Theorem List for Intuitionistic Logic Explorer - 5901-6000 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | cbvmpov 5901* |
Rule to change the bound variable in a maps-to function, using implicit
substitution. With a longer proof analogous to cbvmpt 4059, some distinct
variable requirements could be eliminated. (Contributed by NM,
11-Jun-2013.)
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Theorem | dmoprab 5902* |
The domain of an operation class abstraction. (Contributed by NM,
17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
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Theorem | dmoprabss 5903* |
The domain of an operation class abstraction. (Contributed by NM,
24-Aug-1995.)
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Theorem | rnoprab 5904* |
The range of an operation class abstraction. (Contributed by NM,
30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)
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Theorem | rnoprab2 5905* |
The range of a restricted operation class abstraction. (Contributed by
Scott Fenton, 21-Mar-2012.)
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Theorem | reldmoprab 5906* |
The domain of an operation class abstraction is a relation.
(Contributed by NM, 17-Mar-1995.)
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Theorem | oprabss 5907* |
Structure of an operation class abstraction. (Contributed by NM,
28-Nov-2006.)
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Theorem | eloprabga 5908* |
The law of concretion for operation class abstraction. Compare
elopab 4218. (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.)
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Theorem | eloprabg 5909* |
The law of concretion for operation class abstraction. Compare
elopab 4218. (Contributed by NM, 14-Sep-1999.) (Revised
by David
Abernethy, 19-Jun-2012.)
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Theorem | ssoprab2i 5910* |
Inference of operation class abstraction subclass from implication.
(Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy,
19-Jun-2012.)
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Theorem | mpov 5911* |
Operation with universal domain in maps-to notation. (Contributed by
NM, 16-Aug-2013.)
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Theorem | mpomptx 5912* |
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.)
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Theorem | mpompt 5913* |
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.)
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Theorem | mpodifsnif 5914 |
A mapping with two arguments with the first argument from a difference set
with a singleton and a conditional as result. (Contributed by AV,
13-Feb-2019.)
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Theorem | mposnif 5915 |
A mapping with two arguments with the first argument from a singleton and
a conditional as result. (Contributed by AV, 14-Feb-2019.)
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Theorem | fconstmpo 5916* |
Representation of a constant operation using the mapping operation.
(Contributed by SO, 11-Jul-2018.)
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Theorem | resoprab 5917* |
Restriction of an operation class abstraction. (Contributed by NM,
10-Feb-2007.)
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Theorem | resoprab2 5918* |
Restriction of an operator abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | resmpo 5919* |
Restriction of the mapping operation. (Contributed by Mario Carneiro,
17-Dec-2013.)
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Theorem | funoprabg 5920* |
"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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Theorem | funoprab 5921* |
"At most one" is a sufficient condition for an operation class
abstraction to be a function. (Contributed by NM, 17-Mar-1995.)
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Theorem | fnoprabg 5922* |
Functionality and domain of an operation class abstraction.
(Contributed by NM, 28-Aug-2007.)
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Theorem | mpofun 5923* |
The maps-to notation for an operation is always a function.
(Contributed by Scott Fenton, 21-Mar-2012.)
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Theorem | fnoprab 5924* |
Functionality and domain of an operation class abstraction.
(Contributed by NM, 15-May-1995.)
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Theorem | ffnov 5925* |
An operation maps to a class to which all values belong. (Contributed
by NM, 7-Feb-2004.)
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Theorem | fovcl 5926 |
Closure law for an operation. (Contributed by NM, 19-Apr-2007.)
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Theorem | eqfnov 5927* |
Equality of two operations is determined by their values. (Contributed
by NM, 1-Sep-2005.)
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Theorem | eqfnov2 5928* |
Two operators with the same domain are equal iff their values at each
point in the domain are equal. (Contributed by Jeff Madsen,
7-Jun-2010.)
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Theorem | fnovim 5929* |
Representation of a function in terms of its values. (Contributed by
Jim Kingdon, 16-Jan-2019.)
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Theorem | mpo2eqb 5930* |
Bidirectional equality theorem for a mapping abstraction. Equivalent to
eqfnov2 5928. (Contributed by Mario Carneiro,
4-Jan-2017.)
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Theorem | rnmpo 5931* |
The range of an operation given by the maps-to notation. (Contributed
by FL, 20-Jun-2011.)
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Theorem | reldmmpo 5932* |
The domain of an operation defined by maps-to notation is a relation.
(Contributed by Stefan O'Rear, 27-Nov-2014.)
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Theorem | elrnmpog 5933* |
Membership in the range of an operation class abstraction. (Contributed
by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
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Theorem | elrnmpo 5934* |
Membership in the range of an operation class abstraction.
(Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro,
31-Aug-2015.)
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Theorem | ralrnmpo 5935* |
A restricted quantifier over an image set. (Contributed by Mario
Carneiro, 1-Sep-2015.)
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Theorem | rexrnmpo 5936* |
A restricted quantifier over an image set. (Contributed by Mario
Carneiro, 1-Sep-2015.)
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Theorem | ovid 5937* |
The value of an operation class abstraction. (Contributed by NM,
16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
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Theorem | ovidig 5938* |
The value of an operation class abstraction. Compare ovidi 5939. The
condition is been
removed. (Contributed by
Mario Carneiro, 29-Dec-2014.)
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Theorem | ovidi 5939* |
The value of an operation class abstraction (weak version).
(Contributed by Mario Carneiro, 29-Dec-2014.)
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Theorem | ov 5940* |
The value of an operation class abstraction. (Contributed by NM,
16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
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Theorem | ovigg 5941* |
The value of an operation class abstraction. Compare ovig 5942.
The
condition is been
removed. (Contributed by FL,
24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
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Theorem | ovig 5942* |
The value of an operation class abstraction (weak version).
(Unnecessary distinct variable restrictions were removed by David
Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by
Mario Carneiro, 19-Dec-2013.)
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Theorem | ovmpt4g 5943* |
Value of a function given by the maps-to notation. (This is the
operation analog of fvmpt2 5551.) (Contributed by NM, 21-Feb-2004.)
(Revised by Mario Carneiro, 1-Sep-2015.)
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Theorem | ovmpos 5944* |
Value of a function given by the maps-to notation, expressed using
explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)
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Theorem | ov2gf 5945* |
The value of an operation class abstraction. A version of ovmpog 5955
using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.)
(Revised by Mario Carneiro, 19-Dec-2013.)
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Theorem | ovmpodxf 5946* |
Value of an operation given by a maps-to rule, deduction form.
(Contributed by Mario Carneiro, 29-Dec-2014.)
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Theorem | ovmpodx 5947* |
Value of an operation given by a maps-to rule, deduction form.
(Contributed by Mario Carneiro, 29-Dec-2014.)
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Theorem | ovmpod 5948* |
Value of an operation given by a maps-to rule, deduction form.
(Contributed by Mario Carneiro, 7-Dec-2014.)
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Theorem | ovmpox 5949* |
The value of an operation class abstraction. Variant of ovmpoga 5950 which
does not require and to be
distinct. (Contributed by Jeff
Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
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Theorem | ovmpoga 5950* |
Value of an operation given by a maps-to rule. (Contributed by Mario
Carneiro, 19-Dec-2013.)
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Theorem | ovmpoa 5951* |
Value of an operation given by a maps-to rule. (Contributed by NM,
19-Dec-2013.)
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Theorem | ovmpodf 5952* |
Alternate deduction version of ovmpo 5956, suitable for iteration.
(Contributed by Mario Carneiro, 7-Jan-2017.)
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Theorem | ovmpodv 5953* |
Alternate deduction version of ovmpo 5956, suitable for iteration.
(Contributed by Mario Carneiro, 7-Jan-2017.)
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Theorem | ovmpodv2 5954* |
Alternate deduction version of ovmpo 5956, suitable for iteration.
(Contributed by Mario Carneiro, 7-Jan-2017.)
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Theorem | ovmpog 5955* |
Value of an operation given by a maps-to rule. Special case.
(Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy,
19-Jun-2012.)
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Theorem | ovmpo 5956* |
Value of an operation given by a maps-to rule. Special case.
(Contributed by NM, 16-May-1995.) (Revised by David Abernethy,
19-Jun-2012.)
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Theorem | ovi3 5957* |
The value of an operation class abstraction. Special case.
(Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro,
29-Dec-2014.)
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Theorem | ov6g 5958* |
The value of an operation class abstraction. Special case.
(Contributed by NM, 13-Nov-2006.)
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Theorem | ovg 5959* |
The value of an operation class abstraction. (Contributed by Jeff
Madsen, 10-Jun-2010.)
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Theorem | ovres 5960 |
The value of a restricted operation. (Contributed by FL, 10-Nov-2006.)
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Theorem | ovresd 5961 |
Lemma for converting metric theorems to metric space theorems.
(Contributed by Mario Carneiro, 2-Oct-2015.)
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Theorem | oprssov 5962 |
The value of a member of the domain of a subclass of an operation.
(Contributed by NM, 23-Aug-2007.)
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Theorem | fovrn 5963 |
An operation's value belongs to its codomain. (Contributed by NM,
27-Aug-2006.)
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Theorem | fovrnda 5964 |
An operation's value belongs to its codomain. (Contributed by Mario
Carneiro, 29-Dec-2016.)
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Theorem | fovrnd 5965 |
An operation's value belongs to its codomain. (Contributed by Mario
Carneiro, 29-Dec-2016.)
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Theorem | fnrnov 5966* |
The range of an operation expressed as a collection of the operation's
values. (Contributed by NM, 29-Oct-2006.)
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Theorem | foov 5967* |
An onto mapping of an operation expressed in terms of operation values.
(Contributed by NM, 29-Oct-2006.)
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Theorem | fnovrn 5968 |
An operation's value belongs to its range. (Contributed by NM,
10-Feb-2007.)
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Theorem | ovelrn 5969* |
A member of an operation's range is a value of the operation.
(Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro,
30-Jan-2014.)
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Theorem | funimassov 5970* |
Membership relation for the values of a function whose image is a
subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)
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Theorem | ovelimab 5971* |
Operation value in an image. (Contributed by Mario Carneiro,
23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
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Theorem | ovconst2 5972 |
The value of a constant operation. (Contributed by NM, 5-Nov-2006.)
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Theorem | caovclg 5973* |
Convert an operation closure law to class notation. (Contributed by
Mario Carneiro, 26-May-2014.)
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Theorem | caovcld 5974* |
Convert an operation closure law to class notation. (Contributed by
Mario Carneiro, 30-Dec-2014.)
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Theorem | caovcl 5975* |
Convert an operation closure law to class notation. (Contributed by NM,
4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
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Theorem | caovcomg 5976* |
Convert an operation commutative law to class notation. (Contributed
by Mario Carneiro, 1-Jun-2013.)
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Theorem | caovcomd 5977* |
Convert an operation commutative law to class notation. (Contributed
by Mario Carneiro, 30-Dec-2014.)
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Theorem | caovcom 5978* |
Convert an operation commutative law to class notation. (Contributed
by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)
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Theorem | caovassg 5979* |
Convert an operation associative law to class notation. (Contributed
by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro,
26-May-2014.)
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Theorem | caovassd 5980* |
Convert an operation associative law to class notation. (Contributed
by Mario Carneiro, 30-Dec-2014.)
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Theorem | caovass 5981* |
Convert an operation associative law to class notation. (Contributed
by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
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Theorem | caovcang 5982* |
Convert an operation cancellation law to class notation. (Contributed
by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
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Theorem | caovcand 5983* |
Convert an operation cancellation law to class notation. (Contributed
by Mario Carneiro, 30-Dec-2014.)
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Theorem | caovcanrd 5984* |
Commute the arguments of an operation cancellation law. (Contributed
by Mario Carneiro, 30-Dec-2014.)
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Theorem | caovcan 5985* |
Convert an operation cancellation law to class notation. (Contributed
by NM, 20-Aug-1995.)
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Theorem | caovordig 5986* |
Convert an operation ordering law to class notation. (Contributed by
Mario Carneiro, 31-Dec-2014.)
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Theorem | caovordid 5987* |
Convert an operation ordering law to class notation. (Contributed by
Mario Carneiro, 31-Dec-2014.)
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Theorem | caovordg 5988* |
Convert an operation ordering law to class notation. (Contributed by
NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)
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Theorem | caovordd 5989* |
Convert an operation ordering law to class notation. (Contributed by
Mario Carneiro, 30-Dec-2014.)
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Theorem | caovord2d 5990* |
Operation ordering law with commuted arguments. (Contributed by Mario
Carneiro, 30-Dec-2014.)
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Theorem | caovord3d 5991* |
Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)
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Theorem | caovord 5992* |
Convert an operation ordering law to class notation. (Contributed by
NM, 19-Feb-1996.)
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Theorem | caovord2 5993* |
Operation ordering law with commuted arguments. (Contributed by NM,
27-Feb-1996.)
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Theorem | caovord3 5994* |
Ordering law. (Contributed by NM, 29-Feb-1996.)
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Theorem | caovdig 5995* |
Convert an operation distributive law to class notation. (Contributed
by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)
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Theorem | caovdid 5996* |
Convert an operation distributive law to class notation. (Contributed
by Mario Carneiro, 30-Dec-2014.)
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Theorem | caovdir2d 5997* |
Convert an operation distributive law to class notation. (Contributed
by Mario Carneiro, 30-Dec-2014.)
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Theorem | caovdirg 5998* |
Convert an operation reverse distributive law to class notation.
(Contributed by Mario Carneiro, 19-Oct-2014.)
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Theorem | caovdird 5999* |
Convert an operation distributive law to class notation. (Contributed
by Mario Carneiro, 30-Dec-2014.)
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Theorem | caovdi 6000* |
Convert an operation distributive law to class notation. (Contributed
by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)
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