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Theorem List for Metamath Proof Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcrio 6501 Extend class notation with restricted description binder.

Definitiondf-undef 6502 Define the undefined value function, whose value at set is guaranteed not to be a member of (see pwuninel 6504). (Contributed by NM, 15-Sep-2011.)

Theorempwuninel2 6503 Direct proof of pwuninel 6504 avoiding functions and thus several ZF axioms. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theorempwuninel 6504 The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. See also pwuninel2 6503. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Theoremundefval 6505 Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 6507 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremundefnel2 6506 The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.)

Theoremundefnel 6507 The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.)

Definitiondf-riota 6508 Define restricted description binder. In case it doesn't exist, we return a set which is not a member of the domain of discourse . See also comments for df-iota 5377. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremriotaeqdv 6509* Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.)

Theoremriotabidv 6510* Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.)

Theoremriotaeqbidv 6511* Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)

Theoremriotaex 6512 Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)

Theoremriotav 6513 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)

Theoremriotaiota 6514 Restricted iota in terms of iota. (Contributed by NM, 15-Sep-2011.)

Theoremriotauni 6515 Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)

Theoremnfriota1 6516* The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfriotad 6517 Deduction version of nfriota 6518. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfriota 6518* A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)

Theoremcbvriota 6519* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcbvriotav 6520* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcsbriotag 6521* Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)

Theoremriotacl2 6522 Membership law for "the unique element in such that ."

This can useful for expanding an iota-based definition (see df-iota 5377). If you have an unbounded iota, iotacl 5400 may be useful.

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

Theoremriotacl 6523* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)

Theoremriotasbc 6524 Substitution law for descriptions. Compare iotasbc 27487. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theoremriotabidva 6525* Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2907 analog.) (Contributed by NM, 17-Jan-2012.)

Theoremriotabiia 6526 Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2906 analog.) (Contributed by NM, 16-Jan-2012.)

Theoremriota1 6527* Property of restricted iota. Compare iota1 5391. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremriota1a 6528 Property of iota. (Contributed by NM, 23-Aug-2011.)

Theoremriota2df 6529* A deduction version of riota2f 6530. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremriota2f 6530* 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 6531* 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 6532* Properties of a restricted definite description operator. Todo: can some uses of riota2f 6530 be shortened with this? (Contributed by NM, 23-Nov-2013.)

Theoremriota5f 6533* A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremriota5 6534* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)

Theoremriota5OLD 6535* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremriotass2 6536* Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)

Theoremriotass 6537* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremmoriotass 6538* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)

Theoremsnriota 6539 A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)

Theoremriotaxfrd 6540* Change the variable in the expression for "the unique such that " to another variable contained in expression . Use reuhypd 4709 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremeusvobj2 6541* 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 6542* 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 6543* There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)

Theoremf1ocnvfv3 6544* 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.)

Theoremriotaund 6545* Restricted iota equals the undefined value of its domain of discourse when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremriotaprc 6546* For proper classes, restricted and unrestricted iota are the same. (Contributed by NM, 15-Sep-2011.)

Theoremriotassuni 6547* The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremriotaclbg 6548* Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremriotaclb 6549* Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)

Theoremriotaundb 6550* Restricted iota equals the undefined value of its domain of discourse when not meaningful. (Contributed by NM, 26-Sep-2011.)

Theoremriotasvd 6551* Deduction version of riotasv 6556. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

TheoremriotasvdOLD 6552* Deduction version of riotasv 6556. (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremriotasv2d 6553* Value of description binder for a single-valued class expression (as in e.g. reusv2 4688). Special case of riota2f 6530. (Contributed by NM, 2-Mar-2013.)

Theoremriotasv2dOLD 6554* Value of description binder for a single-valued class expression (as in e.g. reusv2 4688). Special case of riota2f 6530. (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremriotasv2s 6555* The value of description binder for a single-valued class expression (as in e.g. reusv2 4688) in the form of a substitution instance. Special case of riota2f 6530. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Theoremriotasv 6556* Value of description binder for a single-valued class expression (as in e.g. reusv2 4688). Special case of riota2f 6530. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Theoremriotasv3d 6557* A property holding for a representative of a single-valued class expression (see e.g. reusv2 4688) also holds for its description binder (in the form of property ). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremriotasv3dOLD 6558* A property holding for a representative of a single-valued class expression (see e.g. reusv2 4688) also holds for its description binder (in the form of property ). (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

2.4.21  Functions on ordinals; strictly monotone ordinal functions

Theoremiunon 6559* The indexed union of a set of ordinal numbers is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)

TheoremiunonOLD 6560* The indexed union of a set of ordinal numbers is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremiinon 6561* The nonempty indexed intersection of a class of ordinal numbers is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremonfununi 6562* A property of functions on ordinal numbers. Generalization of Theorem Schema 8E of [Enderton] p. 218. (Contributed by Eric Schmidt, 26-May-2009.)

Theoremonovuni 6563* A variant of onfununi 6562 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremonoviun 6564* A variant of onovuni 6563 with indexed unions. (Contributed by Eric Schmidt, 26-May-2009.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremonnseq 6565* There are no length decreasing sequences in the ordinals. See also noinfep 7570 for a stronger version assuming Regularity. (Contributed by Mario Carneiro, 19-May-2015.)

Syntaxwsmo 6566 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.

Definitiondf-smo 6567* Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)

Theoremdfsmo2 6568* Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)

Theoremissmo 6569* Conditions for which is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)

Theoremissmo2 6570* Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremsmoeq 6571 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)

Theoremsmodm 6572 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)

Theoremsmores 6573 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremsmores3 6574 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)

Theoremsmores2 6575 A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)

Theoremsmodm2 6576 The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremsmofvon2 6577 The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremiordsmo 6578 The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)

Theoremsmo0 6579 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)

Theoremsmofvon 6580 If is a strictly monotone ordinal function, and is in the domain of , then the value of the function at is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)

Theoremsmoel 6581 If is less than then a strictly monotone function's value will be strictly less at than at . (Contributed by Andrew Salmon, 22-Nov-2011.)

Theoremsmoiun 6582* The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)

Theoremsmoiso 6583 If is an isomorphism from an ordinal onto , which is a subset of the ordinals, then is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)

Theoremsmoel2 6584 A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremsmo11 6585 A strictly monotone ordinal function is one-to-one. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremsmoord 6586 A strictly monotone ordinal function preserves strict ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)

Theoremsmoword 6587 A strictly monotone ordinal function preserves weak ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)

Theoremsmogt 6588 A strictly monotone ordinal function is greater than or equal to its argument. Exercise 1 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 23-Nov-2011.) (Revised by Mario Carneiro, 28-Feb-2013.)

Theoremsmorndom 6589 The range of a strictly monotone ordinal function dominates the domain. (Contributed by Mario Carneiro, 13-Mar-2013.)

Theoremsmoiso2 6590 The strictly monotone ordinal functions are also epsilon isomorphisms of subclasses of . (Contributed by Mario Carneiro, 20-Mar-2013.)

2.4.22  "Strong" transfinite recursion

Syntaxcrecs 6591 Notation for a function defined by strong transfinite recursion.
recs

Definitiondf-recs 6592* Define a function recs on , the class of ordinal numbers, by transfinite recursion given a rule which sets the next value given all values so far. See df-rdg 6627 for more details on why this definition is desirable. Unlike df-rdg 6627 which restricts the update rule to use only the previous value, this version allows the update rule to use all previous values, which is why it is described as "strong", although it is actually more primitive. See recsfnon 6620 and recsval 6621 for the primary contract of this definition.

EDITORIAL: there are several existing versions of this construction without the definition, notably in ordtype 7457, zorn2 8342, and dfac8alem 7866. (Contributed by Stefan O'Rear, 18-Jan-2015.) (New usage is discouraged.)

recs

Theoremrecseq 6593 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs recs

Theoremnfrecs 6594 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremtfrlem1 6595* A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremtfrlem2 6596* Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 6595 into the main proof. (Contributed by NM, 23-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremtfrlem3 6597* Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.)

Theoremtfrlem3a 6598* Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 22-Jul-2012.)

Theoremtfrlem4 6599* Lemma for transfinite recursion. is the class of all "acceptable" functions, and is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.)

Theoremtfrlem5 6600* Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.)

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