   ((šœ‘ āˆ§ šœ“) ā†’ (š‘„š¹š‘¦) = (š‘„šŗš‘¦))
 
Theoremnfovd 7379 Deduction version of bound-variable hypothesis builder nfov 7380. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(šœ‘ ā†’ ā„²š‘„š“)    &   (šœ‘ ā†’ ā„²š‘„š¹)    &   (šœ‘ ā†’ ā„²š‘„šµ)    ā‡’   (šœ‘ ā†’ ā„²š‘„(š“š¹šµ))
 
Theoremnfov 7380 Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
ā„²š‘„š“    &   ā„²š‘„š¹    &   ā„²š‘„šµ    ā‡’   ā„²š‘„(š“š¹šµ)
 
Theoremoprabidw 7381* The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of oprabid 7382 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.)
(āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆˆ {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ šœ‘} ā†” šœ‘)
 
Theoremoprabid 7382 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 7381 when possible. (Contributed by Mario Carneiro, 20-Mar-2013.) (New usage is discouraged.)
(āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆˆ {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ šœ‘} ā†” šœ‘)
 
Theoremovex 7383 The result of an operation is a set. (Contributed by NM, 13-Mar-1995.)
(š“š¹šµ) āˆˆ V
 
Theoremovexi 7384 The result of an operation is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
š“ = (šµš¹š¶)    ā‡’   š“ āˆˆ V
 
Theoremovexd 7385 The result of an operation is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(šœ‘ ā†’ (š“š¹šµ) āˆˆ V)
 
Theoremovssunirn 7386 The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.)
(š‘‹š¹š‘Œ) āŠ† āˆŖ ran š¹
 
Theorem0ov 7387 Operation value of the empty set. (Contributed by AV, 15-May-2021.)
(š“āˆ…šµ) = āˆ…
 
Theoremovprc 7388 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 š¹    ā‡’   (Ā¬ (š“ āˆˆ V āˆ§ šµ āˆˆ V) ā†’ (š“š¹šµ) = āˆ…)
 
Theoremovprc1 7389 The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.)
Rel dom š¹    ā‡’   (Ā¬ š“ āˆˆ V ā†’ (š“š¹šµ) = āˆ…)
 
Theoremovprc2 7390 The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
Rel dom š¹    ā‡’   (Ā¬ šµ āˆˆ V ā†’ (š“š¹šµ) = āˆ…)
 
Theoremovrcl 7391 Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.)
Rel dom š¹    ā‡’   (š¶ āˆˆ (š“š¹šµ) ā†’ (š“ āˆˆ V āˆ§ šµ āˆˆ V))
 
Theoremcsbov123 7392 Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Revised by NM, 23-Aug-2018.)
ā¦‹š“ / š‘„ā¦Œ(šµš¹š¶) = (ā¦‹š“ / š‘„ā¦Œšµā¦‹š“ / š‘„ā¦Œš¹ā¦‹š“ / š‘„ā¦Œš¶)
 
Theoremcsbov 7393* Move class substitution in and out of an operation. (Contributed by NM, 23-Aug-2018.)
ā¦‹š“ / š‘„ā¦Œ(šµš¹š¶) = (šµā¦‹š“ / š‘„ā¦Œš¹š¶)
 
Theoremcsbov12g 7394* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(š“ āˆˆ š‘‰ ā†’ ā¦‹š“ / š‘„ā¦Œ(šµš¹š¶) = (ā¦‹š“ / š‘„ā¦Œšµš¹ā¦‹š“ / š‘„ā¦Œš¶))
 
Theoremcsbov1g 7395* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(š“ āˆˆ š‘‰ ā†’ ā¦‹š“ / š‘„ā¦Œ(šµš¹š¶) = (ā¦‹š“ / š‘„ā¦Œšµš¹š¶))
 
Theoremcsbov2g 7396* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(š“ āˆˆ š‘‰ ā†’ ā¦‹š“ / š‘„ā¦Œ(šµš¹š¶) = (šµš¹ā¦‹š“ / š‘„ā¦Œš¶))
 
Theoremrspceov 7397* A frequently used special case of rspc2ev 3591 for operation values. (Contributed by NM, 21-Mar-2007.)
((š¶ āˆˆ š“ āˆ§ š· āˆˆ šµ āˆ§ š‘† = (š¶š¹š·)) ā†’ āˆƒš‘„ āˆˆ š“ āˆƒš‘¦ āˆˆ šµ š‘† = (š‘„š¹š‘¦))
 
Theoremelovimad 7398 Elementhood of the image set of an operation value. (Contributed by Thierry Arnoux, 13-Mar-2017.)
(šœ‘ ā†’ š“ āˆˆ š¶)    &   (šœ‘ ā†’ šµ āˆˆ š·)    &   (šœ‘ ā†’ Fun š¹)    &   (šœ‘ ā†’ (š¶ Ɨ š·) āŠ† dom š¹)    ā‡’   (šœ‘ ā†’ (š“š¹šµ) āˆˆ (š¹ ā€œ (š¶ Ɨ š·)))
 
Theoremfnbrovb 7399 Value of a binary operation expressed as a binary relation. See also fnbrfvb 6891 for functions on Cartesian products. (Contributed by BJ, 15-Feb-2022.)
((š¹ Fn (š‘‰ Ɨ š‘Š) āˆ§ (š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š)) ā†’ ((š“š¹šµ) = š¶ ā†” āŸØš“, šµāŸ©š¹š¶))
 
Theoremfnotovb 7400 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6892. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 15-Feb-2022.)
((š¹ Fn (š“ Ɨ šµ) āˆ§ š¶ āˆˆ š“ āˆ§ š· āˆˆ šµ) ā†’ ((š¶š¹š·) = š‘… ā†” āŸØš¶, š·, š‘…āŸ© āˆˆ š¹))
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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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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46966
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