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

Theoremffvelrni 5301 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)

Theoremffvelrnda 5302 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremffvelrnd 5303 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremrexrn 5304* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremralrn 5305* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremelrnrexdm 5306* 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 5307* 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 5308* 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 5309* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexrnmpt 5310* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremdff2 5311 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)

Theoremdff3im 5312* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)

Theoremdff4im 5313* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)

Theoremdffo3 5314* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)

Theoremdffo4 5315* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdffo5 5316* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremfoelrn 5317* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)

Theoremfoco2 5318 If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremfmpt 5319* Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremf1ompt 5320* Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremfmpti 5321* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremfmptd 5322* Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremffnfv 5323* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)

Theoremffnfvf 5324 A function maps to a class to which all values belong. This version of ffnfv 5323 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)

Theoremfnfvrnss 5325* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)

Theoremrnmptss 5326* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Theoremfmpt2d 5327* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)

Theoremffvresb 5328* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremf1oresrab 5329* Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)

Theoremfmptco 5330* 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 5331* Version of fmptco 5330 where needn't be distinct from . (Contributed by NM, 27-Dec-2014.)

Theoremfmptcos 5332* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfcompt 5333* 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 5334 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)

Theoremfsn 5335 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)

Theoremfsng 5336 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)

Theoremfsn2 5337 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)

Theoremxpsng 5338 The cross product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremxpsn 5339 The cross product of two singletons. (Contributed by NM, 4-Nov-2006.)

Theoremdfmpt 5340 Alternate definition for the "maps to" notation df-mpt 3820 (although it requires that be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)

Theoremfnasrn 5341 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 5342 Alternate definition for the "maps to" notation df-mpt 3820 (which requires that be a set). (Contributed by Jim Kingdon, 9-Jan-2019.)

Theoremfnasrng 5343 A function expressed as the range of another function. (Contributed by Jim Kingdon, 9-Jan-2019.)

Theoremressnop0 5344 If is not in , then the restriction of a singleton of to is null. (Contributed by Scott Fenton, 15-Apr-2011.)

Theoremfpr 5345 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremfprg 5346 A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)

Theoremftpg 5347 A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremftp 5348 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 5349 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Theoremfressnfv 5350 The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Theoremfvconst 5351 The value of a constant function. (Contributed by NM, 30-May-1999.)

Theoremfmptsn 5352* 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 5353* Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfmptapd 5354* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremfmptpr 5355* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremfvresi 5356 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)

Theoremfvunsng 5357 Remove an ordered pair not participating in a function value. (Contributed by Jim Kingdon, 7-Jan-2019.)

Theoremfvsn 5358 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)

Theoremfvsng 5359 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)

Theoremfvsnun1 5360 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5361. (Contributed by NM, 23-Sep-2007.)

Theoremfvsnun2 5361 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5360. (Contributed by NM, 23-Sep-2007.)

Theoremfsnunf 5362 Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfsnunfv 5363 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)

Theoremfsnunres 5364 Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfvpr1 5365 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfvpr2 5366 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfvpr1g 5367 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)

Theoremfvpr2g 5368 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)

Theoremfvtp1g 5369 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremfvtp2g 5370 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremfvtp3g 5371 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremfvtp1 5372 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvtp2 5373 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvtp3 5374 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvconst2g 5375 The value of a constant function. (Contributed by NM, 20-Aug-2005.)

Theoremfconst2g 5376 A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)

Theoremfvconst2 5377 The value of a constant function. (Contributed by NM, 16-Apr-2005.)

Theoremfconst2 5378 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)

Theoremfconstfvm 5379* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5378. (Contributed by Jim Kingdon, 8-Jan-2019.)

Theoremfconst3m 5380* Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)

Theoremfconst4m 5381* Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)

Theoremresfunexg 5382 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)

Theoremfnex 5383 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5382. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfunex 5384 If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5383. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)

Theoremopabex 5385* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)

Theoremmptexg 5386* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmptex 5387* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theoremfex 5388 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)

Theoremeufnfv 5389* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)

Theoremfunfvima 5390 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)

Theoremfunfvima2 5391 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)

Theoremfunfvima3 5392 A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)

Theoremfnfvima 5393 The function value of an operand in a set is contained in the image of that set, using the abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)

Theoremrexima 5394* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremralima 5395* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremidref 5396* TODO: This is the same as issref 4707 (which has a much longer proof). Should we replace issref 4707 with this one? - NM 9-May-2016.

Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)

Theoremelabrex 5397* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)

Theoremabrexco 5398* Composition of two image maps and . (Contributed by NM, 27-May-2013.)

Theoremimaiun 5399* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremimauni 5400* The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

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