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Theorem List for Intuitionistic Logic Explorer - 5601-5700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremressnop0 5601 If is not in , then the restriction of a singleton of to is null. (Contributed by Scott Fenton, 15-Apr-2011.)

Theoremfpr 5602 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremfprg 5603 A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)

Theoremftpg 5604 A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremftp 5605 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 5606 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Theoremfressnfv 5607 The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Theoremfvconst 5608 The value of a constant function. (Contributed by NM, 30-May-1999.)

Theoremfmptsn 5609* 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 5610* Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfmptapd 5611* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremfmptpr 5612* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremfvresi 5613 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)

Theoremfvunsng 5614 Remove an ordered pair not participating in a function value. (Contributed by Jim Kingdon, 7-Jan-2019.)

Theoremfvsn 5615 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)

Theoremfvsng 5616 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)

Theoremfvsnun1 5617 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5618. (Contributed by NM, 23-Sep-2007.)

Theoremfvsnun2 5618 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5617. (Contributed by NM, 23-Sep-2007.)

Theoremfnsnsplitss 5619 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 20-Jan-2023.)

Theoremfsnunf 5620 Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfsnunfv 5621 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)

Theoremfsnunres 5622 Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfunresdfunsnss 5623 Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.)

Theoremfvpr1 5624 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfvpr2 5625 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfvpr1g 5626 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)

Theoremfvpr2g 5627 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)

Theoremfvtp1g 5628 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremfvtp2g 5629 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremfvtp3g 5630 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremfvtp1 5631 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvtp2 5632 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvtp3 5633 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvconst2g 5634 The value of a constant function. (Contributed by NM, 20-Aug-2005.)

Theoremfconst2g 5635 A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)

Theoremfvconst2 5636 The value of a constant function. (Contributed by NM, 16-Apr-2005.)

Theoremfconst2 5637 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)

Theoremfconstfvm 5638* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5637. (Contributed by Jim Kingdon, 8-Jan-2019.)

Theoremfconst3m 5639* Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)

Theoremfconst4m 5640* Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)

Theoremresfunexg 5641 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 5642 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 5641. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfunex 5643 If the domain of a function exists, so does 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 5642. (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 5644* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)

Theoremmptexg 5645* 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 5646* 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 5647 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)

Theoremeufnfv 5648* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)

Theoremfunfvima 5649 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)

Theoremfunfvima2 5650 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)

Theoremfunfvima3 5651 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 5652 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.)

Theoremfoima2 5653* Given an onto function, an element is in its codomain if and only if it is the image of an element of its domain (see foima 5350). (Contributed by BJ, 6-Jul-2022.)

Theoremfoelrn 5654* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof shortened by BJ, 6-Jul-2022.)

Theoremfoco2 5655 If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrexima 5656* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremralima 5657* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremidref 5658* TODO: This is the same as issref 4921 (which has a much longer proof). Should we replace issref 4921 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 5659* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)

Theoremabrexco 5660* Composition of two image maps and . (Contributed by NM, 27-May-2013.)

Theoremimaiun 5661* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremimauni 5662* 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.)

Theoremfniunfv 5663* The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)

Theoremfuniunfvdm 5664* The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5663. (Contributed by Jim Kingdon, 10-Jan-2019.)

Theoremfuniunfvdmf 5665* The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5664 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)

Theoremeluniimadm 5666* Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)

Theoremelunirn 5667* Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)

Theoremfnunirn 5668* Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremdff13 5669* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)

Theoremf1veqaeq 5670 If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)

Theoremdff13f 5671* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)

Theoremf1mpt 5672* Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremf1fveq 5673 Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)

Theoremf1elima 5674 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremf1imass 5675 Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imaeq 5676 Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremdff1o6 5677* A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)

Theoremf1ocnvfv1 5678 The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv2 5679 The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv 5680 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremf1ocnvfvb 5681 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvdm 5682 The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)

Theoremf1ocnvfvrneq 5683 If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)

Theoremfcof1 5684 An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremfcofo 5685 An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremcbvfo 5686* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremcbvexfo 5687* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)

Theoremcocan1 5688 An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theoremcocan2 5689 A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremfcof1o 5690 Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremfoeqcnvco 5691 Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremf1eqcocnv 5692 Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremfliftrel 5693* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel 5694* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel1 5695* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftcnv 5696* Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfun 5697* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfund 5698* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfuns 5699* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftf 5700* The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

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