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Theorem List for Intuitionistic Logic Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfndmdifcom 5301 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmin 5302* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfneqeql 5303 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremfneqeql2 5304 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremfnreseql 5305 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremchfnrn 5306* The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)

Theoremfunfvop 5307 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)

Theoremfunfvbrb 5308 Two ways to say that is in the domain of . (Contributed by Mario Carneiro, 1-May-2014.)

Theoremfvimacnvi 5309 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)

Theoremfvimacnv 5310 The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 5005 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass3 5311 A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 5310 would be the special case of being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass5 5312* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)

Theoremfunconstss 5313* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)

Theoremelpreima 5314 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfniniseg 5315 Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfncnvima2 5316* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfniniseg2 5317* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfnniniseg2 5318* Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremrexsupp 5319* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)

Theoremunpreima 5320 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoreminpreima 5321 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)

Theoremdifpreima 5322 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)

Theoremrespreima 5323 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfimacnv 5324 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)

Theoremfnopfv 5325 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)

Theoremfvelrn 5326 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)

Theoremfnfvelrn 5327 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)

Theoremffvelrn 5328 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)

Theoremffvelrni 5329 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)

Theoremffvelrnda 5330 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremffvelrnd 5331 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremrexrn 5332* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremralrn 5333* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremelrnrexdm 5334* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Theoremelrnrexdmb 5335* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.)

Theoremeldmrexrn 5336* For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Theoremralrnmpt 5337* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexrnmpt 5338* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremdff2 5339 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)

Theoremdff3im 5340* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)

Theoremdff4im 5341* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)

Theoremdffo3 5342* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)

Theoremdffo4 5343* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdffo5 5344* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremfoelrn 5345* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)

Theoremfoco2 5346 If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremfmpt 5347* Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremf1ompt 5348* Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremfmpti 5349* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremfmptd 5350* Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremffnfv 5351* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)

Theoremffnfvf 5352 A function maps to a class to which all values belong. This version of ffnfv 5351 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)

Theoremfnfvrnss 5353* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)

Theoremrnmptss 5354* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Theoremfmpt2d 5355* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)

Theoremffvresb 5356* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremf1oresrab 5357* Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)

Theoremfmptco 5358* Composition of two functions expressed as ordered-pair class abstractions. If has the equation ( x + 2 ) and the equation ( 3 * z ) then has the equation ( 3 * ( x + 2 ) ) . (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)

Theoremfmptcof 5359* Version of fmptco 5358 where needn't be distinct from . (Contributed by NM, 27-Dec-2014.)

Theoremfmptcos 5360* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfcompt 5361* Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremfcoconst 5362 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)

Theoremfsn 5363 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)

Theoremfsng 5364 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)

Theoremfsn2 5365 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)

Theoremxpsng 5366 The cross product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremxpsn 5367 The cross product of two singletons. (Contributed by NM, 4-Nov-2006.)

Theoremdfmpt 5368 Alternate definition for the "maps to" notation df-mpt 3848 (although it requires that be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)

Theoremfnasrn 5369 A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremdfmptg 5370 Alternate definition for the "maps to" notation df-mpt 3848 (which requires that be a set). (Contributed by Jim Kingdon, 9-Jan-2019.)

Theoremfnasrng 5371 A function expressed as the range of another function. (Contributed by Jim Kingdon, 9-Jan-2019.)

Theoremressnop0 5372 If is not in , then the restriction of a singleton of to is null. (Contributed by Scott Fenton, 15-Apr-2011.)

Theoremfpr 5373 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremfprg 5374 A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)

Theoremftpg 5375 A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremftp 5376 A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by Alexander van der Vekens, 23-Jan-2018.)

Theoremfnressn 5377 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Theoremfressnfv 5378 The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Theoremfvconst 5379 The value of a constant function. (Contributed by NM, 30-May-1999.)

Theoremfmptsn 5380* Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 28-Feb-2015.)

Theoremfmptap 5381* Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfmptapd 5382* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremfmptpr 5383* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremfvresi 5384 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)

Theoremfvunsng 5385 Remove an ordered pair not participating in a function value. (Contributed by Jim Kingdon, 7-Jan-2019.)

Theoremfvsn 5386 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)

Theoremfvsng 5387 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)

Theoremfvsnun1 5388 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5389. (Contributed by NM, 23-Sep-2007.)

Theoremfvsnun2 5389 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5388. (Contributed by NM, 23-Sep-2007.)

Theoremfsnunf 5390 Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfsnunfv 5391 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)

Theoremfsnunres 5392 Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfvpr1 5393 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfvpr2 5394 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfvpr1g 5395 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)

Theoremfvpr2g 5396 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)

Theoremfvtp1g 5397 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremfvtp2g 5398 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremfvtp3g 5399 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremfvtp1 5400 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

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