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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | isorel 5601 | An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.) |
Theorem | isoresbr 5602* | A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.) |
Theorem | isoid 5603 | Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
Theorem | isocnv 5604 | Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
Theorem | isocnv2 5605 | Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.) |
Theorem | isores2 5606 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Theorem | isores1 5607 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Theorem | isores3 5608 | Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
Theorem | isotr 5609 | 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.) |
Theorem | iso0 5610 | The empty set is an isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.) |
Theorem | isoini 5611 | Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.) |
Theorem | isoini2 5612 | Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.) |
Theorem | isoselem 5613* | Lemma for isose 5614. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Se Se | ||
Theorem | isose 5614 | An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Se Se | ||
Theorem | isopolem 5615 | Lemma for isopo 5616. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isopo 5616 | An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isosolem 5617 | Lemma for isoso 5618. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isoso 5618 | An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | f1oiso 5619* | 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 5620* | Any one-to-one onto function determines an isomorphism with an induced relation . (Contributed by Mario Carneiro, 9-Mar-2013.) |
Syntax | crio 5621 | Extend class notation with restricted description binder. |
Definition | df-riota 5622 | 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 4993. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.) |
Theorem | riotaeqdv 5623* | Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.) |
Theorem | riotabidv 5624* | Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.) |
Theorem | riotaeqbidv 5625* | Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.) |
Theorem | riotaexg 5626* | Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.) |
Theorem | riotav 5627 | An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.) |
Theorem | riotauni 5628 | Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.) |
Theorem | nfriota1 5629* | 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 5630* | Deduction version of nfriota 5631. (Contributed by Jim Kingdon, 12-Jan-2019.) |
Theorem | nfriota 5631* | A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.) |
Theorem | cbvriota 5632* | Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | cbvriotav 5633* | Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | csbriotag 5634* | Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) |
Theorem | riotacl2 5635 |
Membership law for "the unique element in such that ."
(Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | riotacl 5636* | Closure of restricted iota. (Contributed by NM, 21-Aug-2011.) |
Theorem | riotasbc 5637 | Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
Theorem | riotabidva 5638* | Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2608 analog.) (Contributed by NM, 17-Jan-2012.) |
Theorem | riotabiia 5639 | Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2605 analog.) (Contributed by NM, 16-Jan-2012.) |
Theorem | riota1 5640* | Property of restricted iota. Compare iota1 5007. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | riota1a 5641 | Property of iota. (Contributed by NM, 23-Aug-2011.) |
Theorem | riota2df 5642* | A deduction version of riota2f 5643. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | riota2f 5643* | 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 5644* | 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 5645* | Properties of a restricted definite description operator. Todo (df-riota 5622 update): can some uses of riota2f 5643 be shortened with this? (Contributed by NM, 23-Nov-2013.) |
Theorem | riota5f 5646* | A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | riota5 5647* | A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.) |
Theorem | riotass2 5648* | Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.) |
Theorem | riotass 5649* | Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.) |
Theorem | moriotass 5650* | Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.) |
Theorem | snriota 5651 | A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.) |
Theorem | eusvobj2 5652* | 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 5653* | 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 5654* | There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.) |
Theorem | f1ocnvfv3 5655* | 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 5656* | 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 5657* | Lemma for acexmid 5665. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemb 5658* | Lemma for acexmid 5665. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemph 5659* | Lemma for acexmid 5665. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemab 5660* | Lemma for acexmid 5665. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemcase 5661* |
Lemma for acexmid 5665. 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 5030. 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 5662* | Lemma for acexmid 5665. List the cases identified in acexmidlemcase 5661 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.) |
Theorem | acexmidlem2 5663* |
Lemma for acexmid 5665. This builds on acexmidlem1 5662 by noting that every
element of is
inhabited.
(Note that is not quite a function in the df-fun 5030 sense because it uses ordered pairs as described in opthreg 4385 rather than df-op 3459). The set is also found in onsucelsucexmidlem 4358. (Contributed by Jim Kingdon, 5-Aug-2019.) |
Theorem | acexmidlemv 5664* |
Lemma for acexmid 5665.
This is acexmid 5665 with additional distinct variable constraints, most notably between and . (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmid 5665* |
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 nonempty 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 5666 | 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 5667 | Extend class notation to include class abstraction (class builder) of nested ordered pairs. |
Syntax | cmpt2 5668 | Extend the definition of a class to include maps-to notation for defining an operation via a rule. |
Definition | df-ov 5669 | 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 5699 and ovprc2 5700. 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 5670. (Contributed by NM, 28-Feb-1995.) |
Definition | df-oprab 5670* | 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 5669 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 5794. (Contributed by NM, 12-Mar-1995.) |
Definition | df-mpt2 5671* | 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 3907 for two arguments. (Contributed by NM, 17-Feb-2008.) |
Theorem | oveq 5672 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq1 5673 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq2 5674 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq12 5675 | Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.) |
Theorem | oveq1i 5676 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq2i 5677 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq12i 5678 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | oveqi 5679 | Equality inference for operation value. (Contributed by NM, 24-Nov-2007.) |
Theorem | oveq123i 5680 | Equality inference for operation value. (Contributed by FL, 11-Jul-2010.) |
Theorem | oveq1d 5681 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
Theorem | oveq2d 5682 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
Theorem | oveqd 5683 | Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.) |
Theorem | oveq12d 5684 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | oveqan12d 5685 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
Theorem | oveqan12rd 5686 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
Theorem | oveq123d 5687 | Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.) |
Theorem | fvoveq1d 5688 | Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.) |
Theorem | fvoveq1 5689 | Equality theorem for nested function and operation value. Closed form of fvoveq1d 5688. (Contributed by AV, 23-Jul-2022.) |
Theorem | ovanraleqv 5690* | Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.) |
Theorem | imbrov2fvoveq 5691 | Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.) |
Theorem | nfovd 5692 | Deduction version of bound-variable hypothesis builder nfov 5693. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | nfov 5693 | Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.) |
Theorem | oprabidlem 5694* | Slight elaboration of exdistrfor 1729. A lemma for oprabid 5695. (Contributed by Jim Kingdon, 15-Jan-2019.) |
Theorem | oprabid 5695 | 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-bndl 1445 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.) |
Theorem | fnovex 5696 | The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.) |
Theorem | ovexg 5697 | Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.) |
Theorem | ovprc 5698 | 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 5699 | The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.) |
Theorem | ovprc2 5700 | The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.) |
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