Theorem List for Intuitionistic Logic Explorer - 5701-5800 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | caov32d 5701* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
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Theorem | caov12d 5702* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
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Theorem | caov31d 5703* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
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Theorem | caov13d 5704* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
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Theorem | caov4d 5705* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
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Theorem | caov411d 5706* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
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Theorem | caov42d 5707* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
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Theorem | caov32 5708* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.)
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Theorem | caov12 5709* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.)
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Theorem | caov31 5710* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.)
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Theorem | caov13 5711* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.)
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Theorem | caovdilemd 5712* |
Lemma used by real number construction. (Contributed by Jim Kingdon,
16-Sep-2019.)
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Theorem | caovlem2d 5713* |
Rearrangement of expression involving multiplication () and
addition ().
(Contributed by Jim Kingdon, 3-Jan-2020.)
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Theorem | caovimo 5714* |
Uniqueness of inverse element in commutative, associative operation with
identity. The identity element is . (Contributed by Jim Kingdon,
18-Sep-2019.)
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Theorem | grprinvlem 5715* |
Lemma for grprinvd 5716. (Contributed by NM, 9-Aug-2013.)
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Theorem | grprinvd 5716* |
Deduce right inverse from left inverse and left identity in an
associative structure (such as a group). (Contributed by NM,
10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
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Theorem | grpridd 5717* |
Deduce right identity from left inverse and left identity in an
associative structure (such as a group). (Contributed by NM,
10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
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2.6.11 "Maps to" notation
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Theorem | elmpt2cl 5718* |
If a two-parameter class is not empty, constrain the implicit pair.
(Contributed by Stefan O'Rear, 7-Mar-2015.)
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Theorem | elmpt2cl1 5719* |
If a two-parameter class is not empty, the first argument is in its
nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan
O'Rear, 7-Mar-2015.)
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Theorem | elmpt2cl2 5720* |
If a two-parameter class is not empty, the second argument is in its
nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan
O'Rear, 7-Mar-2015.)
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Theorem | elovmpt2 5721* |
Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear,
21-Jan-2015.)
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Theorem | f1ocnvd 5722* |
Describe an implicit one-to-one onto function. (Contributed by Mario
Carneiro, 30-Apr-2015.)
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Theorem | f1od 5723* |
Describe an implicit one-to-one onto function. (Contributed by Mario
Carneiro, 12-May-2014.)
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Theorem | f1ocnv2d 5724* |
Describe an implicit one-to-one onto function. (Contributed by Mario
Carneiro, 30-Apr-2015.)
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Theorem | f1o2d 5725* |
Describe an implicit one-to-one onto function. (Contributed by Mario
Carneiro, 12-May-2014.)
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Theorem | f1opw2 5726* |
A one-to-one mapping induces a one-to-one mapping on power sets. This
version of f1opw 5727 avoids the Axiom of Replacement.
(Contributed by
Mario Carneiro, 26-Jun-2015.)
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Theorem | f1opw 5727* |
A one-to-one mapping induces a one-to-one mapping on power sets.
(Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario
Carneiro, 26-Jun-2015.)
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Theorem | suppssfv 5728* |
Formula building theorem for support restriction, on a function which
preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)
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Theorem | suppssov1 5729* |
Formula building theorem for support restrictions: operator with left
annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
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2.6.12 Function operation
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Syntax | cof 5730 |
Extend class notation to include mapping of an operation to a function
operation.
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Syntax | cofr 5731 |
Extend class notation to include mapping of a binary relation to a
function relation.
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Definition | df-of 5732* |
Define the function operation map. The definition is designed so that
if is a binary
operation, then is the analogous operation
on functions which corresponds to applying pointwise to the values
of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
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Definition | df-ofr 5733* |
Define the function relation map. The definition is designed so that if
is a binary
relation, then is the analogous relation on
functions which is true when each element of the left function relates
to the corresponding element of the right function. (Contributed by
Mario Carneiro, 28-Jul-2014.)
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Theorem | ofeq 5734 |
Equality theorem for function operation. (Contributed by Mario
Carneiro, 20-Jul-2014.)
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Theorem | ofreq 5735 |
Equality theorem for function relation. (Contributed by Mario Carneiro,
28-Jul-2014.)
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Theorem | ofexg 5736 |
A function operation restricted to a set is a set. (Contributed by NM,
28-Jul-2014.)
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Theorem | nfof 5737* |
Hypothesis builder for function operation. (Contributed by Mario
Carneiro, 20-Jul-2014.)
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Theorem | nfofr 5738* |
Hypothesis builder for function relation. (Contributed by Mario
Carneiro, 28-Jul-2014.)
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Theorem | offval 5739* |
Value of an operation applied to two functions. (Contributed by Mario
Carneiro, 20-Jul-2014.)
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Theorem | ofrfval 5740* |
Value of a relation applied to two functions. (Contributed by Mario
Carneiro, 28-Jul-2014.)
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Theorem | fnofval 5741 |
Evaluate a function operation at a point. (Contributed by Mario
Carneiro, 20-Jul-2014.)
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Theorem | ofrval 5742 |
Exhibit a function relation at a point. (Contributed by Mario
Carneiro, 28-Jul-2014.)
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Theorem | ofmresval 5743 |
Value of a restriction of the function operation map. (Contributed by
NM, 20-Oct-2014.)
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Theorem | off 5744* |
The function operation produces a function. (Contributed by Mario
Carneiro, 20-Jul-2014.)
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Theorem | ofres 5745 |
Restrict the operands of a function operation to the same domain as that
of the operation itself. (Contributed by Mario Carneiro,
15-Sep-2014.)
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Theorem | offval2 5746* |
The function operation expressed as a mapping. (Contributed by Mario
Carneiro, 20-Jul-2014.)
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Theorem | ofrfval2 5747* |
The function relation acting on maps. (Contributed by Mario Carneiro,
20-Jul-2014.)
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Theorem | suppssof1 5748* |
Formula building theorem for support restrictions: vector operation with
left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
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Theorem | ofco 5749 |
The composition of a function operation with another function.
(Contributed by Mario Carneiro, 19-Dec-2014.)
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Theorem | offveqb 5750* |
Equivalent expressions for equality with a function operation.
(Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro,
5-Dec-2016.)
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Theorem | ofc12 5751 |
Function operation on two constant functions. (Contributed by Mario
Carneiro, 28-Jul-2014.)
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Theorem | caofref 5752* |
Transfer a reflexive law to the function relation. (Contributed by
Mario Carneiro, 28-Jul-2014.)
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Theorem | caofinvl 5753* |
Transfer a left inverse law to the function operation. (Contributed
by NM, 22-Oct-2014.)
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Theorem | caofcom 5754* |
Transfer a commutative law to the function operation. (Contributed by
Mario Carneiro, 26-Jul-2014.)
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Theorem | caofrss 5755* |
Transfer a relation subset law to the function relation. (Contributed
by Mario Carneiro, 28-Jul-2014.)
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Theorem | caoftrn 5756* |
Transfer a transitivity law to the function relation. (Contributed by
Mario Carneiro, 28-Jul-2014.)
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2.6.13 Functions (continued)
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Theorem | resfunexgALT 5757 |
The restriction of a function to a set exists. Compare Proposition 6.17
of [TakeutiZaring] p. 28. This
version has a shorter proof than
resfunexg 5403 but requires ax-pow 3948 and ax-un 4188. (Contributed by NM,
7-Apr-1995.) (Proof modification is discouraged.)
(New usage is discouraged.)
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Theorem | cofunexg 5758 |
Existence of a composition when the first member is a function.
(Contributed by NM, 8-Oct-2007.)
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Theorem | cofunex2g 5759 |
Existence of a composition when the second member is one-to-one.
(Contributed by NM, 8-Oct-2007.)
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Theorem | fnexALT 5760 |
If the domain of a function is a set, the function is a set. Theorem
6.16(1) of [TakeutiZaring] p. 28.
This theorem is derived using the Axiom
of Replacement in the form of funimaexg 5003. This version of fnex 5404
uses
ax-pow 3948 and ax-un 4188, whereas fnex 5404
does not. (Contributed by NM,
14-Aug-1994.) (Proof modification is discouraged.)
(New usage is discouraged.)
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Theorem | funrnex 5761 |
If the domain of a function exists, so does its range. Part of Theorem
4.15(v) of [Monk1] p. 46. This theorem is
derived using the Axiom of
Replacement in the form of funex 5405. (Contributed by NM, 11-Nov-1995.)
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Theorem | fornex 5762 |
If the domain of an onto function exists, so does its codomain.
(Contributed by NM, 23-Jul-2004.)
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Theorem | f1dmex 5763 |
If the codomain of a one-to-one function exists, so does its domain. This
can be thought of as a form of the Axiom of Replacement. (Contributed by
NM, 4-Sep-2004.)
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Theorem | abrexex 5764* |
Existence of a class abstraction of existentially restricted sets.
is normally a free-variable parameter in the class expression
substituted for , which can be thought of as . This
simple-looking theorem is actually quite powerful and appears to involve
the Axiom of Replacement in an intrinsic way, as can be seen by tracing
back through the path mptexg 5407, funex 5405, fnex 5404, resfunexg 5403, and
funimaexg 5003. See also abrexex2 5771. (Contributed by NM, 16-Oct-2003.)
(Proof shortened by Mario Carneiro, 31-Aug-2015.)
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Theorem | abrexexg 5765* |
Existence of a class abstraction of existentially restricted sets.
is normally a free-variable parameter in . The antecedent assures
us that is a
set. (Contributed by NM, 3-Nov-2003.)
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Theorem | iunexg 5766* |
The existence of an indexed union. is normally a free-variable
parameter in .
(Contributed by NM, 23-Mar-2006.)
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Theorem | abrexex2g 5767* |
Existence of an existentially restricted class abstraction.
(Contributed by Jeff Madsen, 2-Sep-2009.)
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Theorem | opabex3d 5768* |
Existence of an ordered pair abstraction, deduction version.
(Contributed by Alexander van der Vekens, 19-Oct-2017.)
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Theorem | opabex3 5769* |
Existence of an ordered pair abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iunex 5770* |
The existence of an indexed union. is normally a free-variable
parameter in the class expression substituted for , which can be
read informally as . (Contributed by NM, 13-Oct-2003.)
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Theorem | abrexex2 5771* |
Existence of an existentially restricted class abstraction. is
normally has free-variable parameters and . See also
abrexex 5764. (Contributed by NM, 12-Sep-2004.)
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Theorem | abexssex 5772* |
Existence of a class abstraction with an existentially quantified
expression. Both and can be
free in .
(Contributed
by NM, 29-Jul-2006.)
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Theorem | abexex 5773* |
A condition where a class builder continues to exist after its wff is
existentially quantified. (Contributed by NM, 4-Mar-2007.)
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Theorem | oprabexd 5774* |
Existence of an operator abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | oprabex 5775* |
Existence of an operation class abstraction. (Contributed by NM,
19-Oct-2004.)
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Theorem | oprabex3 5776* |
Existence of an operation class abstraction (special case).
(Contributed by NM, 19-Oct-2004.)
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Theorem | oprabrexex2 5777* |
Existence of an existentially restricted operation abstraction.
(Contributed by Jeff Madsen, 11-Jun-2010.)
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Theorem | ab2rexex 5778* |
Existence of a class abstraction of existentially restricted sets.
Variables and
are normally
free-variable parameters in the
class expression substituted for , which can be thought of as
. See comments for abrexex 5764. (Contributed by NM,
20-Sep-2011.)
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Theorem | ab2rexex2 5779* |
Existence of an existentially restricted class abstraction.
normally has free-variable parameters , , and .
Compare abrexex2 5771. (Contributed by NM, 20-Sep-2011.)
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Theorem | xpexgALT 5780 |
The cross product of two sets is a set. Proposition 6.2 of
[TakeutiZaring] p. 23. This
version is proven using Replacement; see
xpexg 4470 for a version that uses the Power Set axiom
instead.
(Contributed by Mario Carneiro, 20-May-2013.)
(Proof modification is discouraged.) (New usage is discouraged.)
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Theorem | offval3 5781* |
General value of with no assumptions on functionality
of and . (Contributed by Stefan
O'Rear, 24-Jan-2015.)
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Theorem | offres 5782 |
Pointwise combination commutes with restriction. (Contributed by Stefan
O'Rear, 24-Jan-2015.)
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Theorem | ofmres 5783* |
Equivalent expressions for a restriction of the function operation map.
Unlike which is a proper class,
can be a set by ofmresex 5784, allowing it to be used as a function or
structure argument. By ofmresval 5743, the restricted operation map
values are the same as the original values, allowing theorems for
to be reused. (Contributed by NM, 20-Oct-2014.)
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Theorem | ofmresex 5784 |
Existence of a restriction of the function operation map. (Contributed
by NM, 20-Oct-2014.)
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2.6.14 First and second members of an ordered
pair
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Syntax | c1st 5785 |
Extend the definition of a class to include the first member an ordered
pair function.
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Syntax | c2nd 5786 |
Extend the definition of a class to include the second member an ordered
pair function.
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Definition | df-1st 5787 |
Define a function that extracts the first member, or abscissa, of an
ordered pair. Theorem op1st 5793 proves that it does this. For example,
( 3 , 4 ) = 3 . Equivalent to Definition
5.13 (i) of
[Monk1] p. 52 (compare op1sta 4822 and op1stb 4227). The notation is the same
as Monk's. (Contributed by NM, 9-Oct-2004.)
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Definition | df-2nd 5788 |
Define a function that extracts the second member, or ordinate, of an
ordered pair. Theorem op2nd 5794 proves that it does this. For example,
3 , 4 ) = 4 . Equivalent to Definition 5.13 (ii)
of [Monk1] p. 52 (compare op2nda 4825 and op2ndb 4824). The notation is the
same as Monk's. (Contributed by NM, 9-Oct-2004.)
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Theorem | 1stvalg 5789 |
The value of the function that extracts the first member of an ordered
pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro,
8-Sep-2013.)
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Theorem | 2ndvalg 5790 |
The value of the function that extracts the second member of an ordered
pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro,
8-Sep-2013.)
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Theorem | 1st0 5791 |
The value of the first-member function at the empty set. (Contributed by
NM, 23-Apr-2007.)
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Theorem | 2nd0 5792 |
The value of the second-member function at the empty set. (Contributed by
NM, 23-Apr-2007.)
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Theorem | op1st 5793 |
Extract the first member of an ordered pair. (Contributed by NM,
5-Oct-2004.)
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Theorem | op2nd 5794 |
Extract the second member of an ordered pair. (Contributed by NM,
5-Oct-2004.)
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Theorem | op1std 5795 |
Extract the first member of an ordered pair. (Contributed by Mario
Carneiro, 31-Aug-2015.)
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Theorem | op2ndd 5796 |
Extract the second member of an ordered pair. (Contributed by Mario
Carneiro, 31-Aug-2015.)
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Theorem | op1stg 5797 |
Extract the first member of an ordered pair. (Contributed by NM,
19-Jul-2005.)
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Theorem | op2ndg 5798 |
Extract the second member of an ordered pair. (Contributed by NM,
19-Jul-2005.)
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Theorem | ot1stg 5799 |
Extract the first member of an ordered triple. (Due to infrequent
usage, it isn't worthwhile at this point to define special extractors
for triples, so we reuse the ordered pair extractors for ot1stg 5799,
ot2ndg 5800, ot3rdgg 5801.) (Contributed by NM, 3-Apr-2015.) (Revised
by
Mario Carneiro, 2-May-2015.)
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Theorem | ot2ndg 5800 |
Extract the second member of an ordered triple. (See ot1stg 5799 comment.)
(Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro,
2-May-2015.)
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