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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | isoini2 5401 | Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.) |
Theorem | isoselem 5402* | Lemma for isose 5403. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Se Se | ||
Theorem | isose 5403 | An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Se Se | ||
Theorem | isopolem 5404 | Lemma for isopo 5405. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isopo 5405 | An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isosolem 5406 | Lemma for isoso 5407. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isoso 5407 | An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | f1oiso 5408* | 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 5409* | Any one-to-one onto function determines an isomorphism with an induced relation . (Contributed by Mario Carneiro, 9-Mar-2013.) |
Syntax | crio 5410 | Extend class notation with restricted description binder. |
Definition | df-riota 5411 | 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 4810. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.) |
Theorem | riotaeqdv 5412* | Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.) |
Theorem | riotabidv 5413* | Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.) |
Theorem | riotaeqbidv 5414* | Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.) |
Theorem | riotaexg 5415* | Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.) |
Theorem | riotav 5416 | An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.) |
Theorem | riotauni 5417 | Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.) |
Theorem | nfriota1 5418* | 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 5419* | Deduction version of nfriota 5420. (Contributed by Jim Kingdon, 12-Jan-2019.) |
Theorem | nfriota 5420* | A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.) |
Theorem | cbvriota 5421* | Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | cbvriotav 5422* | Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | csbriotag 5423* | Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) |
Theorem | riotacl2 5424 |
Membership law for "the unique element in such that ."
(Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | riotacl 5425* | Closure of restricted iota. (Contributed by NM, 21-Aug-2011.) |
Theorem | riotasbc 5426 | Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
Theorem | riotabidva 5427* | Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2542 analog.) (Contributed by NM, 17-Jan-2012.) |
Theorem | riotabiia 5428 | Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2541 analog.) (Contributed by NM, 16-Jan-2012.) |
Theorem | riota1 5429* | Property of restricted iota. Compare iota1 4824. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | riota1a 5430 | Property of iota. (Contributed by NM, 23-Aug-2011.) |
Theorem | riota2df 5431* | A deduction version of riota2f 5432. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | riota2f 5432* | 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 5433* | 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 5434* | Properties of a restricted definite description operator. Todo (df-riota 5411 update): can some uses of riota2f 5432 be shortened with this? (Contributed by NM, 23-Nov-2013.) |
Theorem | riota5f 5435* | A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | riota5 5436* | A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.) |
Theorem | riotass2 5437* | Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.) |
Theorem | riotass 5438* | Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.) |
Theorem | moriotass 5439* | Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.) |
Theorem | snriota 5440 | A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.) |
Theorem | eusvobj2 5441* | 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 5442* | 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 5443* | There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.) |
Theorem | f1ocnvfv3 5444* | 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 5445* | 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 5446* | Lemma for acexmid 5454. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemb 5447* | Lemma for acexmid 5454. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemph 5448* | Lemma for acexmid 5454. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemab 5449* | Lemma for acexmid 5454. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemcase 5450* |
Lemma for acexmid 5454. 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 4847. 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 5451* | Lemma for acexmid 5454. List the cases identified in acexmidlemcase 5450 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.) |
Theorem | acexmidlem2 5452* |
Lemma for acexmid 5454. This builds on acexmidlem1 5451 by noting that every
element of is
inhabited.
(Note that is not quite a function in the df-fun 4847 sense because it uses ordered pairs as described in opthreg 4234 rather than df-op 3376). The set is also found in onsucelsucexmidlem 4214. (Contributed by Jim Kingdon, 5-Aug-2019.) |
Theorem | acexmidlemv 5453* |
Lemma for acexmid 5454.
This is acexmid 5454 with additional distinct variable constraints, most notably between and . (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmid 5454* |
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 5455 | 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 5456 | Extend class notation to include class abstraction (class builder) of nested ordered pairs. |
Syntax | cmpt2 5457 | Extend the definition of a class to include maps-to notation for defining an operation via a rule. |
Definition | df-ov 5458 | 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 5483 and ovprc2 5484. 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 5459. (Contributed by NM, 28-Feb-1995.) |
Definition | df-oprab 5459* | 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 5458 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 5578. (Contributed by NM, 12-Mar-1995.) |
Definition | df-mpt2 5460* | 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 3811 for two arguments. (Contributed by NM, 17-Feb-2008.) |
Theorem | oveq 5461 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq1 5462 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq2 5463 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq12 5464 | Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.) |
Theorem | oveq1i 5465 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq2i 5466 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq12i 5467 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | oveqi 5468 | Equality inference for operation value. (Contributed by NM, 24-Nov-2007.) |
Theorem | oveq123i 5469 | Equality inference for operation value. (Contributed by FL, 11-Jul-2010.) |
Theorem | oveq1d 5470 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
Theorem | oveq2d 5471 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
Theorem | oveqd 5472 | Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.) |
Theorem | oveq12d 5473 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | oveqan12d 5474 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
Theorem | oveqan12rd 5475 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
Theorem | oveq123d 5476 | Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.) |
Theorem | nfovd 5477 | Deduction version of bound-variable hypothesis builder nfov 5478. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | nfov 5478 | Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.) |
Theorem | oprabidlem 5479* | Slight elaboration of exdistrfor 1678. A lemma for oprabid 5480. (Contributed by Jim Kingdon, 15-Jan-2019.) |
Theorem | oprabid 5480 | 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 1396 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.) |
Theorem | fnovex 5481 | The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.) |
Theorem | ovprc 5482 | 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 5483 | The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.) |
Theorem | ovprc2 5484 | The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | csbov123g 5485 | 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 5486* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
Theorem | csbov1g 5487* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
Theorem | csbov2g 5488* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
Theorem | rspceov 5489* | A frequently used special case of rspc2ev 2658 for operation values. (Contributed by NM, 21-Mar-2007.) |
Theorem | fnotovb 5490 | Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5158. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | opabbrex 5491* | 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 5492 | The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) |
Theorem | brabvv 5493* | 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 5494* | Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.) |
Theorem | reloprab 5495* | An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.) |
Theorem | nfoprab1 5496 | 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 5497 | 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 5498 | The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.) |
Theorem | nfoprab 5499* | Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.) |
Theorem | oprabbid 5500* | Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.) |
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