Theorem List for Intuitionistic Logic Explorer - 5701-5800 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | elovmpt2 5701* |
Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear,
21-Jan-2015.)
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Theorem | f1ocnvd 5702* |
Describe an implicit one-to-one onto function. (Contributed by Mario
Carneiro, 30-Apr-2015.)
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Theorem | f1od 5703* |
Describe an implicit one-to-one onto function. (Contributed by Mario
Carneiro, 12-May-2014.)
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Theorem | f1ocnv2d 5704* |
Describe an implicit one-to-one onto function. (Contributed by Mario
Carneiro, 30-Apr-2015.)
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Theorem | f1o2d 5705* |
Describe an implicit one-to-one onto function. (Contributed by Mario
Carneiro, 12-May-2014.)
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Theorem | f1opw2 5706* |
A one-to-one mapping induces a one-to-one mapping on power sets. This
version of f1opw 5707 avoids the Axiom of Replacement.
(Contributed by
Mario Carneiro, 26-Jun-2015.)
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Theorem | f1opw 5707* |
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 5708* |
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 5709* |
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 5710 |
Extend class notation to include mapping of an operation to a function
operation.
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Syntax | cofr 5711 |
Extend class notation to include mapping of a binary relation to a
function relation.
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Definition | df-of 5712* |
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 5713* |
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 5714 |
Equality theorem for function operation. (Contributed by Mario
Carneiro, 20-Jul-2014.)
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Theorem | ofreq 5715 |
Equality theorem for function relation. (Contributed by Mario Carneiro,
28-Jul-2014.)
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Theorem | ofexg 5716 |
A function operation restricted to a set is a set. (Contributed by NM,
28-Jul-2014.)
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Theorem | nfof 5717* |
Hypothesis builder for function operation. (Contributed by Mario
Carneiro, 20-Jul-2014.)
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Theorem | nfofr 5718* |
Hypothesis builder for function relation. (Contributed by Mario
Carneiro, 28-Jul-2014.)
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Theorem | offval 5719* |
Value of an operation applied to two functions. (Contributed by Mario
Carneiro, 20-Jul-2014.)
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Theorem | ofrfval 5720* |
Value of a relation applied to two functions. (Contributed by Mario
Carneiro, 28-Jul-2014.)
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Theorem | fnofval 5721 |
Evaluate a function operation at a point. (Contributed by Mario
Carneiro, 20-Jul-2014.)
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Theorem | ofrval 5722 |
Exhibit a function relation at a point. (Contributed by Mario
Carneiro, 28-Jul-2014.)
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Theorem | ofmresval 5723 |
Value of a restriction of the function operation map. (Contributed by
NM, 20-Oct-2014.)
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Theorem | off 5724* |
The function operation produces a function. (Contributed by Mario
Carneiro, 20-Jul-2014.)
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Theorem | ofres 5725 |
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 5726* |
The function operation expressed as a mapping. (Contributed by Mario
Carneiro, 20-Jul-2014.)
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Theorem | ofrfval2 5727* |
The function relation acting on maps. (Contributed by Mario Carneiro,
20-Jul-2014.)
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Theorem | suppssof1 5728* |
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 5729 |
The composition of a function operation with another function.
(Contributed by Mario Carneiro, 19-Dec-2014.)
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Theorem | offveqb 5730* |
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 5731 |
Function operation on two constant functions. (Contributed by Mario
Carneiro, 28-Jul-2014.)
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Theorem | caofref 5732* |
Transfer a reflexive law to the function relation. (Contributed by
Mario Carneiro, 28-Jul-2014.)
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Theorem | caofinvl 5733* |
Transfer a left inverse law to the function operation. (Contributed
by NM, 22-Oct-2014.)
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Theorem | caofcom 5734* |
Transfer a commutative law to the function operation. (Contributed by
Mario Carneiro, 26-Jul-2014.)
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Theorem | caofrss 5735* |
Transfer a relation subset law to the function relation. (Contributed
by Mario Carneiro, 28-Jul-2014.)
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Theorem | caoftrn 5736* |
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 5737 |
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 5382 but requires ax-pow 3927 and ax-un 4170. (Contributed by NM,
7-Apr-1995.) (Proof modification is discouraged.)
(New usage is discouraged.)
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Theorem | cofunexg 5738 |
Existence of a composition when the first member is a function.
(Contributed by NM, 8-Oct-2007.)
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Theorem | cofunex2g 5739 |
Existence of a composition when the second member is one-to-one.
(Contributed by NM, 8-Oct-2007.)
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Theorem | fnexALT 5740 |
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 4983. This version of fnex 5383
uses
ax-pow 3927 and ax-un 4170, whereas fnex 5383
does not. (Contributed by NM,
14-Aug-1994.) (Proof modification is discouraged.)
(New usage is discouraged.)
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Theorem | funrnex 5741 |
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 5384. (Contributed by NM, 11-Nov-1995.)
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Theorem | fornex 5742 |
If the domain of an onto function exists, so does its codomain.
(Contributed by NM, 23-Jul-2004.)
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Theorem | f1dmex 5743 |
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 5744* |
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 5386, funex 5384, fnex 5383, resfunexg 5382, and
funimaexg 4983. See also abrexex2 5751. (Contributed by NM, 16-Oct-2003.)
(Proof shortened by Mario Carneiro, 31-Aug-2015.)
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Theorem | abrexexg 5745* |
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 5746* |
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 5747* |
Existence of an existentially restricted class abstraction.
(Contributed by Jeff Madsen, 2-Sep-2009.)
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Theorem | opabex3d 5748* |
Existence of an ordered pair abstraction, deduction version.
(Contributed by Alexander van der Vekens, 19-Oct-2017.)
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Theorem | opabex3 5749* |
Existence of an ordered pair abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iunex 5750* |
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 5751* |
Existence of an existentially restricted class abstraction. is
normally has free-variable parameters and . See also
abrexex 5744. (Contributed by NM, 12-Sep-2004.)
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Theorem | abexssex 5752* |
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 5753* |
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 5754* |
Existence of an operator abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | oprabex 5755* |
Existence of an operation class abstraction. (Contributed by NM,
19-Oct-2004.)
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Theorem | oprabex3 5756* |
Existence of an operation class abstraction (special case).
(Contributed by NM, 19-Oct-2004.)
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Theorem | oprabrexex2 5757* |
Existence of an existentially restricted operation abstraction.
(Contributed by Jeff Madsen, 11-Jun-2010.)
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Theorem | ab2rexex 5758* |
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 5744. (Contributed by NM,
20-Sep-2011.)
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Theorem | ab2rexex2 5759* |
Existence of an existentially restricted class abstraction.
normally has free-variable parameters , , and .
Compare abrexex2 5751. (Contributed by NM, 20-Sep-2011.)
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Theorem | xpexgALT 5760 |
The cross product of two sets is a set. Proposition 6.2 of
[TakeutiZaring] p. 23. This
version is proven using Replacement; see
xpexg 4452 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 5761* |
General value of with no assumptions on functionality
of and . (Contributed by Stefan
O'Rear, 24-Jan-2015.)
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Theorem | offres 5762 |
Pointwise combination commutes with restriction. (Contributed by Stefan
O'Rear, 24-Jan-2015.)
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Theorem | ofmres 5763* |
Equivalent expressions for a restriction of the function operation map.
Unlike which is a proper class,
can be a set by ofmresex 5764, allowing it to be used as a function or
structure argument. By ofmresval 5723, 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 5764 |
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 5765 |
Extend the definition of a class to include the first member an ordered
pair function.
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Syntax | c2nd 5766 |
Extend the definition of a class to include the second member an ordered
pair function.
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Definition | df-1st 5767 |
Define a function that extracts the first member, or abscissa, of an
ordered pair. Theorem op1st 5773 proves that it does this. For example,
( 3 , 4 ) = 3 . Equivalent to Definition
5.13 (i) of
[Monk1] p. 52 (compare op1sta 4802 and op1stb 4209). The notation is the same
as Monk's. (Contributed by NM, 9-Oct-2004.)
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Definition | df-2nd 5768 |
Define a function that extracts the second member, or ordinate, of an
ordered pair. Theorem op2nd 5774 proves that it does this. For example,
3 , 4 ) = 4 . Equivalent to Definition 5.13 (ii)
of [Monk1] p. 52 (compare op2nda 4805 and op2ndb 4804). The notation is the
same as Monk's. (Contributed by NM, 9-Oct-2004.)
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Theorem | 1stvalg 5769 |
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 5770 |
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 5771 |
The value of the first-member function at the empty set. (Contributed by
NM, 23-Apr-2007.)
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Theorem | 2nd0 5772 |
The value of the second-member function at the empty set. (Contributed by
NM, 23-Apr-2007.)
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Theorem | op1st 5773 |
Extract the first member of an ordered pair. (Contributed by NM,
5-Oct-2004.)
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Theorem | op2nd 5774 |
Extract the second member of an ordered pair. (Contributed by NM,
5-Oct-2004.)
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Theorem | op1std 5775 |
Extract the first member of an ordered pair. (Contributed by Mario
Carneiro, 31-Aug-2015.)
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Theorem | op2ndd 5776 |
Extract the second member of an ordered pair. (Contributed by Mario
Carneiro, 31-Aug-2015.)
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Theorem | op1stg 5777 |
Extract the first member of an ordered pair. (Contributed by NM,
19-Jul-2005.)
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Theorem | op2ndg 5778 |
Extract the second member of an ordered pair. (Contributed by NM,
19-Jul-2005.)
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Theorem | ot1stg 5779 |
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 5779,
ot2ndg 5780, ot3rdgg 5781.) (Contributed by NM, 3-Apr-2015.) (Revised
by
Mario Carneiro, 2-May-2015.)
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Theorem | ot2ndg 5780 |
Extract the second member of an ordered triple. (See ot1stg 5779 comment.)
(Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro,
2-May-2015.)
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Theorem | ot3rdgg 5781 |
Extract the third member of an ordered triple. (See ot1stg 5779 comment.)
(Contributed by NM, 3-Apr-2015.)
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Theorem | 1stval2 5782 |
Alternate value of the function that extracts the first member of an
ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by
NM, 18-Aug-2006.)
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Theorem | 2ndval2 5783 |
Alternate value of the function that extracts the second member of an
ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by
NM, 18-Aug-2006.)
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Theorem | fo1st 5784 |
The function
maps the universe onto the universe. (Contributed
by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
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Theorem | fo2nd 5785 |
The function
maps the universe onto the universe. (Contributed
by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
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Theorem | f1stres 5786 |
Mapping of a restriction of the (first member of an ordered
pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario
Carneiro, 8-Sep-2013.)
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Theorem | f2ndres 5787 |
Mapping of a restriction of the (second member of an ordered
pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario
Carneiro, 8-Sep-2013.)
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Theorem | fo1stresm 5788* |
Onto mapping of a restriction of the (first member of an ordered
pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
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Theorem | fo2ndresm 5789* |
Onto mapping of a restriction of the (second member of an
ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
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Theorem | 1stcof 5790 |
Composition of the first member function with another function.
(Contributed by NM, 12-Oct-2007.)
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Theorem | 2ndcof 5791 |
Composition of the second member function with another function.
(Contributed by FL, 15-Oct-2012.)
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Theorem | xp1st 5792 |
Location of the first element of a Cartesian product. (Contributed by
Jeff Madsen, 2-Sep-2009.)
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Theorem | xp2nd 5793 |
Location of the second element of a Cartesian product. (Contributed by
Jeff Madsen, 2-Sep-2009.)
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Theorem | 1stexg 5794 |
Existence of the first member of a set. (Contributed by Jim Kingdon,
26-Jan-2019.)
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Theorem | 2ndexg 5795 |
Existence of the first member of a set. (Contributed by Jim Kingdon,
26-Jan-2019.)
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Theorem | elxp6 5796 |
Membership in a cross product. This version requires no quantifiers or
dummy variables. See also elxp4 4808. (Contributed by NM, 9-Oct-2004.)
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Theorem | elxp7 5797 |
Membership in a cross product. This version requires no quantifiers or
dummy variables. See also elxp4 4808. (Contributed by NM, 19-Aug-2006.)
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Theorem | eqopi 5798 |
Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.)
(Revised by Mario Carneiro, 23-Feb-2014.)
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Theorem | xp2 5799* |
Representation of cross product based on ordered pair component
functions. (Contributed by NM, 16-Sep-2006.)
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Theorem | unielxp 5800 |
The membership relation for a cross product is inherited by union.
(Contributed by NM, 16-Sep-2006.)
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