Theorem List for Intuitionistic Logic Explorer - 5301-5400 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | fneu2 5301* |
There is exactly one value of a function. (Contributed by NM,
7-Nov-1995.)
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Theorem | fnun 5302 |
The union of two functions with disjoint domains. (Contributed by NM,
22-Sep-2004.)
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Theorem | fnunsn 5303 |
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 5304 |
Composition of two functions. (Contributed by NM, 22-May-2006.)
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Theorem | fnresdm 5305 |
A function does not change when restricted to its domain. (Contributed by
NM, 5-Sep-2004.)
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Theorem | fnresdisj 5306 |
A function restricted to a class disjoint with its domain is empty.
(Contributed by NM, 23-Sep-2004.)
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Theorem | 2elresin 5307 |
Membership in two functions restricted by each other's domain.
(Contributed by NM, 8-Aug-1994.)
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Theorem | fnssresb 5308 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 10-Oct-2007.)
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Theorem | fnssres 5309 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 2-Aug-1994.)
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Theorem | fnresin1 5310 |
Restriction of a function's domain with an intersection. (Contributed by
NM, 9-Aug-1994.)
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Theorem | fnresin2 5311 |
Restriction of a function's domain with an intersection. (Contributed by
NM, 9-Aug-1994.)
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Theorem | fnres 5312* |
An equivalence for functionality of a restriction. Compare dffun8 5224.
(Contributed by Mario Carneiro, 20-May-2015.)
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Theorem | fnresi 5313 |
Functionality and domain of restricted identity. (Contributed by NM,
27-Aug-2004.)
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Theorem | fnima 5314 |
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 5315 |
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 5316 |
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 5317 |
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 5318 |
Alternate definition for the maps-to notation df-mpt 4050. (Contributed
by Mario Carneiro, 30-Dec-2016.)
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Theorem | fnopabg 5319* |
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 5320* |
Functionality and domain of an ordered-pair class abstraction.
(Contributed by NM, 5-Mar-1996.)
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Theorem | mptfng 5321* |
The maps-to notation defines a function with domain. (Contributed by
Scott Fenton, 21-Mar-2011.)
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Theorem | fnmpt 5322* |
The maps-to notation defines a function with domain. (Contributed by
NM, 9-Apr-2013.)
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Theorem | mpt0 5323 |
A mapping operation with empty domain. (Contributed by Mario Carneiro,
28-Dec-2014.)
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Theorem | fnmpti 5324* |
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 5325* |
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 5326* |
The domain of the mapping operation, deduction form. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
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Theorem | mptun 5327 |
Union of mappings which are mutually compatible. (Contributed by Mario
Carneiro, 31-Aug-2015.)
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Theorem | feq1 5328 |
Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | feq2 5329 |
Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | feq3 5330 |
Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | feq23 5331 |
Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | feq1d 5332 |
Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
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Theorem | feq2d 5333 |
Equality deduction for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq3d 5334 |
Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
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Theorem | feq12d 5335 |
Equality deduction for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq123d 5336 |
Equality deduction for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq123 5337 |
Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
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Theorem | feq1i 5338 |
Equality inference for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq2i 5339 |
Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
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Theorem | feq23i 5340 |
Equality inference for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq23d 5341 |
Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
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Theorem | nff 5342 |
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 5343* |
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 5344* |
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 5345 |
A mapping is a function. (Contributed by NM, 2-Aug-1994.)
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Theorem | ffnd 5346 |
A mapping is a function with domain, deduction form. (Contributed by
Glauco Siliprandi, 17-Aug-2020.)
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Theorem | dffn2 5347 |
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 5348 |
A mapping is a function. (Contributed by NM, 3-Aug-1994.)
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Theorem | ffund 5349 |
A mapping is a function, deduction version. (Contributed by Glauco
Siliprandi, 3-Mar-2021.)
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Theorem | frel 5350 |
A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
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Theorem | fdm 5351 |
The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
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Theorem | fdmd 5352 |
Deduction form of fdm 5351. The domain of a mapping. (Contributed by
Glauco Siliprandi, 26-Jun-2021.)
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Theorem | fdmi 5353 |
The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
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Theorem | frn 5354 |
The range of a mapping. (Contributed by NM, 3-Aug-1994.)
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Theorem | frnd 5355 |
Deduction form of frn 5354. The range of a mapping. (Contributed by
Glauco Siliprandi, 26-Jun-2021.)
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Theorem | dffn3 5356 |
A function maps to its range. (Contributed by NM, 1-Sep-1999.)
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Theorem | fss 5357 |
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 5358 |
Expanding the codomain of a mapping, deduction form. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
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Theorem | fssdmd 5359 |
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 5360 |
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 5361 |
Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fco2 5362 |
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 5363 |
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 5364 |
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 5365 |
Two ways of specifying a partial function from to .
(Contributed by NM, 13-Nov-2007.)
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Theorem | ffdm 5366 |
A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
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Theorem | opelf 5367 |
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 5368 |
The union of two functions with disjoint domains. (Contributed by NM,
22-Sep-2004.)
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Theorem | fun2 5369 |
The union of two functions with disjoint domains. (Contributed by Mario
Carneiro, 12-Mar-2015.)
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Theorem | fnfco 5370 |
Composition of two functions. (Contributed by NM, 22-May-2006.)
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Theorem | fssres 5371 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 23-Sep-2004.)
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Theorem | fssresd 5372 |
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 5373 |
Restriction of a restricted function with a subclass of its domain.
(Contributed by NM, 21-Jul-2005.)
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Theorem | fresin 5374 |
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 5375 |
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 5376 |
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 5377 |
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 5378* |
There is exactly one value of a function in its codomain. (Contributed
by NM, 10-Dec-2003.)
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Theorem | fcnvres 5379 |
The converse of a restriction of a function. (Contributed by NM,
26-Mar-1998.)
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Theorem | fimacnvdisj 5380 |
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 5381* |
Function into an intersection. (Contributed by Jim Kingdon,
28-Dec-2018.)
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Theorem | fin 5382 |
Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fabexg 5383* |
Existence of a set of functions. (Contributed by Paul Chapman,
25-Feb-2008.)
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Theorem | fabex 5384* |
Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
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Theorem | dmfex 5385 |
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 5386 |
The empty function. (Contributed by NM, 14-Aug-1999.)
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Theorem | f00 5387 |
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 5388 |
A function with empty domain is empty. (Contributed by Alexander van der
Vekens, 30-Jun-2018.)
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Theorem | f0dom0 5389 |
A function is empty iff it has an empty domain. (Contributed by AV,
10-Feb-2019.)
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Theorem | f0rn0 5390* |
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 5391 |
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 5392 |
A cross product with a singleton is a constant function. (Contributed
by NM, 19-Oct-2004.)
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Theorem | fnconstg 5393 |
A cross product with a singleton is a constant function. (Contributed by
NM, 24-Jul-2014.)
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Theorem | fconst6g 5394 |
Constant function with loose range. (Contributed by Stefan O'Rear,
1-Feb-2015.)
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Theorem | fconst6 5395 |
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 5396 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1eq2 5397 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | f1eq3 5398 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
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Theorem | nff1 5399 |
Bound-variable hypothesis builder for a one-to-one function.
(Contributed by NM, 16-May-2004.)
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Theorem | dff12 5400* |
Alternate definition of a one-to-one function. (Contributed by NM,
31-Dec-1996.)
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