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Theorem List for Intuitionistic Logic Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisoeq1 5801 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
(๐ป = ๐บ โ†’ (๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†” ๐บ Isom ๐‘…, ๐‘† (๐ด, ๐ต)))
 
Theoremisoeq2 5802 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
(๐‘… = ๐‘‡ โ†’ (๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†” ๐ป Isom ๐‘‡, ๐‘† (๐ด, ๐ต)))
 
Theoremisoeq3 5803 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
(๐‘† = ๐‘‡ โ†’ (๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†” ๐ป Isom ๐‘…, ๐‘‡ (๐ด, ๐ต)))
 
Theoremisoeq4 5804 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
(๐ด = ๐ถ โ†’ (๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†” ๐ป Isom ๐‘…, ๐‘† (๐ถ, ๐ต)))
 
Theoremisoeq5 5805 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
(๐ต = ๐ถ โ†’ (๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†” ๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ถ)))
 
Theoremnfiso 5806 Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
โ„ฒ๐‘ฅ๐ป    &   โ„ฒ๐‘ฅ๐‘…    &   โ„ฒ๐‘ฅ๐‘†    &   โ„ฒ๐‘ฅ๐ด    &   โ„ฒ๐‘ฅ๐ต    โ‡’   โ„ฒ๐‘ฅ ๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต)
 
Theoremisof1o 5807 An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
(๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†’ ๐ป:๐ดโ€“1-1-ontoโ†’๐ต)
 
Theoremisorel 5808 An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
((๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โˆง (๐ถ โˆˆ ๐ด โˆง ๐ท โˆˆ ๐ด)) โ†’ (๐ถ๐‘…๐ท โ†” (๐ปโ€˜๐ถ)๐‘†(๐ปโ€˜๐ท)))
 
Theoremisoresbr 5809* A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)
((๐น โ†พ ๐ด) Isom ๐‘…, ๐‘† (๐ด, (๐น โ€œ ๐ด)) โ†’ โˆ€๐‘ฅ โˆˆ ๐ด โˆ€๐‘ฆ โˆˆ ๐ด (๐‘ฅ๐‘…๐‘ฆ โ†’ (๐นโ€˜๐‘ฅ)๐‘†(๐นโ€˜๐‘ฆ)))
 
Theoremisoid 5810 Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
( I โ†พ ๐ด) Isom ๐‘…, ๐‘… (๐ด, ๐ด)
 
Theoremisocnv 5811 Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
(๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†’ โ—ก๐ป Isom ๐‘†, ๐‘… (๐ต, ๐ด))
 
Theoremisocnv2 5812 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
(๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†” ๐ป Isom โ—ก๐‘…, โ—ก๐‘†(๐ด, ๐ต))
 
Theoremisores2 5813 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
(๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†” ๐ป Isom ๐‘…, (๐‘† โˆฉ (๐ต ร— ๐ต))(๐ด, ๐ต))
 
Theoremisores1 5814 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
(๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†” ๐ป Isom (๐‘… โˆฉ (๐ด ร— ๐ด)), ๐‘†(๐ด, ๐ต))
 
Theoremisores3 5815 Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
((๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โˆง ๐พ โŠ† ๐ด โˆง ๐‘‹ = (๐ป โ€œ ๐พ)) โ†’ (๐ป โ†พ ๐พ) Isom ๐‘…, ๐‘† (๐พ, ๐‘‹))
 
Theoremisotr 5816 Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
((๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โˆง ๐บ Isom ๐‘†, ๐‘‡ (๐ต, ๐ถ)) โ†’ (๐บ โˆ˜ ๐ป) Isom ๐‘…, ๐‘‡ (๐ด, ๐ถ))
 
Theoremiso0 5817 The empty set is an ๐‘…, ๐‘† isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
โˆ… Isom ๐‘…, ๐‘† (โˆ…, โˆ…)
 
Theoremisoini 5818 Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
((๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โˆง ๐ท โˆˆ ๐ด) โ†’ (๐ป โ€œ (๐ด โˆฉ (โ—ก๐‘… โ€œ {๐ท}))) = (๐ต โˆฉ (โ—ก๐‘† โ€œ {(๐ปโ€˜๐ท)})))
 
Theoremisoini2 5819 Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)
๐ถ = (๐ด โˆฉ (โ—ก๐‘… โ€œ {๐‘‹}))    &   ๐ท = (๐ต โˆฉ (โ—ก๐‘† โ€œ {(๐ปโ€˜๐‘‹)}))    โ‡’   ((๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โˆง ๐‘‹ โˆˆ ๐ด) โ†’ (๐ป โ†พ ๐ถ) Isom ๐‘…, ๐‘† (๐ถ, ๐ท))
 
Theoremisoselem 5820* Lemma for isose 5821. (Contributed by Mario Carneiro, 23-Jun-2015.)
(๐œ‘ โ†’ ๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต))    &   (๐œ‘ โ†’ (๐ป โ€œ ๐‘ฅ) โˆˆ V)    โ‡’   (๐œ‘ โ†’ (๐‘… Se ๐ด โ†’ ๐‘† Se ๐ต))
 
Theoremisose 5821 An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
(๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†’ (๐‘… Se ๐ด โ†” ๐‘† Se ๐ต))
 
Theoremisopolem 5822 Lemma for isopo 5823. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†’ (๐‘† Po ๐ต โ†’ ๐‘… Po ๐ด))
 
Theoremisopo 5823 An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†’ (๐‘… Po ๐ด โ†” ๐‘† Po ๐ต))
 
Theoremisosolem 5824 Lemma for isoso 5825. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†’ (๐‘† Or ๐ต โ†’ ๐‘… Or ๐ด))
 
Theoremisoso 5825 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต) โ†’ (๐‘… Or ๐ด โ†” ๐‘† Or ๐ต))
 
Theoremf1oiso 5826* Any one-to-one onto function determines an isomorphism with an induced relation ๐‘†. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)
((๐ป:๐ดโ€“1-1-ontoโ†’๐ต โˆง ๐‘† = {โŸจ๐‘ง, ๐‘คโŸฉ โˆฃ โˆƒ๐‘ฅ โˆˆ ๐ด โˆƒ๐‘ฆ โˆˆ ๐ด ((๐‘ง = (๐ปโ€˜๐‘ฅ) โˆง ๐‘ค = (๐ปโ€˜๐‘ฆ)) โˆง ๐‘ฅ๐‘…๐‘ฆ)}) โ†’ ๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต))
 
Theoremf1oiso2 5827* Any one-to-one onto function determines an isomorphism with an induced relation ๐‘†. (Contributed by Mario Carneiro, 9-Mar-2013.)
๐‘† = {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ((๐‘ฅ โˆˆ ๐ต โˆง ๐‘ฆ โˆˆ ๐ต) โˆง (โ—ก๐ปโ€˜๐‘ฅ)๐‘…(โ—ก๐ปโ€˜๐‘ฆ))}    โ‡’   (๐ป:๐ดโ€“1-1-ontoโ†’๐ต โ†’ ๐ป Isom ๐‘…, ๐‘† (๐ด, ๐ต))
 
2.6.9  Cantor's Theorem
 
Theoremcanth 5828 No set ๐ด is equinumerous to its power set (Cantor's theorem), i.e., no function can map ๐ด onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. (Use nex 1500 if you want the form ยฌ โˆƒ๐‘“๐‘“:๐ดโ€“ontoโ†’๐’ซ ๐ด.) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.)
๐ด โˆˆ V    โ‡’    ยฌ ๐น:๐ดโ€“ontoโ†’๐’ซ ๐ด
 
2.6.10  Restricted iota (description binder)
 
Syntaxcrio 5829 Extend class notation with restricted description binder.
class (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘)
 
Definitiondf-riota 5830 Define restricted description binder. In case there is no unique ๐‘ฅ such that (๐‘ฅ โˆˆ ๐ด โˆง ๐œ‘) holds, it evaluates to the empty set. See also comments for df-iota 5178. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)
(โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) = (โ„ฉ๐‘ฅ(๐‘ฅ โˆˆ ๐ด โˆง ๐œ‘))
 
Theoremriotaeqdv 5831* Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.)
(๐œ‘ โ†’ ๐ด = ๐ต)    โ‡’   (๐œ‘ โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ“) = (โ„ฉ๐‘ฅ โˆˆ ๐ต ๐œ“))
 
Theoremriotabidv 5832* Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)
(๐œ‘ โ†’ (๐œ“ โ†” ๐œ’))    โ‡’   (๐œ‘ โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ“) = (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ’))
 
Theoremriotaeqbidv 5833* Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
(๐œ‘ โ†’ ๐ด = ๐ต)    &   (๐œ‘ โ†’ (๐œ“ โ†” ๐œ’))    โ‡’   (๐œ‘ โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ“) = (โ„ฉ๐‘ฅ โˆˆ ๐ต ๐œ’))
 
Theoremriotaexg 5834* Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
(๐ด โˆˆ ๐‘‰ โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ“) โˆˆ V)
 
Theoremriotav 5835 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
(โ„ฉ๐‘ฅ โˆˆ V ๐œ‘) = (โ„ฉ๐‘ฅ๐œ‘)
 
Theoremriotauni 5836 Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
(โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ‘ โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) = โˆช {๐‘ฅ โˆˆ ๐ด โˆฃ ๐œ‘})
 
Theoremnfriota1 5837* The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
โ„ฒ๐‘ฅ(โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘)
 
Theoremnfriotadxy 5838* Deduction version of nfriota 5839. (Contributed by Jim Kingdon, 12-Jan-2019.)
โ„ฒ๐‘ฆ๐œ‘    &   (๐œ‘ โ†’ โ„ฒ๐‘ฅ๐œ“)    &   (๐œ‘ โ†’ โ„ฒ๐‘ฅ๐ด)    โ‡’   (๐œ‘ โ†’ โ„ฒ๐‘ฅ(โ„ฉ๐‘ฆ โˆˆ ๐ด ๐œ“))
 
Theoremnfriota 5839* A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)
โ„ฒ๐‘ฅ๐œ‘    &   โ„ฒ๐‘ฅ๐ด    โ‡’   โ„ฒ๐‘ฅ(โ„ฉ๐‘ฆ โˆˆ ๐ด ๐œ‘)
 
Theoremcbvriota 5840* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
โ„ฒ๐‘ฆ๐œ‘    &   โ„ฒ๐‘ฅ๐œ“    &   (๐‘ฅ = ๐‘ฆ โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) = (โ„ฉ๐‘ฆ โˆˆ ๐ด ๐œ“)
 
Theoremcbvriotav 5841* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(๐‘ฅ = ๐‘ฆ โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) = (โ„ฉ๐‘ฆ โˆˆ ๐ด ๐œ“)
 
Theoremcsbriotag 5842* Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
(๐ด โˆˆ ๐‘‰ โ†’ โฆ‹๐ด / ๐‘ฅโฆŒ(โ„ฉ๐‘ฆ โˆˆ ๐ต ๐œ‘) = (โ„ฉ๐‘ฆ โˆˆ ๐ต [๐ด / ๐‘ฅ]๐œ‘))
 
Theoremriotacl2 5843 Membership law for "the unique element in ๐ด such that ๐œ‘."

(Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

(โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ‘ โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) โˆˆ {๐‘ฅ โˆˆ ๐ด โˆฃ ๐œ‘})
 
Theoremriotacl 5844* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
(โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ‘ โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) โˆˆ ๐ด)
 
Theoremriotasbc 5845 Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
(โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ‘ โ†’ [(โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) / ๐‘ฅ]๐œ‘)
 
Theoremriotabidva 5846* Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2725 analog.) (Contributed by NM, 17-Jan-2012.)
((๐œ‘ โˆง ๐‘ฅ โˆˆ ๐ด) โ†’ (๐œ“ โ†” ๐œ’))    โ‡’   (๐œ‘ โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ“) = (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ’))
 
Theoremriotabiia 5847 Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2722 analog.) (Contributed by NM, 16-Jan-2012.)
(๐‘ฅ โˆˆ ๐ด โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) = (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ“)
 
Theoremriota1 5848* Property of restricted iota. Compare iota1 5192. (Contributed by Mario Carneiro, 15-Oct-2016.)
(โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ‘ โ†’ ((๐‘ฅ โˆˆ ๐ด โˆง ๐œ‘) โ†” (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) = ๐‘ฅ))
 
Theoremriota1a 5849 Property of iota. (Contributed by NM, 23-Aug-2011.)
((๐‘ฅ โˆˆ ๐ด โˆง โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ‘) โ†’ (๐œ‘ โ†” (โ„ฉ๐‘ฅ(๐‘ฅ โˆˆ ๐ด โˆง ๐œ‘)) = ๐‘ฅ))
 
Theoremriota2df 5850* A deduction version of riota2f 5851. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
โ„ฒ๐‘ฅ๐œ‘    &   (๐œ‘ โ†’ โ„ฒ๐‘ฅ๐ต)    &   (๐œ‘ โ†’ โ„ฒ๐‘ฅ๐œ’)    &   (๐œ‘ โ†’ ๐ต โˆˆ ๐ด)    &   ((๐œ‘ โˆง ๐‘ฅ = ๐ต) โ†’ (๐œ“ โ†” ๐œ’))    โ‡’   ((๐œ‘ โˆง โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ“) โ†’ (๐œ’ โ†” (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ“) = ๐ต))
 
Theoremriota2f 5851* This theorem shows a condition that allows us to represent a descriptor with a class expression ๐ต. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
โ„ฒ๐‘ฅ๐ต    &   โ„ฒ๐‘ฅ๐œ“    &   (๐‘ฅ = ๐ต โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   ((๐ต โˆˆ ๐ด โˆง โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ‘) โ†’ (๐œ“ โ†” (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) = ๐ต))
 
Theoremriota2 5852* This theorem shows a condition that allows us to represent a descriptor with a class expression ๐ต. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.)
(๐‘ฅ = ๐ต โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   ((๐ต โˆˆ ๐ด โˆง โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ‘) โ†’ (๐œ“ โ†” (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) = ๐ต))
 
Theoremriotaprop 5853* Properties of a restricted definite description operator. Todo (df-riota 5830 update): can some uses of riota2f 5851 be shortened with this? (Contributed by NM, 23-Nov-2013.)
โ„ฒ๐‘ฅ๐œ“    &   ๐ต = (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘)    &   (๐‘ฅ = ๐ต โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   (โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ‘ โ†’ (๐ต โˆˆ ๐ด โˆง ๐œ“))
 
Theoremriota5f 5854* A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(๐œ‘ โ†’ โ„ฒ๐‘ฅ๐ต)    &   (๐œ‘ โ†’ ๐ต โˆˆ ๐ด)    &   ((๐œ‘ โˆง ๐‘ฅ โˆˆ ๐ด) โ†’ (๐œ“ โ†” ๐‘ฅ = ๐ต))    โ‡’   (๐œ‘ โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ“) = ๐ต)
 
Theoremriota5 5855* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)
(๐œ‘ โ†’ ๐ต โˆˆ ๐ด)    &   ((๐œ‘ โˆง ๐‘ฅ โˆˆ ๐ด) โ†’ (๐œ“ โ†” ๐‘ฅ = ๐ต))    โ‡’   (๐œ‘ โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ“) = ๐ต)
 
Theoremriotass2 5856* Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
(((๐ด โŠ† ๐ต โˆง โˆ€๐‘ฅ โˆˆ ๐ด (๐œ‘ โ†’ ๐œ“)) โˆง (โˆƒ๐‘ฅ โˆˆ ๐ด ๐œ‘ โˆง โˆƒ!๐‘ฅ โˆˆ ๐ต ๐œ“)) โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) = (โ„ฉ๐‘ฅ โˆˆ ๐ต ๐œ“))
 
Theoremriotass 5857* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
((๐ด โŠ† ๐ต โˆง โˆƒ๐‘ฅ โˆˆ ๐ด ๐œ‘ โˆง โˆƒ!๐‘ฅ โˆˆ ๐ต ๐œ‘) โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) = (โ„ฉ๐‘ฅ โˆˆ ๐ต ๐œ‘))
 
Theoremmoriotass 5858* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
((๐ด โŠ† ๐ต โˆง โˆƒ๐‘ฅ โˆˆ ๐ด ๐œ‘ โˆง โˆƒ*๐‘ฅ โˆˆ ๐ต ๐œ‘) โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘) = (โ„ฉ๐‘ฅ โˆˆ ๐ต ๐œ‘))
 
Theoremsnriota 5859 A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
(โˆƒ!๐‘ฅ โˆˆ ๐ด ๐œ‘ โ†’ {๐‘ฅ โˆˆ ๐ด โˆฃ ๐œ‘} = {(โ„ฉ๐‘ฅ โˆˆ ๐ด ๐œ‘)})
 
Theoremeusvobj2 5860* 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 5861* 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 5862* 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 5863* 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 5864* 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…)
 
Theoremacexmidlema 5865* Lemma for acexmid 5873. (Contributed by Jim Kingdon, 6-Aug-2019.)
๐ด = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = โˆ… โˆจ ๐œ‘)}    &   ๐ต = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = {โˆ…} โˆจ ๐œ‘)}    &   ๐ถ = {๐ด, ๐ต}    โ‡’   ({โˆ…} โˆˆ ๐ด โ†’ ๐œ‘)
 
Theoremacexmidlemb 5866* Lemma for acexmid 5873. (Contributed by Jim Kingdon, 6-Aug-2019.)
๐ด = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = โˆ… โˆจ ๐œ‘)}    &   ๐ต = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = {โˆ…} โˆจ ๐œ‘)}    &   ๐ถ = {๐ด, ๐ต}    โ‡’   (โˆ… โˆˆ ๐ต โ†’ ๐œ‘)
 
Theoremacexmidlemph 5867* Lemma for acexmid 5873. (Contributed by Jim Kingdon, 6-Aug-2019.)
๐ด = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = โˆ… โˆจ ๐œ‘)}    &   ๐ต = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = {โˆ…} โˆจ ๐œ‘)}    &   ๐ถ = {๐ด, ๐ต}    โ‡’   (๐œ‘ โ†’ ๐ด = ๐ต)
 
Theoremacexmidlemab 5868* Lemma for acexmid 5873. (Contributed by Jim Kingdon, 6-Aug-2019.)
๐ด = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = โˆ… โˆจ ๐œ‘)}    &   ๐ต = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = {โˆ…} โˆจ ๐œ‘)}    &   ๐ถ = {๐ด, ๐ต}    โ‡’   (((โ„ฉ๐‘ฃ โˆˆ ๐ด โˆƒ๐‘ข โˆˆ ๐‘ฆ (๐ด โˆˆ ๐‘ข โˆง ๐‘ฃ โˆˆ ๐‘ข)) = โˆ… โˆง (โ„ฉ๐‘ฃ โˆˆ ๐ต โˆƒ๐‘ข โˆˆ ๐‘ฆ (๐ต โˆˆ ๐‘ข โˆง ๐‘ฃ โˆˆ ๐‘ข)) = {โˆ…}) โ†’ ยฌ ๐œ‘)
 
Theoremacexmidlemcase 5869* Lemma for acexmid 5873. Here we divide the proof into cases (based on the disjunction implicit in an unordered pair, not the sort of case elimination which relies on excluded middle).

The cases are (1) the choice function evaluated at ๐ด equals {โˆ…}, (2) the choice function evaluated at ๐ต equals โˆ…, and (3) the choice function evaluated at ๐ด equals โˆ… and the choice function evaluated at ๐ต equals {โˆ…}.

Because of the way we represent the choice function ๐‘ฆ, the choice function evaluated at ๐ด is (โ„ฉ๐‘ฃ โˆˆ ๐ดโˆƒ๐‘ข โˆˆ ๐‘ฆ(๐ด โˆˆ ๐‘ข โˆง ๐‘ฃ โˆˆ ๐‘ข)) and the choice function evaluated at ๐ต is (โ„ฉ๐‘ฃ โˆˆ ๐ตโˆƒ๐‘ข โˆˆ ๐‘ฆ(๐ต โˆˆ ๐‘ข โˆง ๐‘ฃ โˆˆ ๐‘ข)). Other than the difference in notation these work just as (๐‘ฆโ€˜๐ด) and (๐‘ฆโ€˜๐ต) would if ๐‘ฆ were a function as defined by df-fun 5218.

Although it isn't exactly about the division into cases, it is also convenient for this lemma to also include the step that if the choice function evaluated at ๐ด equals {โˆ…}, then {โˆ…} โˆˆ ๐ด and likewise for ๐ต.

(Contributed by Jim Kingdon, 7-Aug-2019.)

๐ด = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = โˆ… โˆจ ๐œ‘)}    &   ๐ต = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = {โˆ…} โˆจ ๐œ‘)}    &   ๐ถ = {๐ด, ๐ต}    โ‡’   (โˆ€๐‘ง โˆˆ ๐ถ โˆƒ!๐‘ฃ โˆˆ ๐‘ง โˆƒ๐‘ข โˆˆ ๐‘ฆ (๐‘ง โˆˆ ๐‘ข โˆง ๐‘ฃ โˆˆ ๐‘ข) โ†’ ({โˆ…} โˆˆ ๐ด โˆจ โˆ… โˆˆ ๐ต โˆจ ((โ„ฉ๐‘ฃ โˆˆ ๐ด โˆƒ๐‘ข โˆˆ ๐‘ฆ (๐ด โˆˆ ๐‘ข โˆง ๐‘ฃ โˆˆ ๐‘ข)) = โˆ… โˆง (โ„ฉ๐‘ฃ โˆˆ ๐ต โˆƒ๐‘ข โˆˆ ๐‘ฆ (๐ต โˆˆ ๐‘ข โˆง ๐‘ฃ โˆˆ ๐‘ข)) = {โˆ…})))
 
Theoremacexmidlem1 5870* Lemma for acexmid 5873. List the cases identified in acexmidlemcase 5869 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.)
๐ด = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = โˆ… โˆจ ๐œ‘)}    &   ๐ต = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = {โˆ…} โˆจ ๐œ‘)}    &   ๐ถ = {๐ด, ๐ต}    โ‡’   (โˆ€๐‘ง โˆˆ ๐ถ โˆƒ!๐‘ฃ โˆˆ ๐‘ง โˆƒ๐‘ข โˆˆ ๐‘ฆ (๐‘ง โˆˆ ๐‘ข โˆง ๐‘ฃ โˆˆ ๐‘ข) โ†’ (๐œ‘ โˆจ ยฌ ๐œ‘))
 
Theoremacexmidlem2 5871* Lemma for acexmid 5873. This builds on acexmidlem1 5870 by noting that every element of ๐ถ is inhabited.

(Note that ๐‘ฆ is not quite a function in the df-fun 5218 sense because it uses ordered pairs as described in opthreg 4555 rather than df-op 3601).

The set ๐ด is also found in onsucelsucexmidlem 4528.

(Contributed by Jim Kingdon, 5-Aug-2019.)

๐ด = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = โˆ… โˆจ ๐œ‘)}    &   ๐ต = {๐‘ฅ โˆˆ {โˆ…, {โˆ…}} โˆฃ (๐‘ฅ = {โˆ…} โˆจ ๐œ‘)}    &   ๐ถ = {๐ด, ๐ต}    โ‡’   (โˆ€๐‘ง โˆˆ ๐ถ โˆ€๐‘ค โˆˆ ๐‘ง โˆƒ!๐‘ฃ โˆˆ ๐‘ง โˆƒ๐‘ข โˆˆ ๐‘ฆ (๐‘ง โˆˆ ๐‘ข โˆง ๐‘ฃ โˆˆ ๐‘ข) โ†’ (๐œ‘ โˆจ ยฌ ๐œ‘))
 
Theoremacexmidlemv 5872* Lemma for acexmid 5873.

This is acexmid 5873 with additional disjoint variable conditions, most notably between ๐œ‘ and ๐‘ฅ.

(Contributed by Jim Kingdon, 6-Aug-2019.)

โˆƒ๐‘ฆโˆ€๐‘ง โˆˆ ๐‘ฅ โˆ€๐‘ค โˆˆ ๐‘ง โˆƒ!๐‘ฃ โˆˆ ๐‘ง โˆƒ๐‘ข โˆˆ ๐‘ฆ (๐‘ง โˆˆ ๐‘ข โˆง ๐‘ฃ โˆˆ ๐‘ข)    โ‡’   (๐œ‘ โˆจ ยฌ ๐œ‘)
 
Theoremacexmid 5873* The axiom of choice implies excluded middle. Theorem 1.3 in [Bauer] p. 483.

The statement of the axiom of choice given here is ac2 in the Metamath Proof Explorer (version of 3-Aug-2019). In particular, note that the choice function ๐‘ฆ provides a value when ๐‘ง is inhabited (as opposed to nonempty as in some statements of the axiom of choice).

Essentially the same proof can also be found at "The axiom of choice implies instances of EM", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill theorem, after Radu Diaconescu who discovered it in 1975 in the framework of topos theory and N. D. Goodman and John Myhill in 1978 in the framework of set theory (although it already appeared as an exercise in Errett Bishop's book Foundations of Constructive Analysis from 1967).

For this theorem stated using the df-ac 7204 and df-exmid 4195 syntaxes, see exmidac 7207. (Contributed by Jim Kingdon, 4-Aug-2019.)

โˆƒ๐‘ฆโˆ€๐‘ง โˆˆ ๐‘ฅ โˆ€๐‘ค โˆˆ ๐‘ง โˆƒ!๐‘ฃ โˆˆ ๐‘ง โˆƒ๐‘ข โˆˆ ๐‘ฆ (๐‘ง โˆˆ ๐‘ข โˆง ๐‘ฃ โˆˆ ๐‘ข)    โ‡’   (๐œ‘ โˆจ ยฌ ๐œ‘)
 
2.6.11  Operations
 
Syntaxco 5874 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.
class (๐ด๐น๐ต)
 
Syntaxcoprab 5875 Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ๐œ‘}
 
Syntaxcmpo 5876 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 5877 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 . 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 5910 and ovprc2 5911. On the other hand, we often find uses for this definition when ๐น is a proper class. ๐น is normally equal to a class of nested ordered pairs of the form defined by df-oprab 5878. (Contributed by NM, 28-Feb-1995.)
(๐ด๐น๐ต) = (๐นโ€˜โŸจ๐ด, ๐ตโŸฉ)
 
Definitiondf-oprab 5878* 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 5877 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 the most common operation class builder is given by ovmpo 6009. (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 5879* 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ต(๐‘ฅ, ๐‘ฆ)". An extension of df-mpt 4066 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 5880 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 5881 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 5882 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 5883 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 5884 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 5885 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 5886 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 5887 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 5888 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 5889 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 5890 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 5891 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 5892 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 5893 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 5894 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 5895 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 5896 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 5897 Equality theorem for nested function and operation value. Closed form of fvoveq1d 5896. (Contributed by AV, 23-Jul-2022.)
(๐ด = ๐ต โ†’ (๐นโ€˜(๐ด๐‘‚๐ถ)) = (๐นโ€˜(๐ต๐‘‚๐ถ)))
 
Theoremovanraleqv 5898* 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 5899 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 5900* 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)
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