Theorem List for Intuitionistic Logic Explorer - 5201-5300 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | funimaexglem 5201 |
Lemma for funimaexg 5202. It constitutes the interesting part of
funimaexg 5202, in which
. (Contributed by Jim
Kingdon,
27-Dec-2018.)
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Theorem | funimaexg 5202 |
Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284.
Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM,
10-Sep-2006.)
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Theorem | funimaex 5203 |
The image of a set under any function is also a set. Equivalent of
Axiom of Replacement. Axiom 39(vi) of [Quine] p. 284. Compare Exercise
9 of [TakeutiZaring] p. 29.
(Contributed by NM, 17-Nov-2002.)
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Theorem | isarep1 5204* |
Part of a study of the Axiom of Replacement used by the Isabelle prover.
The object PrimReplace is apparently the image of the function encoded
by i.e. the class .
If so, we can prove Isabelle's "Axiom of Replacement"
conclusion without
using the Axiom of Replacement, for which I (N. Megill) currently have
no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by
Mario Carneiro, 4-Dec-2016.)
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Theorem | isarep2 5205* |
Part of a study of the Axiom of Replacement used by the Isabelle prover.
In Isabelle, the sethood of PrimReplace is apparently postulated
implicitly by its type signature " i, i, i
=> o
=> i", which automatically asserts that it is a set without
using any
axioms. To prove that it is a set in Metamath, we need the hypotheses
of Isabelle's "Axiom of Replacement" as well as the Axiom of
Replacement
in the form funimaex 5203. (Contributed by NM, 26-Oct-2006.)
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Theorem | fneq1 5206 |
Equality theorem for function predicate with domain. (Contributed by NM,
1-Aug-1994.)
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Theorem | fneq2 5207 |
Equality theorem for function predicate with domain. (Contributed by NM,
1-Aug-1994.)
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Theorem | fneq1d 5208 |
Equality deduction for function predicate with domain. (Contributed by
Paul Chapman, 22-Jun-2011.)
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Theorem | fneq2d 5209 |
Equality deduction for function predicate with domain. (Contributed by
Paul Chapman, 22-Jun-2011.)
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Theorem | fneq12d 5210 |
Equality deduction for function predicate with domain. (Contributed by
NM, 26-Jun-2011.)
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Theorem | fneq12 5211 |
Equality theorem for function predicate with domain. (Contributed by
Thierry Arnoux, 31-Jan-2017.)
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Theorem | fneq1i 5212 |
Equality inference for function predicate with domain. (Contributed by
Paul Chapman, 22-Jun-2011.)
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Theorem | fneq2i 5213 |
Equality inference for function predicate with domain. (Contributed by
NM, 4-Sep-2011.)
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Theorem | nffn 5214 |
Bound-variable hypothesis builder for a function with domain.
(Contributed by NM, 30-Jan-2004.)
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Theorem | fnfun 5215 |
A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
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Theorem | fnrel 5216 |
A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
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Theorem | fndm 5217 |
The domain of a function. (Contributed by NM, 2-Aug-1994.)
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Theorem | funfni 5218 |
Inference to convert a function and domain antecedent. (Contributed by
NM, 22-Apr-2004.)
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Theorem | fndmu 5219 |
A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
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Theorem | fnbr 5220 |
The first argument of binary relation on a function belongs to the
function's domain. (Contributed by NM, 7-May-2004.)
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Theorem | fnop 5221 |
The first argument of an ordered pair in a function belongs to the
function's domain. (Contributed by NM, 8-Aug-1994.)
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Theorem | fneu 5222* |
There is exactly one value of a function. (Contributed by NM,
22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fneu2 5223* |
There is exactly one value of a function. (Contributed by NM,
7-Nov-1995.)
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Theorem | fnun 5224 |
The union of two functions with disjoint domains. (Contributed by NM,
22-Sep-2004.)
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Theorem | fnunsn 5225 |
Extension of a function with a new ordered pair. (Contributed by NM,
28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
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Theorem | fnco 5226 |
Composition of two functions. (Contributed by NM, 22-May-2006.)
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Theorem | fnresdm 5227 |
A function does not change when restricted to its domain. (Contributed by
NM, 5-Sep-2004.)
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Theorem | fnresdisj 5228 |
A function restricted to a class disjoint with its domain is empty.
(Contributed by NM, 23-Sep-2004.)
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Theorem | 2elresin 5229 |
Membership in two functions restricted by each other's domain.
(Contributed by NM, 8-Aug-1994.)
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Theorem | fnssresb 5230 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 10-Oct-2007.)
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Theorem | fnssres 5231 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 2-Aug-1994.)
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Theorem | fnresin1 5232 |
Restriction of a function's domain with an intersection. (Contributed by
NM, 9-Aug-1994.)
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Theorem | fnresin2 5233 |
Restriction of a function's domain with an intersection. (Contributed by
NM, 9-Aug-1994.)
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Theorem | fnres 5234* |
An equivalence for functionality of a restriction. Compare dffun8 5146.
(Contributed by Mario Carneiro, 20-May-2015.)
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Theorem | fnresi 5235 |
Functionality and domain of restricted identity. (Contributed by NM,
27-Aug-2004.)
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Theorem | fnima 5236 |
The image of a function's domain is its range. (Contributed by NM,
4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fn0 5237 |
A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.)
(Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fnimadisj 5238 |
A class that is disjoint with the domain of a function has an empty image
under the function. (Contributed by FL, 24-Jan-2007.)
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Theorem | fnimaeq0 5239 |
Images under a function never map nonempty sets to empty sets.
(Contributed by Stefan O'Rear, 21-Jan-2015.)
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Theorem | dfmpt3 5240 |
Alternate definition for the maps-to notation df-mpt 3986. (Contributed
by Mario Carneiro, 30-Dec-2016.)
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Theorem | fnopabg 5241* |
Functionality and domain of an ordered-pair class abstraction.
(Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro,
4-Dec-2016.)
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Theorem | fnopab 5242* |
Functionality and domain of an ordered-pair class abstraction.
(Contributed by NM, 5-Mar-1996.)
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Theorem | mptfng 5243* |
The maps-to notation defines a function with domain. (Contributed by
Scott Fenton, 21-Mar-2011.)
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Theorem | fnmpt 5244* |
The maps-to notation defines a function with domain. (Contributed by
NM, 9-Apr-2013.)
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Theorem | mpt0 5245 |
A mapping operation with empty domain. (Contributed by Mario Carneiro,
28-Dec-2014.)
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Theorem | fnmpti 5246* |
Functionality and domain of an ordered-pair class abstraction.
(Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro,
31-Aug-2015.)
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Theorem | dmmpti 5247* |
Domain of an ordered-pair class abstraction that specifies a function.
(Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro,
31-Aug-2015.)
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Theorem | dmmptd 5248* |
The domain of the mapping operation, deduction form. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
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Theorem | mptun 5249 |
Union of mappings which are mutually compatible. (Contributed by Mario
Carneiro, 31-Aug-2015.)
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Theorem | feq1 5250 |
Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | feq2 5251 |
Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | feq3 5252 |
Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | feq23 5253 |
Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | feq1d 5254 |
Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
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Theorem | feq2d 5255 |
Equality deduction for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq3d 5256 |
Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
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Theorem | feq12d 5257 |
Equality deduction for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq123d 5258 |
Equality deduction for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq123 5259 |
Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
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Theorem | feq1i 5260 |
Equality inference for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq2i 5261 |
Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
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Theorem | feq23i 5262 |
Equality inference for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq23d 5263 |
Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
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Theorem | nff 5264 |
Bound-variable hypothesis builder for a mapping. (Contributed by NM,
29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | sbcfng 5265* |
Distribute proper substitution through the function predicate with a
domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
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Theorem | sbcfg 5266* |
Distribute proper substitution through the function predicate with
domain and codomain. (Contributed by Alexander van der Vekens,
15-Jul-2018.)
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Theorem | ffn 5267 |
A mapping is a function. (Contributed by NM, 2-Aug-1994.)
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Theorem | ffnd 5268 |
A mapping is a function with domain, deduction form. (Contributed by
Glauco Siliprandi, 17-Aug-2020.)
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Theorem | dffn2 5269 |
Any function is a mapping into . (Contributed by NM, 31-Oct-1995.)
(Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | ffun 5270 |
A mapping is a function. (Contributed by NM, 3-Aug-1994.)
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Theorem | ffund 5271 |
A mapping is a function, deduction version. (Contributed by Glauco
Siliprandi, 3-Mar-2021.)
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Theorem | frel 5272 |
A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
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Theorem | fdm 5273 |
The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
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Theorem | fdmd 5274 |
Deduction form of fdm 5273. The domain of a mapping. (Contributed by
Glauco Siliprandi, 26-Jun-2021.)
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Theorem | fdmi 5275 |
The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
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Theorem | frn 5276 |
The range of a mapping. (Contributed by NM, 3-Aug-1994.)
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Theorem | frnd 5277 |
Deduction form of frn 5276. The range of a mapping. (Contributed by
Glauco Siliprandi, 26-Jun-2021.)
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Theorem | dffn3 5278 |
A function maps to its range. (Contributed by NM, 1-Sep-1999.)
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Theorem | fss 5279 |
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 5280 |
Expanding the codomain of a mapping, deduction form. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
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Theorem | fssdmd 5281 |
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 5282 |
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 5283 |
Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fco2 5284 |
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 5285 |
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 5286 |
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 5287 |
Two ways of specifying a partial function from to .
(Contributed by NM, 13-Nov-2007.)
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Theorem | ffdm 5288 |
A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
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Theorem | opelf 5289 |
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 5290 |
The union of two functions with disjoint domains. (Contributed by NM,
22-Sep-2004.)
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Theorem | fun2 5291 |
The union of two functions with disjoint domains. (Contributed by Mario
Carneiro, 12-Mar-2015.)
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Theorem | fnfco 5292 |
Composition of two functions. (Contributed by NM, 22-May-2006.)
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Theorem | fssres 5293 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 23-Sep-2004.)
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Theorem | fssresd 5294 |
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 5295 |
Restriction of a restricted function with a subclass of its domain.
(Contributed by NM, 21-Jul-2005.)
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Theorem | fresin 5296 |
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 5297 |
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 5298 |
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 5299 |
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 5300* |
There is exactly one value of a function in its codomain. (Contributed
by NM, 10-Dec-2003.)
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