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Type | Label | Description |
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Statement | ||
Theorem | f1ocnvfv3 5501* | 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 5502* | 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 5503* | Lemma for acexmid 5511. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemb 5504* | Lemma for acexmid 5511. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemph 5505* | Lemma for acexmid 5511. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemab 5506* | Lemma for acexmid 5511. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmidlemcase 5507* |
Lemma for acexmid 5511. 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 4904. 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 5508* | Lemma for acexmid 5511. List the cases identified in acexmidlemcase 5507 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.) |
Theorem | acexmidlem2 5509* |
Lemma for acexmid 5511. This builds on acexmidlem1 5508 by noting that every
element of is
inhabited.
(Note that is not quite a function in the df-fun 4904 sense because it uses ordered pairs as described in opthreg 4280 rather than df-op 3384). The set is also found in onsucelsucexmidlem 4254. (Contributed by Jim Kingdon, 5-Aug-2019.) |
Theorem | acexmidlemv 5510* |
Lemma for acexmid 5511.
This is acexmid 5511 with additional distinct variable constraints, most notably between and . (Contributed by Jim Kingdon, 6-Aug-2019.) |
Theorem | acexmid 5511* |
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 5512 | 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 5513 | Extend class notation to include class abstraction (class builder) of nested ordered pairs. |
Syntax | cmpt2 5514 | Extend the definition of a class to include maps-to notation for defining an operation via a rule. |
Definition | df-ov 5515 | 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 5541 and ovprc2 5542. 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 5516. (Contributed by NM, 28-Feb-1995.) |
Definition | df-oprab 5516* | 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 5515 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 5636. (Contributed by NM, 12-Mar-1995.) |
Definition | df-mpt2 5517* | 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 3820 for two arguments. (Contributed by NM, 17-Feb-2008.) |
Theorem | oveq 5518 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq1 5519 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq2 5520 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq12 5521 | Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.) |
Theorem | oveq1i 5522 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq2i 5523 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
Theorem | oveq12i 5524 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | oveqi 5525 | Equality inference for operation value. (Contributed by NM, 24-Nov-2007.) |
Theorem | oveq123i 5526 | Equality inference for operation value. (Contributed by FL, 11-Jul-2010.) |
Theorem | oveq1d 5527 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
Theorem | oveq2d 5528 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
Theorem | oveqd 5529 | Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.) |
Theorem | oveq12d 5530 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | oveqan12d 5531 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
Theorem | oveqan12rd 5532 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
Theorem | oveq123d 5533 | Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.) |
Theorem | nfovd 5534 | Deduction version of bound-variable hypothesis builder nfov 5535. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | nfov 5535 | Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.) |
Theorem | oprabidlem 5536* | Slight elaboration of exdistrfor 1681. A lemma for oprabid 5537. (Contributed by Jim Kingdon, 15-Jan-2019.) |
Theorem | oprabid 5537 | 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 1399 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.) |
Theorem | fnovex 5538 | The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.) |
Theorem | ovexg 5539 | Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.) |
Theorem | ovprc 5540 | 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 5541 | The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.) |
Theorem | ovprc2 5542 | The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | csbov123g 5543 | 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 5544* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
Theorem | csbov1g 5545* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
Theorem | csbov2g 5546* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
Theorem | rspceov 5547* | A frequently used special case of rspc2ev 2664 for operation values. (Contributed by NM, 21-Mar-2007.) |
Theorem | fnotovb 5548 | Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5215. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | opabbrex 5549* | 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 5550 | The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) |
Theorem | brabvv 5551* | 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 5552* | Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.) |
Theorem | reloprab 5553* | An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.) |
Theorem | nfoprab1 5554 | 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 5555 | 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 5556 | The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.) |
Theorem | nfoprab 5557* | Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.) |
Theorem | oprabbid 5558* | 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 5559* | Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) |
Theorem | oprabbii 5560* | Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Theorem | ssoprab2 5561 | Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4012. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Theorem | ssoprab2b 5562 | Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4013. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Theorem | eqoprab2b 5563 | Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4016. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Theorem | mpt2eq123 5564* | An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.) |
Theorem | mpt2eq12 5565* | An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Theorem | mpt2eq123dva 5566* | An equality deduction for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) |
Theorem | mpt2eq123dv 5567* | An equality deduction for the maps to notation. (Contributed by NM, 12-Sep-2011.) |
Theorem | mpt2eq123i 5568 | An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.) |
Theorem | mpt2eq3dva 5569* | Slightly more general equality inference for the maps to notation. (Contributed by NM, 17-Oct-2013.) |
Theorem | mpt2eq3ia 5570 | An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Theorem | nfmpt21 5571 | Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.) |
Theorem | nfmpt22 5572 | Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.) |
Theorem | nfmpt2 5573* | Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.) |
Theorem | mpt20 5574 | A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.) |
Theorem | oprab4 5575* | Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.) |
Theorem | cbvoprab1 5576* | 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 5577* | 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 5578* | 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 5579* | Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) |
Theorem | cbvoprab3 5580* | Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.) |
Theorem | cbvoprab3v 5581* | 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.) |
Theorem | cbvmpt2x 5582* | Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5583 allows to be a function of . (Contributed by NM, 29-Dec-2014.) |
Theorem | cbvmpt2 5583* | Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.) |
Theorem | cbvmpt2v 5584* | Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 3851, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.) |
Theorem | dmoprab 5585* | The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Theorem | dmoprabss 5586* | The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.) |
Theorem | rnoprab 5587* | The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.) |
Theorem | rnoprab2 5588* | The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.) |
Theorem | reldmoprab 5589* | The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.) |
Theorem | oprabss 5590* | Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.) |
Theorem | eloprabga 5591* | The law of concretion for operation class abstraction. Compare elopab 3995. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Theorem | eloprabg 5592* | The law of concretion for operation class abstraction. Compare elopab 3995. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.) |
Theorem | ssoprab2i 5593* | Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Theorem | mpt2v 5594* | Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.) |
Theorem | mpt2mptx 5595* | Express a two-argument function as a one-argument function, or vice-versa. In this version is not assumed to be constant w.r.t . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Theorem | mpt2mpt 5596* | Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.) |
Theorem | resoprab 5597* | Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.) |
Theorem | resoprab2 5598* | Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | resmpt2 5599* | Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.) |
Theorem | funoprabg 5600* | "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.) |
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