Theorem List for Intuitionistic Logic Explorer - 5201-5300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | fdm 5201 |
The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
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Theorem | fdmd 5202 |
Deduction form of fdm 5201. The domain of a mapping. (Contributed by
Glauco Siliprandi, 26-Jun-2021.)
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Theorem | fdmi 5203 |
The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
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Theorem | frn 5204 |
The range of a mapping. (Contributed by NM, 3-Aug-1994.)
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Theorem | frnd 5205 |
Deduction form of frn 5204. The range of a mapping. (Contributed by
Glauco Siliprandi, 26-Jun-2021.)
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Theorem | dffn3 5206 |
A function maps to its range. (Contributed by NM, 1-Sep-1999.)
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Theorem | fss 5207 |
Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.)
(Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fssd 5208 |
Expanding the codomain of a mapping, deduction form. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
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Theorem | fssdmd 5209 |
Expressing that a class is a subclass of the domain of a function
expressed in maps-to notation, deduction form. (Contributed by AV,
21-Aug-2022.)
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Theorem | fssdm 5210 |
Expressing that a class is a subclass of the domain of a function
expressed in maps-to notation, semi-deduction form. (Contributed by AV,
21-Aug-2022.)
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Theorem | fco 5211 |
Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fco2 5212 |
Functionality of a composition with weakened out of domain condition on
the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
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Theorem | fssxp 5213 |
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 5214 |
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 5215 |
Two ways of specifying a partial function from to .
(Contributed by NM, 13-Nov-2007.)
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Theorem | ffdm 5216 |
A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
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Theorem | opelf 5217 |
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 5218 |
The union of two functions with disjoint domains. (Contributed by NM,
22-Sep-2004.)
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Theorem | fun2 5219 |
The union of two functions with disjoint domains. (Contributed by Mario
Carneiro, 12-Mar-2015.)
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Theorem | fnfco 5220 |
Composition of two functions. (Contributed by NM, 22-May-2006.)
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Theorem | fssres 5221 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 23-Sep-2004.)
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Theorem | fssresd 5222 |
Restriction of a function with a subclass of its domain, deduction form.
(Contributed by Glauco Siliprandi, 11-Dec-2019.)
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Theorem | fssres2 5223 |
Restriction of a restricted function with a subclass of its domain.
(Contributed by NM, 21-Jul-2005.)
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Theorem | fresin 5224 |
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 5225 |
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 5226 |
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 5227 |
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 5228* |
There is exactly one value of a function in its codomain. (Contributed
by NM, 10-Dec-2003.)
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Theorem | fcnvres 5229 |
The converse of a restriction of a function. (Contributed by NM,
26-Mar-1998.)
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Theorem | fimacnvdisj 5230 |
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 5231* |
Function into an intersection. (Contributed by Jim Kingdon,
28-Dec-2018.)
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Theorem | fin 5232 |
Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fabexg 5233* |
Existence of a set of functions. (Contributed by Paul Chapman,
25-Feb-2008.)
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Theorem | fabex 5234* |
Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
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Theorem | dmfex 5235 |
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 5236 |
The empty function. (Contributed by NM, 14-Aug-1999.)
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Theorem | f00 5237 |
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 | f0bi 5238 |
A function with empty domain is empty. (Contributed by Alexander van der
Vekens, 30-Jun-2018.)
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Theorem | f0dom0 5239 |
A function is empty iff it has an empty domain. (Contributed by AV,
10-Feb-2019.)
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Theorem | f0rn0 5240* |
If there is no element in the range of a function, its domain must be
empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
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Theorem | fconst 5241 |
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 5242 |
A cross product with a singleton is a constant function. (Contributed
by NM, 19-Oct-2004.)
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Theorem | fnconstg 5243 |
A cross product with a singleton is a constant function. (Contributed by
NM, 24-Jul-2014.)
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Theorem | fconst6g 5244 |
Constant function with loose range. (Contributed by Stefan O'Rear,
1-Feb-2015.)
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Theorem | fconst6 5245 |
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 5246 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1eq2 5247 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1eq3 5248 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | nff1 5249 |
Bound-variable hypothesis builder for a one-to-one function.
(Contributed by NM, 16-May-2004.)
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Theorem | dff12 5250* |
Alternate definition of a one-to-one function. (Contributed by NM,
31-Dec-1996.)
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Theorem | f1f 5251 |
A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
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Theorem | f1rn 5252 |
The range of a one-to-one mapping. (Contributed by BJ, 6-Jul-2022.)
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Theorem | f1fn 5253 |
A one-to-one mapping is a function on its domain. (Contributed by NM,
8-Mar-2014.)
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Theorem | f1fun 5254 |
A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
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Theorem | f1rel 5255 |
A one-to-one onto mapping is a relation. (Contributed by NM,
8-Mar-2014.)
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Theorem | f1dm 5256 |
The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
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Theorem | f1ss 5257 |
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 5258 |
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 | f1ff1 5259 |
If a function is one-to-one from A to B and is also a function from A to
C, then it is a one-to-one function from A to C. (Contributed by BJ,
4-Jul-2022.)
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Theorem | f1ssres 5260 |
A function that is one-to-one is also one-to-one on any subclass of its
domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
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Theorem | f1resf1 5261 |
The restriction of an injective function is injective. (Contributed by
AV, 28-Jun-2022.)
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Theorem | f1cnvcnv 5262 |
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 5263 |
Composition of one-to-one functions. Exercise 30 of [TakeutiZaring]
p. 25. (Contributed by NM, 28-May-1998.)
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Theorem | foeq1 5264 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | foeq2 5265 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | foeq3 5266 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | nffo 5267 |
Bound-variable hypothesis builder for an onto function. (Contributed by
NM, 16-May-2004.)
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Theorem | fof 5268 |
An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
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Theorem | fofun 5269 |
An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
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Theorem | fofn 5270 |
An onto mapping is a function on its domain. (Contributed by NM,
16-Dec-2008.)
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Theorem | forn 5271 |
The codomain of an onto function is its range. (Contributed by NM,
3-Aug-1994.)
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Theorem | dffo2 5272 |
Alternate definition of an onto function. (Contributed by NM,
22-Mar-2006.)
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Theorem | foima 5273 |
The image of the domain of an onto function. (Contributed by NM,
29-Nov-2002.)
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Theorem | dffn4 5274 |
A function maps onto its range. (Contributed by NM, 10-May-1998.)
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Theorem | funforn 5275 |
A function maps its domain onto its range. (Contributed by NM,
23-Jul-2004.)
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Theorem | fodmrnu 5276 |
An onto function has unique domain and range. (Contributed by NM,
5-Nov-2006.)
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Theorem | fores 5277 |
Restriction of a function. (Contributed by NM, 4-Mar-1997.)
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Theorem | foco 5278 |
Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
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Theorem | f1oeq1 5279 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1oeq2 5280 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1oeq3 5281 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1oeq23 5282 |
Equality theorem for one-to-one onto functions. (Contributed by FL,
14-Jul-2012.)
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Theorem | f1eq123d 5283 |
Equality deduction for one-to-one functions. (Contributed by Mario
Carneiro, 27-Jan-2017.)
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Theorem | foeq123d 5284 |
Equality deduction for onto functions. (Contributed by Mario Carneiro,
27-Jan-2017.)
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Theorem | f1oeq123d 5285 |
Equality deduction for one-to-one onto functions. (Contributed by Mario
Carneiro, 27-Jan-2017.)
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Theorem | nff1o 5286 |
Bound-variable hypothesis builder for a one-to-one onto function.
(Contributed by NM, 16-May-2004.)
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Theorem | f1of1 5287 |
A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM,
12-Dec-2003.)
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Theorem | f1of 5288 |
A one-to-one onto mapping is a mapping. (Contributed by NM,
12-Dec-2003.)
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Theorem | f1ofn 5289 |
A one-to-one onto mapping is function on its domain. (Contributed by NM,
12-Dec-2003.)
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Theorem | f1ofun 5290 |
A one-to-one onto mapping is a function. (Contributed by NM,
12-Dec-2003.)
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Theorem | f1orel 5291 |
A one-to-one onto mapping is a relation. (Contributed by NM,
13-Dec-2003.)
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Theorem | f1odm 5292 |
The domain of a one-to-one onto mapping. (Contributed by NM,
8-Mar-2014.)
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Theorem | dff1o2 5293 |
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 5294 |
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 5295 |
A one-to-one onto function is an onto function. (Contributed by NM,
28-Apr-2004.)
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Theorem | dff1o4 5296 |
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 5297 |
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 5298 |
A one-to-one function maps onto its range. (Contributed by NM,
13-Aug-2004.)
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Theorem | f1f1orn 5299 |
A one-to-one function maps one-to-one onto its range. (Contributed by NM,
4-Sep-2004.)
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Theorem | f1oabexg 5300* |
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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