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Theorem List for Metamath Proof Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfnsnsplit 5801 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremfsnunf 5802 Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfsnunf2 5803 Adjoining a point to a punctured function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfsnunfv 5804 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)

Theoremfsnunres 5805 Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfvpr1 5806 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfvpr2 5807 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfvtp1 5808 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvtp2 5809 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvtp3 5810 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvconst2g 5811 The value of a constant function. (Contributed by NM, 20-Aug-2005.)

Theoremfconst2g 5812 A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)

Theoremfvconst2 5813 The value of a constant function. (Contributed by NM, 16-Apr-2005.)

Theoremfconst2 5814 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)

Theoremfconst5 5815 Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.)

Theoremfnpr 5816 Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) (Revised by NM, 10-Dec-2017.)

TheoremfnprOLD 5817 Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) Obsolete version of fnpr 5816 as of 10-Dec-2017. (New usage is discouraged.)

Theoremfnsuppres 5818 Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.)

Theoremfnsuppeq0 5819 The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.)

Theoremfconstfv 5820* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5814. (Contributed by NM, 27-Aug-2004.)

Theoremfconst3 5821 Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)

Theoremfconst4 5822 Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)

Theoremresfunexg 5823 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.)

TheoremresfunexgALT 5824 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5823 but requires ax-pow 4269. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcofunexg 5825 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)

Theoremcofunex2g 5826 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)

Theoremfnex 5827 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 5823. See fnexALT 5828 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

TheoremfnexALT 5828 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 funimaexg 5411. This version of fnex 5827 uses ax-pow 4269, whereas fnex 5827 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremfunex 5829 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 5827. (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 5830* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)

Theoremmptexg 5831* 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 5832* 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.)

Theoremfunrnex 5833 If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 5829. (Contributed by NM, 11-Nov-1995.)

Theoremzfrep6 5834* A version of the Axiom of Replacement. Normally would have free variables and . Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4222 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 4212. (Contributed by NM, 10-Oct-2003.)

Theoremfex 5835 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)

Theoremfornex 5836 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)

Theoremf1dmex 5837 If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 4212. (Contributed by NM, 4-Sep-2004.)

Theoremeufnfv 5838* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)

Theoremfunfvima 5839 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)

Theoremfunfvima2 5840 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)

Theoremfunfvima3 5841 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 5842 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 5843* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremralima 5844* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremidref 5845* TODO: This is the same as issref 5138 (which has a much longer proof). Should we replace issref 5138 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.)

Theoremfvclss 5846* Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)

Theoremfvclex 5847* Existence of the class of values of a set. (Contributed by NM, 9-Nov-1995.)

Theoremfvresex 5848* Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremabrexex 5849* Existence of a class abstraction of existentially restricted sets. is normally a free-variable parameter in the class expression substituted for , which can be thought of as . This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5831, funex 5829, fnex 5827, resfunexg 5823, and funimaexg 5411. See also abrexex2 5867. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremabrexexg 5850* Existence of a class abstraction of existentially restricted sets. is normally a free-variable parameter in . The antecedent assures us that is a set. (Contributed by NM, 3-Nov-2003.)

Theoremelabrex 5851* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)

Theoremabrexco 5852* Composition of two image maps and . (Contributed by NM, 27-May-2013.)

Theoremiunexg 5853* The existence of an indexed union. is normally a free-variable parameter in . (Contributed by NM, 23-Mar-2006.)

Theoremabrexex2g 5854* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremopabex3d 5855* Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)

Theoremopabex3 5856* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiunex 5857* The existence of an indexed union. is normally a free-variable parameter in the class expression substituted for , which can be read informally as . (Contributed by NM, 13-Oct-2003.)

Theoremimaiun 5858* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremimauni 5859* 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 5860* 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.)

Theoremfuniunfv 5861* The indexed union of a function's values is the union of its image under the index class.

Note: This theorem depends on the fact that our function value is the empty set outside of its domain. If the antecedent is changed to , the theorem can be proved without this dependency. (Contributed by NM, 26-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfuniunfvf 5862* The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 5861 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) (Revised by David Abernethy, 15-Apr-2013.)

Theoremeluniima 5863* Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.)

Theoremelunirn 5864* Membership in the union of the range of a function. See elunirnALT 5866 for alternate proof. (Contributed by NM, 24-Sep-2006.)

Theoremfnunirn 5865* Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)

TheoremelunirnALT 5866* Membership in the union of the range of a function, proved directly. Unlike elunirn 5864, it doesn't appeal to ndmfv 5635 (via funiunfv 5861). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremabrexex2 5867* Existence of an existentially restricted class abstraction. is normally has free-variable parameters and . See also abrexex 5849. (Contributed by NM, 12-Sep-2004.)

Theoremabexssex 5868* Existence of a class abstraction with an existentially quantified expression. Both and can be free in . (Contributed by NM, 29-Jul-2006.)

Theoremabexex 5869* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)

Theoremdff13 5870* 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.)

Theoremdff13f 5871* 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 5872* Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremf1fveq 5873 Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)

Theoremf1elima 5874 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremf1imass 5875 Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imaeq 5876 Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imapss 5877 Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremdff1o6 5878* A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)

Theoremf1ocnvfv1 5879 The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv2 5880 The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv 5881 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 5882 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvdm 5883 The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)

Theoremfcof1 5884 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 5885 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 5886* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremcbvexfo 5887* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)

Theoremcocan1 5888 An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theoremcocan2 5889 A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremfcof1o 5890 Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremfoeqcnvco 5891 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 5892 Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremfveqf1o 5893 Given a bijection , produce another bijection which additionally maps two specified points. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremfliftrel 5894* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel 5895* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel1 5896* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftcnv 5897* Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfun 5898* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfund 5899* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfuns 5900* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

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