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Theorem List for Intuitionistic Logic Explorer - 5401-5500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfniunfv 5401* The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)

Theoremfuniunfvdm 5402* The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5401. (Contributed by Jim Kingdon, 10-Jan-2019.)

Theoremfuniunfvdmf 5403* The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5402 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)

Theoremeluniimadm 5404* Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)

Theoremelunirn 5405* Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)

Theoremfnunirn 5406* Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremdff13 5407* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)

Theoremf1veqaeq 5408 If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)

Theoremdff13f 5409* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)

Theoremf1mpt 5410* Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremf1fveq 5411 Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)

Theoremf1elima 5412 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremf1imass 5413 Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imaeq 5414 Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imapss 5415 Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremdff1o6 5416* A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)

Theoremf1ocnvfv1 5417 The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv2 5418 The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv 5419 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremf1ocnvfvb 5420 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvdm 5421 The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)

Theoremf1ocnvfvrneq 5422 If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)

Theoremfcof1 5423 An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremfcofo 5424 An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremcbvfo 5425* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremcbvexfo 5426* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)

Theoremcocan1 5427 An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theoremcocan2 5428 A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremfcof1o 5429 Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremfoeqcnvco 5430 Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremf1eqcocnv 5431 Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremfliftrel 5432* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel 5433* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel1 5434* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftcnv 5435* Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfun 5436* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfund 5437* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfuns 5438* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftf 5439* The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftval 5440* The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremisoeq1 5441 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq2 5442 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq3 5443 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq4 5444 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq5 5445 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremnfiso 5446 Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremisof1o 5447 An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)

Theoremisorel 5448 An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)

Theoremisoresbr 5449* A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)

Theoremisoid 5450 Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)

Theoremisocnv 5451 Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)

Theoremisocnv2 5452 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)

Theoremisores2 5453 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremisores1 5454 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremisores3 5455 Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisotr 5456 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.)

Theoremisoini 5457 Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)

Theoremisoini2 5458 Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)

Theoremisoselem 5459* Lemma for isose 5460. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se Se

Theoremisose 5460 An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se Se

Theoremisopolem 5461 Lemma for isopo 5462. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisopo 5462 An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisosolem 5463 Lemma for isoso 5464. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisoso 5464 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremf1oiso 5465* Any one-to-one onto function determines an isomorphism with an induced relation . Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)

Theoremf1oiso2 5466* Any one-to-one onto function determines an isomorphism with an induced relation . (Contributed by Mario Carneiro, 9-Mar-2013.)

2.6.9  Restricted iota (description binder)

Syntaxcrio 5467 Extend class notation with restricted description binder.

Definitiondf-riota 5468 Define restricted description binder. In case there is no unique such that holds, it evaluates to the empty set. See also comments for df-iota 4867. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)

Theoremriotaeqdv 5469* Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.)

Theoremriotabidv 5470* Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.)

Theoremriotaeqbidv 5471* Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)

Theoremriotaexg 5472* Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)

Theoremriotav 5473 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)

Theoremriotauni 5474 Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)

Theoremnfriota1 5475* The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfriotadxy 5476* Deduction version of nfriota 5477. (Contributed by Jim Kingdon, 12-Jan-2019.)

Theoremnfriota 5477* A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)

Theoremcbvriota 5478* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcbvriotav 5479* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcsbriotag 5480* Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)

Theoremriotacl2 5481 Membership law for "the unique element in such that ."

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

Theoremriotacl 5482* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)

Theoremriotasbc 5483 Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theoremriotabidva 5484* Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2548 analog.) (Contributed by NM, 17-Jan-2012.)

Theoremriotabiia 5485 Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2547 analog.) (Contributed by NM, 16-Jan-2012.)

Theoremriota1 5486* Property of restricted iota. Compare iota1 4881. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremriota1a 5487 Property of iota. (Contributed by NM, 23-Aug-2011.)

Theoremriota2df 5488* A deduction version of riota2f 5489. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremriota2f 5489* This theorem shows a condition that allows us to represent a descriptor with a class expression . (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremriota2 5490* This theorem shows a condition that allows us to represent a descriptor with a class expression . (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.)

Theoremriotaprop 5491* Properties of a restricted definite description operator. Todo (df-riota 5468 update): can some uses of riota2f 5489 be shortened with this? (Contributed by NM, 23-Nov-2013.)

Theoremriota5f 5492* A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremriota5 5493* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)

Theoremriotass2 5494* Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)

Theoremriotass 5495* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremmoriotass 5496* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)

Theoremsnriota 5497 A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)

Theoremeusvobj2 5498* Specify the same property in two ways when class is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theoremeusvobj1 5499* Specify the same object in two ways when class is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremf1ofveu 5500* There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)

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