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Type | Label | Description |
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Statement | ||
Theorem | iotabii 5201 | Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | iotacl 5202 |
Membership law for descriptions.
This can useful for expanding an unbounded iota-based definition (see df-iota 5179). (Contributed by Andrew Salmon, 1-Aug-2011.) |
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Theorem | iota2df 5203 |
A condition that allows us to represent "the unique element such that
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Theorem | iota2d 5204* |
A condition that allows us to represent "the unique element such that
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Theorem | eliota 5205* | An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.) |
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Theorem | eliotaeu 5206 | An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.) |
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Theorem | iota2 5207* |
The unique element such that ![]() |
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Theorem | sniota 5208 | A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.) |
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Theorem | iotam 5209* |
Representation of "the unique element such that ![]() ![]() ![]() |
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Theorem | csbiotag 5210* | Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) |
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Syntax | wfun 5211 |
Extend the definition of a wff to include the function predicate. (Read:
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Syntax | wfn 5212 |
Extend the definition of a wff to include the function predicate with a
domain. (Read: ![]() ![]() |
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Syntax | wf 5213 |
Extend the definition of a wff to include the function predicate with
domain and codomain. (Read: ![]() ![]() ![]() |
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Syntax | wf1 5214 |
Extend the definition of a wff to include one-to-one functions. (Read:
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Syntax | wfo 5215 |
Extend the definition of a wff to include onto functions. (Read: ![]() ![]() ![]() |
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Syntax | wf1o 5216 |
Extend the definition of a wff to include one-to-one onto functions.
(Read: ![]() ![]() ![]() |
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Syntax | cfv 5217 |
Extend the definition of a class to include the value of a function.
(Read: The value of ![]() ![]() ![]() ![]() |
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Syntax | wiso 5218 |
Extend the definition of a wff to include the isomorphism property.
(Read: ![]() ![]() ![]() ![]() ![]() |
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Definition | df-fun 5219 |
Define predicate that determines if some class ![]() ![]() ![]() |
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Definition | df-fn 5220 | Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) |
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Definition | df-f 5221 | Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) |
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Definition | df-f1 5222 | Define a one-to-one function. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow). (Contributed by NM, 1-Aug-1994.) |
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Definition | df-fo 5223 | Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). (Contributed by NM, 1-Aug-1994.) |
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Definition | df-f1o 5224 | Define a one-to-one onto function. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow). (Contributed by NM, 1-Aug-1994.) |
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Definition | df-fv 5225* |
Define the value of a function, ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-isom 5226* |
Define the isomorphism predicate. We read this as "![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dffun2 5227* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) |
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Theorem | dffun4 5228* | Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.) |
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Theorem | dffun5r 5229* | A way of proving a relation is a function, analogous to mo2r 2078. (Contributed by Jim Kingdon, 27-May-2020.) |
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Theorem | dffun6f 5230* | Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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Theorem | dffun6 5231* | Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) |
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Theorem | funmo 5232* | A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) |
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Theorem | dffun4f 5233* | Definition of function like dffun4 5228 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.) |
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Theorem | funrel 5234 | A function is a relation. (Contributed by NM, 1-Aug-1994.) |
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Theorem | 0nelfun 5235 | A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.) |
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Theorem | funss 5236 | Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) |
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Theorem | funeq 5237 | Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
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Theorem | funeqi 5238 | Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
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Theorem | funeqd 5239 | Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
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Theorem | nffun 5240 | Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.) |
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Theorem | sbcfung 5241 | Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
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Theorem | funeu 5242* | 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 | funeu2 5243* | There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.) |
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Theorem | dffun7 5244* | Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one". However, dffun8 5245 shows that it does not matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.) |
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Theorem | dffun8 5245* | Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5244. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
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Theorem | dffun9 5246* | Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
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Theorem | funfn 5247 | An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.) |
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Theorem | funfnd 5248 | A function is a function over its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
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Theorem | funi 5249 | The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.) |
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Theorem | nfunv 5250 | The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.) |
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Theorem | funopg 5251 | A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | funopab 5252* | A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.) |
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Theorem | funopabeq 5253* | A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.) |
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Theorem | funopab4 5254* | A class of ordered pairs of values in the form used by df-mpt 4067 is a function. (Contributed by NM, 17-Feb-2013.) |
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Theorem | funmpt 5255 | A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.) |
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Theorem | funmpt2 5256 | Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.) |
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Theorem | funco 5257 | The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
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Theorem | funres 5258 | A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.) |
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Theorem | funssres 5259 | The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.) |
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Theorem | fun2ssres 5260 | Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.) |
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Theorem | funun 5261 | The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.) |
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Theorem | funcnvsn 5262 |
The converse singleton of an ordered pair is a function. This is
equivalent to funsn 5265 via cnvsn 5112, but stating it this way allows us to
skip the sethood assumptions on ![]() ![]() |
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Theorem | funsng 5263 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.) |
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Theorem | fnsng 5264 | Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.) |
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Theorem | funsn 5265 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.) |
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Theorem | funinsn 5266 |
A function based on the singleton of an ordered pair. Unlike funsng 5263,
this holds even if ![]() ![]() |
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Theorem | funprg 5267 | A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) |
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Theorem | funtpg 5268 | A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
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Theorem | funpr 5269 | A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
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Theorem | funtp 5270 | A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
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Theorem | fnsn 5271 | Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
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Theorem | fnprg 5272 | Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | fntpg 5273 | Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
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Theorem | fntp 5274 | A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | fun0 5275 | The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.) |
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Theorem | funcnvcnv 5276 | The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.) |
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Theorem | funcnv2 5277* | A simpler equivalence for single-rooted (see funcnv 5278). (Contributed by NM, 9-Aug-2004.) |
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Theorem | funcnv 5278* |
The converse of a class is a function iff the class is single-rooted,
which means that for any ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | funcnv3 5279* | A condition showing a class is single-rooted. (See funcnv 5278). (Contributed by NM, 26-May-2006.) |
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Theorem | funcnveq 5280* | Another way of expressing that a class is single-rooted. Counterpart to dffun2 5227. (Contributed by Jim Kingdon, 24-Dec-2018.) |
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Theorem | fun2cnv 5281* |
The double converse of a class is a function iff the class is
single-valued. Each side is equivalent to Definition 6.4(2) of
[TakeutiZaring] p. 23, who use the
notation "Un(A)" for single-valued.
Note that ![]() |
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Theorem | svrelfun 5282 | A single-valued relation is a function. (See fun2cnv 5281 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.) |
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Theorem | fncnv 5283* | Single-rootedness (see funcnv 5278) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.) |
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Theorem | fun11 5284* |
Two ways of stating that ![]() |
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Theorem | fununi 5285* | The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.) |
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Theorem | funcnvuni 5286* | The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5278 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.) |
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Theorem | fun11uni 5287* | The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.) |
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Theorem | funin 5288 | The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
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Theorem | funres11 5289 | The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.) |
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Theorem | funcnvres 5290 | The converse of a restricted function. (Contributed by NM, 27-Mar-1998.) |
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Theorem | cnvresid 5291 | Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.) |
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Theorem | funcnvres2 5292 | The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.) |
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Theorem | funimacnv 5293 | The image of the preimage of a function. (Contributed by NM, 25-May-2004.) |
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Theorem | funimass1 5294 | A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.) |
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Theorem | funimass2 5295 | A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.) |
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Theorem | imadiflem 5296 |
One direction of imadif 5297. This direction does not require
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Theorem | imadif 5297 | The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.) |
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Theorem | imainlem 5298 |
One direction of imain 5299. This direction does not require
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Theorem | imain 5299 | The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.) |
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Theorem | funimaexglem 5300 |
Lemma for funimaexg 5301. It constitutes the interesting part of
funimaexg 5301, in which ![]() ![]() ![]() ![]() |
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