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

Theoremfvtp2 5401 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvtp3 5402 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvconst2g 5403 The value of a constant function. (Contributed by NM, 20-Aug-2005.)

Theoremfconst2g 5404 A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)

Theoremfvconst2 5405 The value of a constant function. (Contributed by NM, 16-Apr-2005.)

Theoremfconst2 5406 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)

Theoremfconstfvm 5407* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5406. (Contributed by Jim Kingdon, 8-Jan-2019.)

Theoremfconst3m 5408* Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)

Theoremfconst4m 5409* Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)

Theoremresfunexg 5410 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)

Theoremfnex 5411 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5410. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfunex 5412 If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5411. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)

Theoremopabex 5413* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)

Theoremmptexg 5414* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmptex 5415* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theoremfex 5416 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)

Theoremeufnfv 5417* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)

Theoremfunfvima 5418 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)

Theoremfunfvima2 5419 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)

Theoremfunfvima3 5420 A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)

Theoremfnfvima 5421 The function value of an operand in a set is contained in the image of that set, using the abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)

Theoremrexima 5422* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremralima 5423* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremidref 5424* TODO: This is the same as issref 4735 (which has a much longer proof). Should we replace issref 4735 with this one? - NM 9-May-2016.

Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)

Theoremelabrex 5425* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)

Theoremabrexco 5426* Composition of two image maps and . (Contributed by NM, 27-May-2013.)

Theoremimaiun 5427* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremimauni 5428* The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

Theoremfniunfv 5429* 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 5430* 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 5429. (Contributed by Jim Kingdon, 10-Jan-2019.)

Theoremfuniunfvdmf 5431* The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5430 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)

Theoremeluniimadm 5432* Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)

Theoremelunirn 5433* Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)

Theoremfnunirn 5434* Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremdff13 5435* 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 5436 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 5437* 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 5438* Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremf1fveq 5439 Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)

Theoremf1elima 5440 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremf1imass 5441 Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imaeq 5442 Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imapss 5443 Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremdff1o6 5444* A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)

Theoremf1ocnvfv1 5445 The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv2 5446 The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv 5447 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 5448 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvdm 5449 The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)

Theoremf1ocnvfvrneq 5450 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 5451 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 5452 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 5453* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremcbvexfo 5454* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)

Theoremcocan1 5455 An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theoremcocan2 5456 A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremfcof1o 5457 Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremfoeqcnvco 5458 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 5459 Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremfliftrel 5460* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel 5461* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel1 5462* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftcnv 5463* Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfun 5464* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfund 5465* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfuns 5466* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftf 5467* The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftval 5468* The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremisoeq1 5469 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq2 5470 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq3 5471 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq4 5472 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq5 5473 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremnfiso 5474 Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremisof1o 5475 An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)

Theoremisorel 5476 An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)

Theoremisoresbr 5477* A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)

Theoremisoid 5478 Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)

Theoremisocnv 5479 Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)

Theoremisocnv2 5480 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)

Theoremisores2 5481 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremisores1 5482 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremisores3 5483 Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisotr 5484 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 5485 Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)

Theoremisoini2 5486 Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)

Theoremisoselem 5487* Lemma for isose 5488. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se Se

Theoremisose 5488 An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se Se

Theoremisopolem 5489 Lemma for isopo 5490. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisopo 5490 An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisosolem 5491 Lemma for isoso 5492. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisoso 5492 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremf1oiso 5493* 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 5494* 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 5495 Extend class notation with restricted description binder.

Definitiondf-riota 5496 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 4895. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)

Theoremriotaeqdv 5497* Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.)

Theoremriotabidv 5498* Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.)

Theoremriotaeqbidv 5499* Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)

Theoremriotaexg 5500* Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)

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