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

Theoremriotabidv 6501* Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.)

Theoremriotaeqbidv 6502* Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)

Theoremriotaex 6503 Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)

Theoremriotav 6504 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)

Theoremriotaiota 6505 Restricted iota in terms of iota. (Contributed by NM, 15-Sep-2011.)

Theoremriotauni 6506 Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)

Theoremnfriota1 6507* 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 6508 Deduction version of nfriota 6509. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfriota 6509* A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)

Theoremcbvriota 6510* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcbvriotav 6511* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcsbriotag 6512* Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)

Theoremriotacl2 6513 Membership law for "the unique element in such that ."

This can useful for expanding an iota-based definition (see df-iota 5372). If you have an unbounded iota, iotacl 5395 may be useful.

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

Theoremriotacl 6514* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)

Theoremriotasbc 6515 Substitution law for descriptions. Compare iotasbc 27304. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theoremriotabidva 6516* Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2904 analog.) (Contributed by NM, 17-Jan-2012.)

Theoremriotabiia 6517 Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2903 analog.) (Contributed by NM, 16-Jan-2012.)

Theoremriota1 6518* Property of restricted iota. Compare iota1 5386. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremriota1a 6519 Property of iota. (Contributed by NM, 23-Aug-2011.)

Theoremriota2df 6520* A deduction version of riota2f 6521. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremriota2f 6521* 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 6522* 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 6523* Properties of a restricted definite description operator. Todo: can some uses of riota2f 6521 be shortened with this? (Contributed by NM, 23-Nov-2013.)

Theoremriota5f 6524* A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremriota5 6525* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)

Theoremriota5OLD 6526* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremriotass2 6527* Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)

Theoremriotass 6528* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremmoriotass 6529* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)

Theoremsnriota 6530 A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)

Theoremriotaxfrd 6531* Change the variable in the expression for "the unique such that " to another variable contained in expression . Use reuhypd 4704 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremeusvobj2 6532* 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 6533* 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 6534* There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)

Theoremf1ocnvfv3 6535* 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 6536* 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 6537* For proper classes, restricted and unrestricted iota are the same. (Contributed by NM, 15-Sep-2011.)

Theoremriotassuni 6538* The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremriotaclbg 6539* Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremriotaclb 6540* Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)

Theoremriotaundb 6541* Restricted iota equals the undefined value of its domain of discourse when not meaningful. (Contributed by NM, 26-Sep-2011.)

Theoremriotasvd 6542* Deduction version of riotasv 6547. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

TheoremriotasvdOLD 6543* Deduction version of riotasv 6547. (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremriotasv2d 6544* Value of description binder for a single-valued class expression (as in e.g. reusv2 4683). Special case of riota2f 6521. (Contributed by NM, 2-Mar-2013.)

Theoremriotasv2dOLD 6545* Value of description binder for a single-valued class expression (as in e.g. reusv2 4683). Special case of riota2f 6521. (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremriotasv2s 6546* The value of description binder for a single-valued class expression (as in e.g. reusv2 4683) in the form of a substitution instance. Special case of riota2f 6521. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Theoremriotasv 6547* Value of description binder for a single-valued class expression (as in e.g. reusv2 4683). Special case of riota2f 6521. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Theoremriotasv3d 6548* A property holding for a representative of a single-valued class expression (see e.g. reusv2 4683) 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 6549* A property holding for a representative of a single-valued class expression (see e.g. reusv2 4683) 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 6550* 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 6551* 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 6552* 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 6553* A property of functions on ordinal numbers. Generalization of Theorem Schema 8E of [Enderton] p. 218. (Contributed by Eric Schmidt, 26-May-2009.)

Theoremonovuni 6554* A variant of onfununi 6553 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremonoviun 6555* A variant of onovuni 6554 with indexed unions. (Contributed by Eric Schmidt, 26-May-2009.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremonnseq 6556* There are no length decreasing sequences in the ordinals. See also noinfep 7561 for a stronger version assuming Regularity. (Contributed by Mario Carneiro, 19-May-2015.)

Syntaxwsmo 6557 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.

Definitiondf-smo 6558* Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)

Theoremdfsmo2 6559* Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)

Theoremissmo 6560* Conditions for which is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)

Theoremissmo2 6561* Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremsmoeq 6562 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)

Theoremsmodm 6563 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)

Theoremsmores 6564 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 6565 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)

Theoremsmores2 6566 A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)

Theoremsmodm2 6567 The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremsmofvon2 6568 The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremiordsmo 6569 The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)

Theoremsmo0 6570 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)

Theoremsmofvon 6571 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 6572 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 6573* The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)

Theoremsmoiso 6574 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 6575 A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremsmo11 6576 A strictly monotone ordinal function is one-to-one. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremsmoord 6577 A strictly monotone ordinal function preserves strict ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)

Theoremsmoword 6578 A strictly monotone ordinal function preserves weak ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)

Theoremsmogt 6579 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 6580 The range of a strictly monotone ordinal function dominates the domain. (Contributed by Mario Carneiro, 13-Mar-2013.)

Theoremsmoiso2 6581 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 6582 Notation for a function defined by strong transfinite recursion.
recs

Definitiondf-recs 6583* 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 6618 for more details on why this definition is desirable. Unlike df-rdg 6618 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 6611 and recsval 6612 for the primary contract of this definition.

EDITORIAL: there are several existing versions of this construction without the definition, notably in ordtype 7448, zorn2 8333, and dfac8alem 7857. (Contributed by Stefan O'Rear, 18-Jan-2015.) (New usage is discouraged.)

recs

Theoremrecseq 6584 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs recs

Theoremnfrecs 6585 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremtfrlem1 6586* 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 6587* Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 6586 into the main proof. (Contributed by NM, 23-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremtfrlem3 6588* 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 6589* 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 6590* 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 6591* Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.)

Theoremrecsfval 6592* Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.)
recs

Theoremtfrlem6 6593* Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremtfrlem7 6594* Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.)
recs

Theoremtfrlem8 6595* Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)
recs

Theoremtfrlem9 6596* Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)
recs recs recs

Theoremtfrlem9a 6597* Lemma for transfinite recursion. Without using ax-rep 4275, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
recs recs

Theoremtfrlem10 6598* Lemma for transfinite recursion. We define class by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to, . Using this assumption we will prove facts about that will lead to a contradiction in tfrlem14 6602, thus showing the domain of recs does in fact equal . Here we show (under the false assumption) that is a function extending the domain of recs by one. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs recs recs       recs recs

Theoremtfrlem11 6599* Lemma for transfinite recursion. Compute the value of . (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs recs recs       recs recs

Theoremtfrlem12 6600* Lemma for transfinite recursion. Show is an acceptable function. (Contributed by NM, 15-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs recs recs       recs

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