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Theorem List for Metamath Proof Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremffnfv 5701* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)

Theoremffnfvf 5702 A function maps to a class to which all values belong. This version of ffnfv 5701 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)

Theoremfnfvrnss 5703* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)

Theoremfmpt2d 5704* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)

Theoremfmpt2dOLD 5705* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 9-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremffvresb 5706* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremfmptco 5707* Composition of two functions expressed as ordered-pair class abstractions. If has the equation and the equation then has the equation . (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)

Theoremfmptcof 5708* Version of fmptco 5707 where needn't be distinct from . (Contributed by NM, 27-Dec-2014.)

Theoremfmptcos 5709* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfcompt 5710* 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 5711 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)

Theoremfsn 5712 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)

Theoremfsng 5713 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)

Theoremfsn2 5714 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)

Theoremxpsng 5715 The cross product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremxpsn 5716 The cross product of two singletons. (Contributed by NM, 4-Nov-2006.)

Theoremdfmpt 5717 Alternate definition for the "maps to" notation df-mpt 4095 (although it requires that be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)

Theoremfnasrn 5718 A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremressnop0 5719 If is not in , then the restriction of a singleton of to is null. (Contributed by Scott Fenton, 15-Apr-2011.)

Theoremfpr 5720 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremfnressn 5721 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Theoremfunressn 5722 A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremfressnfv 5723 The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Theoremfvconst 5724 The value of a constant function. (Contributed by NM, 30-May-1999.)

Theoremfmptsn 5725* 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 5726* Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvresi 5727 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)

Theoremfvunsn 5728 Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.)

Theoremfvsn 5729 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)

Theoremfvsng 5730 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)

Theoremfvsnun1 5731 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5732. (Contributed by NM, 23-Sep-2007.)

Theoremfvsnun2 5732 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5731. (Contributed by NM, 23-Sep-2007.)

Theoremfnsnsplit 5733 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremfsnunf 5734 Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfsnunf2 5735 Adjoining a point to a punctured function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfsnunfv 5736 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)

Theoremfsnunres 5737 Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfvpr1 5738 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfvpr2 5739 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfvtp1 5740 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvtp2 5741 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvtp3 5742 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvconst2g 5743 The value of a constant function. (Contributed by NM, 20-Aug-2005.)

Theoremfconst2g 5744 A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)

Theoremfvconst2 5745 The value of a constant function. (Contributed by NM, 16-Apr-2005.)

Theoremfconst2 5746 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)

Theoremfconst5 5747 Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.)

Theoremfnsuppres 5748 Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.)

Theoremfnsuppeq0 5749 The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.)

Theoremfconstfv 5750* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5746. (Contributed by NM, 27-Aug-2004.)

Theoremfconst3 5751 Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)

Theoremfconst4 5752 Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)

Theoremresfunexg 5753 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 5754 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 5753 but requires ax-pow 4204. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcofunexg 5755 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)

Theoremcofunex2g 5756 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)

Theoremfnex 5757 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 5753. See fnexALT 5758 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

TheoremfnexALT 5758 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 5345. This version of fnex 5757 uses ax-pow 4204, whereas fnex 5757 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremfunex 5759 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 5757. (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 5760* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)

Theoremmptexg 5761* 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 5762* 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 5763 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 5759. (Contributed by NM, 11-Nov-1995.)

Theoremzfrep6 5764* A version of the Axiom of Replacement. Normally would have free variables and . Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4157 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 4147. (Contributed by NM, 10-Oct-2003.)

Theoremfex 5765 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)

Theoremfornex 5766 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)

Theoremf1dmex 5767 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 4147. (Contributed by NM, 4-Sep-2004.)

Theoremeufnfv 5768* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)

Theoremfunfvima 5769 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)

Theoremfunfvima2 5770 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)

Theoremfunfvima3 5771 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 5772 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 5773* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremralima 5774* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremidref 5775* TODO: This is the same as issref 5072 (which has a much longer proof). Should we replace issref 5072 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 5776* Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)

Theoremfvclex 5777* Existence of the class of values of a set. (Contributed by NM, 9-Nov-1995.)

Theoremfvresex 5778* Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremabrexex 5779* 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 5761, funex 5759, fnex 5757, resfunexg 5753, and funimaexg 5345. See also abrexex2 5796. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremabrexexg 5780* 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 5781* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)

Theoremabrexco 5782* Composition of two image maps and . (Contributed by NM, 27-May-2013.)

Theoremiunexg 5783* The existence of an indexed union. is normally a free-variable parameter in . (Contributed by NM, 23-Mar-2006.)

Theoremabrexex2g 5784* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremopabex3 5785* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiunex 5786* 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 5787* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremimauni 5788* 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 5789* 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 5790* 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 5791* The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 5790 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 5792* Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.)

Theoremelunirn 5793* Membership in the union of the range of a function. See elunirnALT 5795 for alternate proof. (Contributed by NM, 24-Sep-2006.)

Theoremfnunirn 5794* Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)

TheoremelunirnALT 5795* Membership in the union of the range of a function, proved directly. Unlike elunirn 5793, it doesn't appeal to ndmfv 5568 (via funiunfv 5790). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremabrexex2 5796* Existence of an existentially restricted class abstraction. is normally has free-variable parameters and . See also abrexex 5779. (Contributed by NM, 12-Sep-2004.)

Theoremabexssex 5797* Existence of a class abstraction with an existentially quantified expression. Both and can be free in . (Contributed by NM, 29-Jul-2006.)

Theoremabexex 5798* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)

Theoremdff13 5799* 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 5800* 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.)

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