Theorem List for Intuitionistic Logic Explorer - 5001-5100 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | fssxp 5001 |
A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.)
(Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fex2 5002 |
A function with bounded domain and range is a set. This version is proven
without the Axiom of Replacement. (Contributed by Mario Carneiro,
24-Jun-2015.)
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Theorem | funssxp 5003 |
Two ways of specifying a partial function from to .
(Contributed by NM, 13-Nov-2007.)
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Theorem | ffdm 5004 |
A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
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Theorem | opelf 5005 |
The members of an ordered pair element of a mapping belong to the
mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.)
(Revised by Mario Carneiro, 26-Apr-2015.)
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Theorem | fun 5006 |
The union of two functions with disjoint domains. (Contributed by NM,
22-Sep-2004.)
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Theorem | fun2 5007 |
The union of two functions with disjoint domains. (Contributed by Mario
Carneiro, 12-Mar-2015.)
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Theorem | fnfco 5008 |
Composition of two functions. (Contributed by NM, 22-May-2006.)
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Theorem | fssres 5009 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 23-Sep-2004.)
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Theorem | fssres2 5010 |
Restriction of a restricted function with a subclass of its domain.
(Contributed by NM, 21-Jul-2005.)
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Theorem | fresin 5011 |
An identity for the mapping relationship under restriction. (Contributed
by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro,
26-May-2016.)
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Theorem | resasplitss 5012 |
If two functions agree on their common domain, their union contains a
union of three functions with pairwise disjoint domains. If we assumed
the law of the excluded middle, this would be equality rather than
subset. (Contributed by Jim Kingdon, 28-Dec-2018.)
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Theorem | fcoi1 5013 |
Composition of a mapping and restricted identity. (Contributed by NM,
13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fcoi2 5014 |
Composition of restricted identity and a mapping. (Contributed by NM,
13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | feu 5015* |
There is exactly one value of a function in its codomain. (Contributed
by NM, 10-Dec-2003.)
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Theorem | fcnvres 5016 |
The converse of a restriction of a function. (Contributed by NM,
26-Mar-1998.)
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Theorem | fimacnvdisj 5017 |
The preimage of a class disjoint with a mapping's codomain is empty.
(Contributed by FL, 24-Jan-2007.)
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Theorem | fintm 5018* |
Function into an intersection. (Contributed by Jim Kingdon,
28-Dec-2018.)
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Theorem | fin 5019 |
Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fabexg 5020* |
Existence of a set of functions. (Contributed by Paul Chapman,
25-Feb-2008.)
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Theorem | fabex 5021* |
Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
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Theorem | dmfex 5022 |
If a mapping is a set, its domain is a set. (Contributed by NM,
27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | f0 5023 |
The empty function. (Contributed by NM, 14-Aug-1999.)
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Theorem | f00 5024 |
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 5025 |
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 5026 |
A cross product with a singleton is a constant function. (Contributed
by NM, 19-Oct-2004.)
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Theorem | fnconstg 5027 |
A cross product with a singleton is a constant function. (Contributed by
NM, 24-Jul-2014.)
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Theorem | fconst6g 5028 |
Constant function with loose range. (Contributed by Stefan O'Rear,
1-Feb-2015.)
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Theorem | fconst6 5029 |
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 5030 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1eq2 5031 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1eq3 5032 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | nff1 5033 |
Bound-variable hypothesis builder for a one-to-one function.
(Contributed by NM, 16-May-2004.)
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Theorem | dff12 5034* |
Alternate definition of a one-to-one function. (Contributed by NM,
31-Dec-1996.)
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Theorem | f1f 5035 |
A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
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Theorem | f1fn 5036 |
A one-to-one mapping is a function on its domain. (Contributed by NM,
8-Mar-2014.)
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Theorem | f1fun 5037 |
A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
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Theorem | f1rel 5038 |
A one-to-one onto mapping is a relation. (Contributed by NM,
8-Mar-2014.)
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Theorem | f1dm 5039 |
The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
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Theorem | f1ss 5040 |
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 5041 |
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 5042 |
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 5043 |
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 5044 |
Composition of one-to-one functions. Exercise 30 of [TakeutiZaring]
p. 25. (Contributed by NM, 28-May-1998.)
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Theorem | foeq1 5045 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | foeq2 5046 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | foeq3 5047 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | nffo 5048 |
Bound-variable hypothesis builder for an onto function. (Contributed by
NM, 16-May-2004.)
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Theorem | fof 5049 |
An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
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Theorem | fofun 5050 |
An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
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Theorem | fofn 5051 |
An onto mapping is a function on its domain. (Contributed by NM,
16-Dec-2008.)
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Theorem | forn 5052 |
The codomain of an onto function is its range. (Contributed by NM,
3-Aug-1994.)
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Theorem | dffo2 5053 |
Alternate definition of an onto function. (Contributed by NM,
22-Mar-2006.)
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Theorem | foima 5054 |
The image of the domain of an onto function. (Contributed by NM,
29-Nov-2002.)
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Theorem | dffn4 5055 |
A function maps onto its range. (Contributed by NM, 10-May-1998.)
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Theorem | funforn 5056 |
A function maps its domain onto its range. (Contributed by NM,
23-Jul-2004.)
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Theorem | fodmrnu 5057 |
An onto function has unique domain and range. (Contributed by NM,
5-Nov-2006.)
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Theorem | fores 5058 |
Restriction of a function. (Contributed by NM, 4-Mar-1997.)
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Theorem | foco 5059 |
Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
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Theorem | f1oeq1 5060 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1oeq2 5061 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1oeq3 5062 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1oeq23 5063 |
Equality theorem for one-to-one onto functions. (Contributed by FL,
14-Jul-2012.)
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Theorem | f1eq123d 5064 |
Equality deduction for one-to-one functions. (Contributed by Mario
Carneiro, 27-Jan-2017.)
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Theorem | foeq123d 5065 |
Equality deduction for onto functions. (Contributed by Mario Carneiro,
27-Jan-2017.)
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Theorem | f1oeq123d 5066 |
Equality deduction for one-to-one onto functions. (Contributed by Mario
Carneiro, 27-Jan-2017.)
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Theorem | nff1o 5067 |
Bound-variable hypothesis builder for a one-to-one onto function.
(Contributed by NM, 16-May-2004.)
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Theorem | f1of1 5068 |
A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM,
12-Dec-2003.)
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Theorem | f1of 5069 |
A one-to-one onto mapping is a mapping. (Contributed by NM,
12-Dec-2003.)
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Theorem | f1ofn 5070 |
A one-to-one onto mapping is function on its domain. (Contributed by NM,
12-Dec-2003.)
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Theorem | f1ofun 5071 |
A one-to-one onto mapping is a function. (Contributed by NM,
12-Dec-2003.)
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Theorem | f1orel 5072 |
A one-to-one onto mapping is a relation. (Contributed by NM,
13-Dec-2003.)
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Theorem | f1odm 5073 |
The domain of a one-to-one onto mapping. (Contributed by NM,
8-Mar-2014.)
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Theorem | dff1o2 5074 |
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 5075 |
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 5076 |
A one-to-one onto function is an onto function. (Contributed by NM,
28-Apr-2004.)
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Theorem | dff1o4 5077 |
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 5078 |
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 5079 |
A one-to-one function maps onto its range. (Contributed by NM,
13-Aug-2004.)
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Theorem | f1f1orn 5080 |
A one-to-one function maps one-to-one onto its range. (Contributed by NM,
4-Sep-2004.)
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Theorem | f1oabexg 5081* |
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 5082 |
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 5083 |
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 5084 |
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 5085 |
The converse of a one-to-one-onto restricted function. (Contributed by
Paul Chapman, 21-Apr-2008.)
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Theorem | f1imacnv 5086 |
Preimage of an image. (Contributed by NM, 30-Sep-2004.)
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Theorem | foimacnv 5087 |
A reverse version of f1imacnv 5086. (Contributed by Jeff Hankins,
16-Jul-2009.)
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Theorem | foun 5088 |
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 5089 |
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 5090* |
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 5091 |
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 5092 |
Composition of one-to-one onto functions. (Contributed by NM,
19-Mar-1998.)
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Theorem | f1cnv 5093 |
The converse of an injective function is bijective. (Contributed by FL,
11-Nov-2011.)
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Theorem | funcocnv2 5094 |
Composition with the converse. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | fococnv2 5095 |
The composition of an onto function and its converse. (Contributed by
Stefan O'Rear, 12-Feb-2015.)
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Theorem | f1ococnv2 5096 |
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 5097 |
Composition of an injective function with its converse. (Contributed by
FL, 11-Nov-2011.)
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Theorem | f1ococnv1 5098 |
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 5099 |
Composition of an injective function with its converse. (Contributed by
FL, 11-Nov-2011.)
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Theorem | funcoeqres 5100 |
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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