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Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
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Statement | ||
Theorem | fniunfv 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.) |
Theorem | funiunfvdm 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.) |
Theorem | funiunfvdmf 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.) |
Theorem | eluniimadm 5404* | Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | elunirn 5405* | Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.) |
Theorem | fnunirn 5406* | Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Theorem | dff13 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.) |
Theorem | f1veqaeq 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.) |
Theorem | dff13f 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.) |
Theorem | f1mpt 5410* | Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Theorem | f1fveq 5411 | Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.) |
Theorem | f1elima 5412 | Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | f1imass 5413 | Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Theorem | f1imaeq 5414 | Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Theorem | f1imapss 5415 | Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Theorem | dff1o6 5416* | A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.) |
Theorem | f1ocnvfv1 5417 | The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
Theorem | f1ocnvfv2 5418 | The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
Theorem | f1ocnvfv 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.) |
Theorem | f1ocnvfvb 5420 | Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.) |
Theorem | f1ocnvdm 5421 | The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.) |
Theorem | f1ocnvfvrneq 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.) |
Theorem | fcof1 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.) |
Theorem | fcofo 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.) |
Theorem | cbvfo 5425* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
Theorem | cbvexfo 5426* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) |
Theorem | cocan1 5427 | An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.) |
Theorem | cocan2 5428 | A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
Theorem | fcof1o 5429 | Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Theorem | foeqcnvco 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.) |
Theorem | f1eqcocnv 5431 | Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
Theorem | fliftrel 5432* | , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftel 5433* | Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftel1 5434* | Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftcnv 5435* | Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfun 5436* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfund 5437* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfuns 5438* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftf 5439* | The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftval 5440* | The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | isoeq1 5441 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq2 5442 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq3 5443 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq4 5444 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq5 5445 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | nfiso 5446 | Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | isof1o 5447 | An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.) |
Theorem | isorel 5448 | An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.) |
Theorem | isoresbr 5449* | A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.) |
Theorem | isoid 5450 | Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
Theorem | isocnv 5451 | Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
Theorem | isocnv2 5452 | Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.) |
Theorem | isores2 5453 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Theorem | isores1 5454 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Theorem | isores3 5455 | Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
Theorem | isotr 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.) |
Theorem | isoini 5457 | Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.) |
Theorem | isoini2 5458 | Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.) |
Theorem | isoselem 5459* | Lemma for isose 5460. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Se Se | ||
Theorem | isose 5460 | An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Se Se | ||
Theorem | isopolem 5461 | Lemma for isopo 5462. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isopo 5462 | An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isosolem 5463 | Lemma for isoso 5464. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isoso 5464 | An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | f1oiso 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.) |
Theorem | f1oiso2 5466* | Any one-to-one onto function determines an isomorphism with an induced relation . (Contributed by Mario Carneiro, 9-Mar-2013.) |
Syntax | crio 5467 | Extend class notation with restricted description binder. |
Definition | df-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.) |
Theorem | riotaeqdv 5469* | Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.) |
Theorem | riotabidv 5470* | Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.) |
Theorem | riotaeqbidv 5471* | Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.) |
Theorem | riotaexg 5472* | Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.) |
Theorem | riotav 5473 | An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.) |
Theorem | riotauni 5474 | Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.) |
Theorem | nfriota1 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.) |
Theorem | nfriotadxy 5476* | Deduction version of nfriota 5477. (Contributed by Jim Kingdon, 12-Jan-2019.) |
Theorem | nfriota 5477* | A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.) |
Theorem | cbvriota 5478* | Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | cbvriotav 5479* | Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | csbriotag 5480* | Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) |
Theorem | riotacl2 5481 |
Membership law for "the unique element in such that ."
(Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | riotacl 5482* | Closure of restricted iota. (Contributed by NM, 21-Aug-2011.) |
Theorem | riotasbc 5483 | Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
Theorem | riotabidva 5484* | Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2548 analog.) (Contributed by NM, 17-Jan-2012.) |
Theorem | riotabiia 5485 | Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2547 analog.) (Contributed by NM, 16-Jan-2012.) |
Theorem | riota1 5486* | Property of restricted iota. Compare iota1 4881. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | riota1a 5487 | Property of iota. (Contributed by NM, 23-Aug-2011.) |
Theorem | riota2df 5488* | A deduction version of riota2f 5489. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | riota2f 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.) |
Theorem | riota2 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.) |
Theorem | riotaprop 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.) |
Theorem | riota5f 5492* | A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | riota5 5493* | A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.) |
Theorem | riotass2 5494* | Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.) |
Theorem | riotass 5495* | Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.) |
Theorem | moriotass 5496* | Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.) |
Theorem | snriota 5497 | A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.) |
Theorem | eusvobj2 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.) |
Theorem | eusvobj1 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.) |
Theorem | f1ofveu 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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