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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | riotacl2 5401 |
Membership law for "the unique element in such that ."
(Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | riotacl 5402* | Closure of restricted iota. (Contributed by NM, 21-Aug-2011.) |
Theorem | riotasbc 5403 | Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
Theorem | riotabidva 5404* | Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2522 analog.) (Contributed by NM, 17-Jan-2012.) |
Theorem | riotabiia 5405 | Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2521 analog.) (Contributed by NM, 16-Jan-2012.) |
Theorem | riota1 5406* | Property of restricted iota. Compare iota1 4804. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | riota1a 5407 | Property of iota. (Contributed by NM, 23-Aug-2011.) |
Theorem | riota2df 5408* | A deduction version of riota2f 5409. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | riota2f 5409* | 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 5410* | 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 5411* | Properties of a restricted definite description operator. Todo (df-riota 5389 update): can some uses of riota2f 5409 be shortened with this? (Contributed by NM, 23-Nov-2013.) |
Theorem | riota5f 5412* | A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | riota5 5413* | A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.) |
Theorem | riotass2 5414* | Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.) |
Theorem | riotass 5415* | Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.) |
Theorem | moriotass 5416* | Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.) |
Theorem | snriota 5417 | A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.) |
Theorem | eusvobj2 5418* | 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 5419* | 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 5420* | There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.) |
Theorem | f1ocnvfv3 5421* | 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.) |
Theorem | riotaund 5422* | Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.) |
Theorem | acexmidlema 5423* | Lemma for acexmid 5431. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemb 5424* | Lemma for acexmid 5431. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemph 5425* | Lemma for acexmid 5431. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemab 5426* | Lemma for acexmid 5431. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemcase 5427* |
Lemma for acexmid 5431. Here we divide the proof into cases (based
on the
disjunction implicit in an unordered pair, not the sort of case
elimination which relies on excluded middle).
The cases are (1) the choice function evaluated at equals , (2) the choice function evaluated at equals , and (3) the choice function evaluated at equals and the choice function evaluated at equals . Because of the way we represent the choice function , the choice function evaluated at is and the choice function evaluated at is . Other than the difference in notation these work just as and would if were a function as defined by df-fun 4827. Although it isn't exactly about the division into cases, it is also convenient for this lemma to also include the step that if the choice function evaluated at equals , then and likewise for . (Contributed by Jim Kingdon, 7-Aug-2019.) |
Theorem | acexmidlem1 5428* | Lemma for acexmid 5431. List the cases identified in acexmidlemcase 5427 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.) |
Theorem | acexmidlem2 5429* |
Lemma for acexmid 5431. This builds on acexmidlem1 5428 by noting that every
element of is
inhabited.
(Note that is not quite a function in the df-fun 4827 sense because it uses ordered pairs as described in opthreg 4214 rather than df-op 3355). The set is also found in onsucelsucexmidlem 4194. (Contributed by Jim Kingdon, 5-Aug-2019.) |
Theorem | acexmidlemv 5430* |
Lemma for acexmid 5431.
This is acexmid 5431 with additional distinct variable constraints, most notably between and . (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmid 5431* |
The axiom of choice implies excluded middle. Theorem 1.3 in [Bauer]
p. 483.
The statement of the axiom of choice given here is ac2 in the Metamath Proof Explorer (version of 3-Aug-2019). In particular, note that the choice function provides a value when is inhabited (as opposed to non-empty as in some statements of the axiom of choice). Essentially the same proof can also be found at "The axiom of choice implies instances of EM", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill theorem, after Radu Diaconescu who discovered it in 1975 in the framework of topos theory and N. D. Goodman and John Myhill in 1978 in the framework of set theory (although it already appeared as an exercise in Errett Bishop's book Foundations of Constructive Analysis from 1967). (Contributed by Jim Kingdon, 4-Aug-2019.) |
Syntax | co 5432 | Extend class notation to include the value of an operation (such as + ) for two arguments and . Note that the syntax is simply three class symbols in a row surrounded by parentheses. Since operation values are the only possible class expressions consisting of three class expressions in a row surrounded by parentheses, the syntax is unambiguous. |
Syntax | coprab 5433 | Extend class notation to include class abstraction (class builder) of nested ordered pairs. |
Syntax | cmpt2 5434 | Extend the definition of a class to include maps-to notation for defining an operation via a rule. |
Definition | df-ov 5435 | Define the value of an operation. Definition of operation value in [Enderton] p. 79. Note that the syntax is simply three class expressions in a row bracketed by parentheses. There are no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation and its arguments and - will be useful for proving meaningful theorems. For example, if class is the operation + and arguments and are 3 and 2 , the expression ( 3 + 2 ) can be proved to equal 5 . This definition is well-defined, although not very meaningful, when classes and/or are proper classes (i.e. are not sets); see ovprc1 5460 and ovprc2 5461. On the other hand, we often find uses for this definition when is a proper class. is normally equal to a class of nested ordered pairs of the form defined by df-oprab 5436. (Contributed by NM, 28-Feb-1995.) |
Definition | df-oprab 5436* | Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally , , and are distinct, although the definition doesn't strictly require it. See df-ov 5435 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of the most common operation class builder is given by ovmpt2 5555. (Contributed by NM, 12-Mar-1995.) |
Definition | df-mpt2 5437* | Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from (in ) to ." An extension of df-mpt 3790 for two arguments. (Contributed by NM, 17-Feb-2008.) |
Theorem | oveq 5438 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq1 5439 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq2 5440 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq12 5441 | Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.) |
Theorem | oveq1i 5442 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq2i 5443 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq12i 5444 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | oveqi 5445 | Equality inference for operation value. (Contributed by NM, 24-Nov-2007.) |
Theorem | oveq123i 5446 | Equality inference for operation value. (Contributed by FL, 11-Jul-2010.) |
Theorem | oveq1d 5447 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
Theorem | oveq2d 5448 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
Theorem | oveqd 5449 | Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.) |
Theorem | oveq12d 5450 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | oveqan12d 5451 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
Theorem | oveqan12rd 5452 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
Theorem | oveq123d 5453 | Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.) |
Theorem | nfovd 5454 | Deduction version of bound-variable hypothesis builder nfov 5455. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | nfov 5455 | Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.) |
Theorem | oprabidlem 5456* | Slight elaboration of exdistrfor 1659. A lemma for oprabid 5457. (Contributed by Jim Kingdon, 15-Jan-2019.) |
Theorem | oprabid 5457 | The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable constraint between , , and , we use ax-bnd 1376 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.) |
Theorem | fnovex 5458 | The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.) |
Theorem | ovprc 5459 | The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | ovprc1 5460 | The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.) |
Theorem | ovprc2 5461 | The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | csbov123g 5462 | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
Theorem | csbov12g 5463* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
Theorem | csbov1g 5464* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
Theorem | csbov2g 5465* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
Theorem | rspceov 5466* | A frequently used special case of rspc2ev 2637 for operation values. (Contributed by NM, 21-Mar-2007.) |
Theorem | fnotovb 5467 | Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5136. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | opabbrex 5468* | A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) |
Theorem | 0neqopab 5469 | The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) |
Theorem | brabvv 5470* | If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Jim Kingdon, 16-Jan-2019.) |
Theorem | dfoprab2 5471* | Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.) |
Theorem | reloprab 5472* | An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.) |
Theorem | nfoprab1 5473 | The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Theorem | nfoprab2 5474 | The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.) |
Theorem | nfoprab3 5475 | The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.) |
Theorem | nfoprab 5476* | Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.) |
Theorem | oprabbid 5477* | Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.) |
Theorem | oprabbidv 5478* | Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) |
Theorem | oprabbii 5479* | Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Theorem | ssoprab2 5480 | Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 3982. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Theorem | ssoprab2b 5481 | Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 3983. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Theorem | eqoprab2b 5482 | Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 3986. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Theorem | mpt2eq123 5483* | An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.) |
Theorem | mpt2eq12 5484* | An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Theorem | mpt2eq123dva 5485* | An equality deduction for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) |
Theorem | mpt2eq123dv 5486* | An equality deduction for the maps to notation. (Contributed by NM, 12-Sep-2011.) |
Theorem | mpt2eq123i 5487 | An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.) |
Theorem | mpt2eq3dva 5488* | Slightly more general equality inference for the maps to notation. (Contributed by NM, 17-Oct-2013.) |
Theorem | mpt2eq3ia 5489 | An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Theorem | nfmpt21 5490 | Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.) |
Theorem | nfmpt22 5491 | Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.) |
Theorem | nfmpt2 5492* | Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.) |
Theorem | mpt20 5493 | A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.) |
Theorem | oprab4 5494* | Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.) |
Theorem | cbvoprab1 5495* | Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.) |
Theorem | cbvoprab2 5496* | Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.) |
Theorem | cbvoprab12 5497* | Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | cbvoprab12v 5498* | Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) |
Theorem | cbvoprab3 5499* | Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.) |
Theorem | cbvoprab3v 5500* | Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.) |
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