Home Metamath Proof ExplorerTheorem List (p. 279 of 325) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22374) Hilbert Space Explorer (22375-23897) Users' Mathboxes (23898-32447)

Theorem List for Metamath Proof Explorer - 27801-27900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmdandyvrx8 27801 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx9 27802 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx10 27803 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx11 27804 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx12 27805 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx13 27806 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx14 27807 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx15 27808 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

TheoremH15NH16TH15IH16 27809 Given 15 hypotheses and a 16th hypothesis, there exists a proof the 15 imply the 16th. (Contributed by Jarvin Udandy, 8-Sep-2016.)
jph       jps       jch       jth       jph jps jch jth

Theoremdandysum2p2e4 27810

CONTRADICTION PROVED AT 1 + 1 = 2 .

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit. (Contributed by Jarvin Udandy, 6-Sep-2016.)

jph        jps        jch        jph jps jch jph jps jch

Theoremmdandysum2p2e4 27811 CONTRADICTION PROVED AT 1 + 1 = 2 . Luckily Mario Carneiro did a successful version of his own.

See Mario's Relevant Work: 1.3.14 Half-adders and full adders in propositional calculus

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F. ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit.

In mdandysum2p2e4, one might imagine what jth or jta could be then do the math with their truths. Also limited to the restriction jth, jta are having opposite truths equivalent to the stated truth constants.

(Contributed by Jarvin Udandy, 6-Sep-2016.)

jth        jta                             jth       jth       jta       jta       jth       jth       jth       jth              jph        jps        jch        jph jps jch jph jps jch

19.22  Mathbox for Alexander van der Vekens

19.22.1  Double restricted existential uniqueness

19.22.1.1  Restricted quantification (extension)

Theoremr19.32 27812 Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 2814. (Contributed by Alexander van der Vekens, 29-Jun-2017.)

Theoremrexsb 27813* An equivalent expression for restricted existence, analogous to exsb 2180. (Contributed by Alexander van der Vekens, 1-Jul-2017.)

Theoremrexrsb 27814* An equivalent expression for restricted existence, analogous to exsb 2180. (Contributed by Alexander van der Vekens, 1-Jul-2017.)

Theorem2rexsb 27815* An equivalent expression for double restricted existence, analogous to rexsb 27813. (Contributed by Alexander van der Vekens, 1-Jul-2017.)

Theorem2rexrsb 27816* An equivalent expression for double restricted existence, analogous to 2exsb 2182. (Contributed by Alexander van der Vekens, 1-Jul-2017.)

Theoremcbvral2 27817* Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 2900. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Theoremcbvrex2 27818* Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 2901. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Theorem2ralbiim 27819 Split a biconditional and distribute 2 quantifiers, analogous to 2albiim 1619 and ralbiim 2803. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

19.22.1.2  The empty set (extension)

Theoremraaan2 27820* Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 3695. It is necessary that either both restricting classes are empty or both are not empty. (Contributed by Alexander van der Vekens, 29-Jun-2017.)

19.22.1.3  Restricted uniqueness and "at most one" quantification

Theoremrmoimi 27821 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theorem2reu5a 27822 Double restricted existential uniqueness in terms of restricted existence and restricted "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremreuimrmo 27823 Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2303. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Theoremrmoanim 27824* Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2310. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Theoremreuan 27825* Introduction of a conjunct into restricted uniqueness quantifier, analogous to euan 2311. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

19.22.1.4  Analogs to Existential uniqueness (double quantification)

Theorem2reurex 27826* Double restricted quantification with existential uniqueness, analogous to 2euex 2326. (Contributed by Alexander van der Vekens, 24-Jun-2017.)

Theorem2reurmo 27827* Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2327. (Contributed by Alexander van der Vekens, 24-Jun-2017.)

Theorem2reu2rex 27828* Double restricted existential uniqueness, analogous to 2eu2ex 2328. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Theorem2rmoswap 27829* A condition allowing swap of restricted "at most one" and restricted existential quantifiers, analogous to 2moswap 2329. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Theorem2rexreu 27830* Double restricted existential uniqueness implies double restricted uniqueness quantification, analogous to 2exeu 2331. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Theorem2reu1 27831* Double restricted existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one, analogous to 2eu1 2334. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Theorem2reu2 27832* Double restricted existential uniqueness, analogous to 2eu2 2335. (Contributed by Alexander van der Vekens, 29-Jun-2017.)

Theorem2reu3 27833* Double restricted existential uniqueness, analogous to 2eu3 2336. (Contributed by Alexander van der Vekens, 29-Jun-2017.)

Theorem2reu4a 27834* Definition of double restricted existential uniqueness ("exactly one and exactly one "), analogous to 2eu4 2337 with the additional requirement that the restricting classes are not empty (which is not necessary as shown in 2reu4 27835). (Contributed by Alexander van der Vekens, 1-Jul-2017.)

Theorem2reu4 27835* Definition of double restricted existential uniqueness ("exactly one and exactly one "), analogous to 2eu4 2337. (Contributed by Alexander van der Vekens, 1-Jul-2017.)

Theorem2reu7 27836* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2340. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Theorem2reu8 27837* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2341. Curiously, we can put on either of the internal conjuncts but not both. We can also commute using 2reu7 27836. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

19.22.2  Alternative definitions of function's and operation's values

The current definition of the value of a function for an argument (see df-fv 5421) assures that this value is always a set, see fex 5928. This is because this definition can be applied to any classes and , and evaluates to the empty set when it is not meaningful (as shown by ndmfv 5714 and fvprc 5681).

Although it is very convenient for many theorems on functions and their proofs, there are some cases in which from alone it cannot be decided/derived if is meaningful ( is actually a function which is defined for and really has the function value ) or not. Therefore, additional assumptions are required, such as , or (see, for example, ndmfvrcl 5715).

To avoid such an ambiguity, an alternative definition ''' ( see df-afv 27842) would be possible which evaluates to the universal class (''' ) if it is not meaningful (see afvnfundmuv 27870, ndmafv 27871, afvprc 27875 and nfunsnafv 27873), and which corresponds to the current definition ( ''') if it is (see afvfundmfveq 27869). That means ''' (see afvpcfv0 27877), but ''' is not generally valid.

In the theory of partial functions, it is a common case that is not defined at , which also would result in ''' . In this context we say ''' "is not defined" instead of "is not meaningful".

With this definition the following intuitive equivalence holds: ''' <-> "''' is meaningful/defined".

An interesting question would be if could be replaced by ''' in most of the theorems based on function's values. If we look at the (currently 19) proofs using the definition df-fv 5421 of , we see that analogons for the following 8 theorems can be proven using the alternative definition: fveq1 5686-> afveq1 27865, fveq2 5687-> afveq2 27866, nffv 5694-> nfafv 27867, csbfv12g 5697-> csbafv12g , fvres 5704-> afvres 27903, rlimdm 12300-> rlimdmafv 27908, tz6.12-1 5706-> tz6.12-1-afv 27905, fveu 5679-> afveu 27884.

3 theorems proved by directly using df-fv 5421 are within a mathbox (fvsb 27522) or not used (isumclim3 12498, avril1 21710).

However, the remaining 8 theorems proved by directly using df-fv 5421 are used more or less often:

* fvex 5701: used in about 1750 proofs.

* tz6.12-1 5706: root theorem of many theorems which have not a strict analogon, and which are used many times: fvprc 5681 (used in about 127 proofs), tz6.12i 5710 (used - indirectly via fvbr0 5711 and fvrn0 5712- in 18 proofs, and in fvclss 5939 used in fvclex 5940 used in fvresex 5941, which is not used!), dcomex 8283 (used in 4 proofs), ndmfv 5714 (used in 86 proofs) and nfunsn 5720 (used by dffv2 5755 which is not used).

* fv2 5682: only used by elfv 5685, which is only used by fv3 5703, which is not used.

* dffv3 5683: used by dffv4 5684 (the previous "df-fv"), which now is only used in deprecated (usage discouraged) theorems or within mathboxes (csbfv12gALT 5698, csbfv12gALTVD 28720), by shftval 11844 (itself used in 9 proofs), by dffv5 25677 (mathbox) and by fvco2 5757, which has the analogon afvco2 27907.

* fvopab5 6493: used only by ajval 22316 (not used) and by adjval 23346 ( used - indirectly - in 9 proofs).

* zsum 12467: used (via isum 12468, sum0 12470 and fsumsers 12477) in more than 90 proofs.

* isumshft 12574: used in pserdv2 20299 and (via logtayl 20504) 4 other proofs.

* ovtpos 6453: used in 14 proofs.

As a result of this analysis we can say that the current definition of a function's value is crucial for Metamath and cannot be exchanged easily with an alternative definition. While fv2 5682, dffv3 5683, fvopab5 6493, zsum 12467, isumshft 12574 and ovtpos 6453 are not critical or are, hopefully, also valid for the alternative definition, fvex 5701 and tz6.12-1 5706 (and the theorems based on them) are essential for the current definition of function values.

With the same arguments, an alternatvie definition of operation's values (()) could be meaningful to avoid ambiguities, see df-aov 27843.

Syntaxwdfat 27838 Extend the definition of a wff to include the "defined at" predicate. (Read: (The Function) is defined at (the argument) ). In a previous version, the token "def@" was used. However, since the @ is used (informally) as a replacement for \$ in commented out sections that may be deleted some day. While there is no violation of any standard to use the @ in a token, it could make the search for such commented-out sections slightly more difficult. (See remark of Norman Megill at https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4).
defAt

Syntaxcafv 27839 Extend the definition of a class to include the value of a function. (Read: The value of at , or " of ."). In a previous version, the symbol " ' " was used. However, since the similarity with the symbol used for the current definition of a function's value (see df-fv 5421), which, by the way, was intended to visualize that in many cases and " ' " are exchangeable, makes reading the theorems, especially those which uses both definitions as dfafv2 27863, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 5421 and df-ima 4850. And not three backticks ( three times ) since that would be annoying to escape in a comment. (See remark of Norman Megill and Gerard Lang at https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4).
'''

Syntaxcaov 27840 Extend class notation to include the value of an operation (such as ) for two arguments and . Note that the syntax is simply three class symbols in a row surrounded by special parentheses (exclamation mark with underscore) in contrast to the current definition, see df-ov 6043.
(())

Definitiondf-dfat 27841 Definition of the predicate that determines if some class is defined as function for an argument or, in other words, if the function value for some class for an argument is defined. We say that is defined at if a is a function restricted to the member of its domain. (Contributed by Alexander van der Vekens, 25-May-2017.)
defAt

Definitiondf-afv 27842* Alternative definition of the value of a function, ''', also known as function application. In contrast to (see df-fv 5421 and ndmfv 5714), ''' if F is not defined for A! (Contributed by Alexander van der Vekens, 25-May-2017.)
''' defAt

Definitiondf-aov 27843 Define the value of an operation. In contrast to df-ov 6043, the alternative definition for a function value ( see df-afv 27842) is used. By this, the value of the operation applied to two arguments is the universal class if the operation is not defined for these two arguments. There are still no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation and its arguments and - will be useful for proving meaningful theorems. (Contributed by Alexander van der Vekens, 26-May-2017.)
(()) '''

19.22.2.1  Restricted quantification (extension)

Theoremralbinrald 27844* Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)

19.22.2.2  The universal class (extension)

Theoremnvelim 27845 If a class is the universal class it doesn't belong to any class, generalisation of nvel 4302. (Contributed by Alexander van der Vekens, 26-May-2017.)

19.22.2.3  Introduce the Axiom of Power Sets (extension)

Theoremalneu 27846 If a statement holds for all sets, there is not a unique set for which the statement holds. (Contributed by Alexander van der Vekens, 28-Nov-2017.)

Theoremeu2ndop1stv 27847* If there is a unique second component in an ordered pair contained in a given set, the first component must be a set. (Contributed by Alexander van der Vekens, 29-Nov-2017.)

19.22.2.4  Relations (extension)

Theoremsbcrel 27848 Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Theoremcsbdmg 27849 Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Theoremdmmpt2g 27850* Domain of a class given by the "maps to" notation, closed form of dmmpt2 6380. (Contributed by Alexander van der Vekens, 1-Jun-2017.)

Theoremeldmressn 27851 Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Theoremdmressnsn 27852 The domain of a restriction to a singleton is a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Theoremeldmressnsn 27853 The element of the domain of a restriction to a singleton is the element of the singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

19.22.2.5  Functions (extension)

Theoremsbcfun 27854 Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Theoremfveqvfvv 27855 If a function's value at an argument is the universal class (which can never be the case because of fvex 5701), the function's value at this argument is any set (especially the empty set). In short "If a function's value is a proper class, it is a set", which sounds strange/contradictory, but which is a consequence of that a contradiction implies anything ( see pm2.21i 125). (Contributed by Alexander van der Vekens, 26-May-2017.)

Theoremfunresfunco 27856 Composition of two functions, generalization of funco 5450. (Contributed by Alexander van der Vekens, 25-Jul-2017.)

Theoremfnresfnco 27857 Composition of two functions, similar to fnco 5512. (Contributed by Alexander van der Vekens, 25-Jul-2017.)

Theoremfuncoressn 27858 A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)

Theoremfunressnfv 27859 A restriction to a singleton with a function value is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)

19.22.2.6  Predicate "defined at"

Theoremdfateq12d 27860 Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
defAt defAt

Theoremnfdfat 27861 Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g. for Fun/Rel, dom, C_, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
defAt

Theoremdfdfat2 27862* Alternate definition of the predicate "defined at" not using the predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
defAt

19.22.2.7  Alternative definition of the value of a function

Theoremdfafv2 27863 Alternative definition of ''' using directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
''' defAt

Theoremafveq12d 27864 Equality deduction for function value, analogous to fveq12d 5693. (Contributed by Alexander van der Vekens, 26-May-2017.)
''' '''

Theoremafveq1 27865 Equality theorem for function value, analogous to fveq1 5686. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
''' '''

Theoremafveq2 27866 Equality theorem for function value, analogous to fveq1 5686. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
''' '''

Theoremnfafv 27867 Bound-variable hypothesis builder for function value, analogous to nffv 5694. To prove a deduction version of this analogous to nffvd 5696 is not easily possible because a deduction version of nfdfat 27861 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
'''

Theoremcsbafv12g 27868 Move class substitution in and out of a function value, analogous to csbfv12g 5697, with a direct proof proposed by Mario Carneiro, analogous to csbovg 6071. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
''' '''

Theoremafvfundmfveq 27869 If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017.)
defAt '''

Theoremafvnfundmuv 27870 If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017.)
defAt '''

Theoremndmafv 27871 The value of a class outside its domain is the universe, compare with ndmfv 5714. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafvvdm 27872 If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremnfunsnafv 27873 If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 5720 (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafvvfunressn 27874 If the function value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafvprc 27875 A function's value at a proper class is the universe, compare with fvprc 5681. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafvvv 27876 If a function's value at an argument is a set, the argument is also a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafvpcfv0 27877 If the value of the alternative function at an argument is the universe, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafvnufveq 27878 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
''' '''

Theoremafvvfveq 27879 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
''' '''

Theoremafv0fv0 27880 If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafvfvn0fveq 27881 If the function's value at an argument is not the empty set, it equals the value of the alternative function at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafv0nbfvbi 27882 The function's value at an argument is an element of a set if and only if the value of the alternative function at this argument is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafvfv0bi 27883 The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.)
''' '''

Theoremafveu 27884* The value of a function at a unique point, analogous to fveu 5679. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
'''

Theoremfnbrafvb 27885 Equivalence of function value and binary relation, analogous to fnbrfvb 5726. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremfnopafvb 27886 Equivalence of function value and ordered pair membership, analogous to fnopfvb 5727. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremfunbrafvb 27887 Equivalence of function value and binary relation, analogous to funbrfvb 5728. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremfunopafvb 27888 Equivalence of function value and ordered pair membership, analogous to funopfvb 5729. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremfunbrafv 27889 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 5724. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremfunbrafv2b 27890 Function value in terms of a binary relation, analogous to funbrfv2b 5730. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremdfafn5a 27891* Representation of a function in terms of its values, analogous to dffn5 5731 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremdfafn5b 27892* Representation of a function in terms of its values, analogous to dffn5 5731 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)
''' '''

Theoremfnrnafv 27893* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 5732. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafvelrnb 27894* A member of a function's range is a value of the function, analogous to fvelrnb 5733 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafvelrnb0 27895* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 5733. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
'''

Theoremdfaimafn 27896* Alternate definition of the image of a function, analogous to dfimafn 5734. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremdfaimafn2 27897* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 5735. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafvelima 27898* Function value in an image, analogous to fvelima 5737. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremafvelrn 27899 A function's value belongs to its range, analogous to fvelrn 5825. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Theoremfnafvelrn 27900 A function's value belongs to its range, analogous to fnfvelrn 5826. (Contributed by Alexander van der Vekens, 25-May-2017.)
'''

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32447
 Copyright terms: Public domain < Previous  Next >