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

Theoremfvmpt3i 5801* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptd 5802* Deduction version of fvmpt 5798. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpt2i 5803* Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Theoremfvmpt2 5804* Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)

Theoremfvmptss 5805* If all the values of the mapping are subsets of a class , then so is any evaluation of the mapping, even if is not in the base set . (Contributed by Mario Carneiro, 13-Feb-2015.)

Theoremfvmpt2d 5806* Deduction version of fvmpt2 5804. (Contributed by Thierry Arnoux, 8-Dec-2016.)

Theoremfvmptex 5807* Express a function whose value may not always be a set in terms of another function for which sethood is guaranteed. (Note that is just shorthand for , and it is always a set by fvex 5734.) Note also that these functions are not the same; wherever is not a set, is not in the domain of (so it evaluates to the empty set), but is in the domain of , and is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremfvmptdf 5808* Alternate deduction version of fvmpt 5798, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv 5809* Alternate deduction version of fvmpt 5798, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv2 5810* Alternate deduction version of fvmpt 5798, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremmpteqb 5811* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5819. (Contributed by Mario Carneiro, 14-Nov-2014.)

Theoremfvmptt 5812* Closed theorem form of fvmpt 5798. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptf 5813* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5796 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremfvmptnf 5814* The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 5815 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptn 5815* This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvmptg 5796. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)

Theoremfvmptss2 5816* A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.)

Theoremfvopab4ndm 5817* Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)

Theoremfvopab6 5818* Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremeqfnfv 5819* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremeqfnfv2 5820* Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremeqfnfv3 5821* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremeqfnfvd 5822* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremeqfnfv2f 5823* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5819 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)

Theoremeqfunfv 5824* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)

Theoremfvreseq 5825* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)

Theoremfndmdif 5826* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmdifcom 5827 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmdifeq0 5828 The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmin 5829* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfneqeql 5830 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremfneqeql2 5831 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremfnreseql 5832 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremchfnrn 5833* 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 5834 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)

Theoremfunfvbrb 5835 Two ways to say that is in the domain of . (Contributed by Mario Carneiro, 1-May-2014.)

Theoremfvimacnvi 5836 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)

Theoremfvimacnv 5837 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 5519 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass3 5838 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 5837 would be the special case of being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass5 5839* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)

Theoremfunconstss 5840* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)

TheoremfvimacnvALT 5841 Another proof of fvimacnv 5837, based on funimass3 5838. If funimass3 5838 is ever proved directly, as opposed to using funimacnv 5517 pointwise, then the proof of funimacnv 5517 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremelpreima 5842 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfniniseg 5843 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 5844* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfniniseg2 5845* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfnniniseg2 5846* Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremrexsupp 5847* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)

Theoremunpreima 5848 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoreminpreima 5849 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)

Theoremdifpreima 5850 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)

Theoremrespreima 5851 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiinpreima 5852* Preimage of an intersection. (Contributed by FL, 16-Apr-2012.)

Theoremintpreima 5853* Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)

Theoremfimacnv 5854 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)

Theoremsuppss 5855* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.)

Theoremsuppssr 5856 A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.)

Theoremfnopfv 5857 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)

Theoremfvelrn 5858 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)

Theoremfnfvelrn 5859 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)

Theoremffvelrn 5860 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)

Theoremffvelrni 5861 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)

Theoremffvelrnda 5862 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremffvelrnd 5863 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremrexrn 5864* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremralrn 5865* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremelrnrexdm 5866* 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 5867* 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 5868* 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.)

Theoremeldmrexrnb 5869* 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. Because of the special definition of a function value, the theorem is only valid in general if the empty set is not contained in the range of the function. Otherwise, it cannot be distiguished between the empty set as a valid function value, or as an indication that the function is not defined. (Contributed by Alexander van der Vekens, 17-Dec-2017.)

Theoremralrnmpt 5870* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexrnmpt 5871* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremf0cli 5872 Unconditional closure of a function when the range includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.)

Theoremdff2 5873 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)

Theoremdff3 5874* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdff4 5875* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdffo3 5876* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)

Theoremdffo4 5877* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdffo5 5878* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremexfo 5879* A relation equivalent to the existence of an onto mapping. The right-hand is not necessarily a function. (Contributed by NM, 20-Mar-2007.)

Theoremfoelrn 5880* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)

Theoremfoco2 5881 If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremfmpt 5882* Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremf1ompt 5883* Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremfmpti 5884* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremfmptd 5885* Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremffnfv 5886* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)

Theoremffnfvf 5887 A function maps to a class to which all values belong. This version of ffnfv 5886 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)

Theoremfnfvrnss 5888* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)

Theoremrnmptss 5889* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Theoremfmpt2d 5890* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)

Theoremfmpt2dOLD 5891* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 9-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremffvresb 5892* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremfmptco 5893* 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 5894* Version of fmptco 5893 where needn't be distinct from . (Contributed by NM, 27-Dec-2014.)

Theoremfmptcos 5895* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfcompt 5896* 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 5897 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)

Theoremfsn 5898 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)

Theoremfsng 5899 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)

Theoremfsn2 5900 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)

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