Theorem List for Intuitionistic Logic Explorer - 5101-5200 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | f00 5101 |
A class is a function with empty codomain iff it and its domain are empty.
(Contributed by NM, 10-Dec-2003.)
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Theorem | fconst 5102 |
A cross product with a singleton is a constant function. (Contributed
by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon,
17-Sep-2011.)
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Theorem | fconstg 5103 |
A cross product with a singleton is a constant function. (Contributed
by NM, 19-Oct-2004.)
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Theorem | fnconstg 5104 |
A cross product with a singleton is a constant function. (Contributed by
NM, 24-Jul-2014.)
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Theorem | fconst6g 5105 |
Constant function with loose range. (Contributed by Stefan O'Rear,
1-Feb-2015.)
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Theorem | fconst6 5106 |
A constant function as a mapping. (Contributed by Jeff Madsen,
30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
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Theorem | f1eq1 5107 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1eq2 5108 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1eq3 5109 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | nff1 5110 |
Bound-variable hypothesis builder for a one-to-one function.
(Contributed by NM, 16-May-2004.)
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Theorem | dff12 5111* |
Alternate definition of a one-to-one function. (Contributed by NM,
31-Dec-1996.)
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Theorem | f1f 5112 |
A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
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Theorem | f1fn 5113 |
A one-to-one mapping is a function on its domain. (Contributed by NM,
8-Mar-2014.)
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Theorem | f1fun 5114 |
A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
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Theorem | f1rel 5115 |
A one-to-one onto mapping is a relation. (Contributed by NM,
8-Mar-2014.)
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Theorem | f1dm 5116 |
The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
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Theorem | f1ss 5117 |
A function that is one-to-one is also one-to-one on some superset of its
range. (Contributed by Mario Carneiro, 12-Jan-2013.)
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Theorem | f1ssr 5118 |
Combine a one-to-one function with a restriction on the domain.
(Contributed by Stefan O'Rear, 20-Feb-2015.)
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Theorem | f1ssres 5119 |
A function that is one-to-one is also one-to-one on some aubset of its
domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
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Theorem | f1cnvcnv 5120 |
Two ways to express that a set (not necessarily a function) is
one-to-one. Each side is equivalent to Definition 6.4(3) of
[TakeutiZaring] p. 24, who use the
notation "Un2 (A)" for one-to-one.
We
do not introduce a separate notation since we rarely use it. (Contributed
by NM, 13-Aug-2004.)
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Theorem | f1co 5121 |
Composition of one-to-one functions. Exercise 30 of [TakeutiZaring]
p. 25. (Contributed by NM, 28-May-1998.)
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Theorem | foeq1 5122 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | foeq2 5123 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | foeq3 5124 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | nffo 5125 |
Bound-variable hypothesis builder for an onto function. (Contributed by
NM, 16-May-2004.)
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Theorem | fof 5126 |
An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
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Theorem | fofun 5127 |
An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
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Theorem | fofn 5128 |
An onto mapping is a function on its domain. (Contributed by NM,
16-Dec-2008.)
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Theorem | forn 5129 |
The codomain of an onto function is its range. (Contributed by NM,
3-Aug-1994.)
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Theorem | dffo2 5130 |
Alternate definition of an onto function. (Contributed by NM,
22-Mar-2006.)
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Theorem | foima 5131 |
The image of the domain of an onto function. (Contributed by NM,
29-Nov-2002.)
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Theorem | dffn4 5132 |
A function maps onto its range. (Contributed by NM, 10-May-1998.)
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Theorem | funforn 5133 |
A function maps its domain onto its range. (Contributed by NM,
23-Jul-2004.)
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Theorem | fodmrnu 5134 |
An onto function has unique domain and range. (Contributed by NM,
5-Nov-2006.)
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Theorem | fores 5135 |
Restriction of a function. (Contributed by NM, 4-Mar-1997.)
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Theorem | foco 5136 |
Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
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Theorem | f1oeq1 5137 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1oeq2 5138 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1oeq3 5139 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1oeq23 5140 |
Equality theorem for one-to-one onto functions. (Contributed by FL,
14-Jul-2012.)
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Theorem | f1eq123d 5141 |
Equality deduction for one-to-one functions. (Contributed by Mario
Carneiro, 27-Jan-2017.)
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Theorem | foeq123d 5142 |
Equality deduction for onto functions. (Contributed by Mario Carneiro,
27-Jan-2017.)
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Theorem | f1oeq123d 5143 |
Equality deduction for one-to-one onto functions. (Contributed by Mario
Carneiro, 27-Jan-2017.)
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Theorem | nff1o 5144 |
Bound-variable hypothesis builder for a one-to-one onto function.
(Contributed by NM, 16-May-2004.)
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Theorem | f1of1 5145 |
A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM,
12-Dec-2003.)
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Theorem | f1of 5146 |
A one-to-one onto mapping is a mapping. (Contributed by NM,
12-Dec-2003.)
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Theorem | f1ofn 5147 |
A one-to-one onto mapping is function on its domain. (Contributed by NM,
12-Dec-2003.)
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Theorem | f1ofun 5148 |
A one-to-one onto mapping is a function. (Contributed by NM,
12-Dec-2003.)
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Theorem | f1orel 5149 |
A one-to-one onto mapping is a relation. (Contributed by NM,
13-Dec-2003.)
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Theorem | f1odm 5150 |
The domain of a one-to-one onto mapping. (Contributed by NM,
8-Mar-2014.)
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Theorem | dff1o2 5151 |
Alternate definition of one-to-one onto function. (Contributed by NM,
10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
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Theorem | dff1o3 5152 |
Alternate definition of one-to-one onto function. (Contributed by NM,
25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
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Theorem | f1ofo 5153 |
A one-to-one onto function is an onto function. (Contributed by NM,
28-Apr-2004.)
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Theorem | dff1o4 5154 |
Alternate definition of one-to-one onto function. (Contributed by NM,
25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
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Theorem | dff1o5 5155 |
Alternate definition of one-to-one onto function. (Contributed by NM,
10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
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Theorem | f1orn 5156 |
A one-to-one function maps onto its range. (Contributed by NM,
13-Aug-2004.)
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Theorem | f1f1orn 5157 |
A one-to-one function maps one-to-one onto its range. (Contributed by NM,
4-Sep-2004.)
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Theorem | f1oabexg 5158* |
The class of all 1-1-onto functions mapping one set to another is a set.
(Contributed by Paul Chapman, 25-Feb-2008.)
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Theorem | f1ocnv 5159 |
The converse of a one-to-one onto function is also one-to-one onto.
(Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon,
22-Oct-2011.)
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Theorem | f1ocnvb 5160 |
A relation is a one-to-one onto function iff its converse is a one-to-one
onto function with domain and range interchanged. (Contributed by NM,
8-Dec-2003.)
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Theorem | f1ores 5161 |
The restriction of a one-to-one function maps one-to-one onto the image.
(Contributed by NM, 25-Mar-1998.)
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Theorem | f1orescnv 5162 |
The converse of a one-to-one-onto restricted function. (Contributed by
Paul Chapman, 21-Apr-2008.)
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Theorem | f1imacnv 5163 |
Preimage of an image. (Contributed by NM, 30-Sep-2004.)
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Theorem | foimacnv 5164 |
A reverse version of f1imacnv 5163. (Contributed by Jeff Hankins,
16-Jul-2009.)
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Theorem | foun 5165 |
The union of two onto functions with disjoint domains is an onto function.
(Contributed by Mario Carneiro, 22-Jun-2016.)
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Theorem | f1oun 5166 |
The union of two one-to-one onto functions with disjoint domains and
ranges. (Contributed by NM, 26-Mar-1998.)
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Theorem | fun11iun 5167* |
The union of a chain (with respect to inclusion) of one-to-one functions
is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.)
(Revised by Mario Carneiro, 24-Jun-2015.)
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Theorem | resdif 5168 |
The restriction of a one-to-one onto function to a difference maps onto
the difference of the images. (Contributed by Paul Chapman,
11-Apr-2009.)
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Theorem | f1oco 5169 |
Composition of one-to-one onto functions. (Contributed by NM,
19-Mar-1998.)
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Theorem | f1cnv 5170 |
The converse of an injective function is bijective. (Contributed by FL,
11-Nov-2011.)
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Theorem | funcocnv2 5171 |
Composition with the converse. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | fococnv2 5172 |
The composition of an onto function and its converse. (Contributed by
Stefan O'Rear, 12-Feb-2015.)
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Theorem | f1ococnv2 5173 |
The composition of a one-to-one onto function and its converse equals the
identity relation restricted to the function's range. (Contributed by NM,
13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)
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Theorem | f1cocnv2 5174 |
Composition of an injective function with its converse. (Contributed by
FL, 11-Nov-2011.)
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Theorem | f1ococnv1 5175 |
The composition of a one-to-one onto function's converse and itself equals
the identity relation restricted to the function's domain. (Contributed
by NM, 13-Dec-2003.)
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Theorem | f1cocnv1 5176 |
Composition of an injective function with its converse. (Contributed by
FL, 11-Nov-2011.)
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Theorem | funcoeqres 5177 |
Re-express a constraint on a composition as a constraint on the composand.
(Contributed by Stefan O'Rear, 7-Mar-2015.)
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Theorem | ffoss 5178* |
Relationship between a mapping and an onto mapping. Figure 38 of
[Enderton] p. 145. (Contributed by NM,
10-May-1998.)
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Theorem | f11o 5179* |
Relationship between one-to-one and one-to-one onto function.
(Contributed by NM, 4-Apr-1998.)
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Theorem | f10 5180 |
The empty set maps one-to-one into any class. (Contributed by NM,
7-Apr-1998.)
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Theorem | f1o00 5181 |
One-to-one onto mapping of the empty set. (Contributed by NM,
15-Apr-1998.)
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Theorem | fo00 5182 |
Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
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Theorem | f1o0 5183 |
One-to-one onto mapping of the empty set. (Contributed by NM,
10-Sep-2004.)
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Theorem | f1oi 5184 |
A restriction of the identity relation is a one-to-one onto function.
(Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon,
22-Oct-2011.)
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Theorem | f1ovi 5185 |
The identity relation is a one-to-one onto function on the universe.
(Contributed by NM, 16-May-2004.)
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Theorem | f1osn 5186 |
A singleton of an ordered pair is one-to-one onto function.
(Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon,
22-Oct-2011.)
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Theorem | f1osng 5187 |
A singleton of an ordered pair is one-to-one onto function.
(Contributed by Mario Carneiro, 12-Jan-2013.)
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Theorem | f1oprg 5188 |
An unordered pair of ordered pairs with different elements is a one-to-one
onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
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Theorem | tz6.12-2 5189* |
Function value when
is not a function. Theorem 6.12(2) of
[TakeutiZaring] p. 27.
(Contributed by NM, 30-Apr-2004.) (Proof
shortened by Mario Carneiro, 31-Aug-2015.)
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Theorem | fveu 5190* |
The value of a function at a unique point. (Contributed by Scott
Fenton, 6-Oct-2017.)
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Theorem | brprcneu 5191* |
If is a proper class
and is any class,
then there is no
unique set which is related to through the binary relation .
(Contributed by Scott Fenton, 7-Oct-2017.)
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Theorem | fvprc 5192 |
A function's value at a proper class is the empty set. (Contributed by
NM, 20-May-1998.)
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Theorem | fv2 5193* |
Alternate definition of function value. Definition 10.11 of [Quine]
p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew
Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
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Theorem | dffv3g 5194* |
A definition of function value in terms of iota. (Contributed by Jim
Kingdon, 29-Dec-2018.)
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Theorem | dffv4g 5195* |
The previous definition of function value, from before the
operator was introduced. Although based on the idea embodied by
Definition 10.2 of [Quine] p. 65 (see args 4714), this definition
apparently does not appear in the literature. (Contributed by NM,
1-Aug-1994.)
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Theorem | elfv 5196* |
Membership in a function value. (Contributed by NM, 30-Apr-2004.)
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Theorem | fveq1 5197 |
Equality theorem for function value. (Contributed by NM,
29-Dec-1996.)
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Theorem | fveq2 5198 |
Equality theorem for function value. (Contributed by NM,
29-Dec-1996.)
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Theorem | fveq1i 5199 |
Equality inference for function value. (Contributed by NM,
2-Sep-2003.)
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Theorem | fveq1d 5200 |
Equality deduction for function value. (Contributed by NM,
2-Sep-2003.)
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