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Theorem List for Metamath Proof Explorer - 5501-5600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfvmpt 5501* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)

Theoremfvmpts 5502* Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpt3 5503* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremfvmpt3i 5504* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptd 5505* Deduction version of fvmpt 5501. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpt2i 5506* Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Theoremfvmpt2 5507* Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)

Theoremfvmptss 5508* 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.)

Theoremfvmptex 5509* 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 5437.) 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 5510* Alternate deduction version of fvmpt 5501, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv 5511* Alternate deduction version of fvmpt 5501, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv2 5512* Alternate deduction version of fvmpt 5501, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremmpteqb 5513* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5521. (Contributed by Mario Carneiro, 14-Nov-2014.)

Theoremfvmptt 5514* Closed theorem form of fvmpt 5501. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptf 5515* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5499 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremfvmptnf 5516* 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 5517 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptn 5517* 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 5499. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)

Theoremfvmptss2 5518* 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 5519* Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)

Theoremfvopab6 5520* 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 5521* 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 5522* 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 5523* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremeqfnfvd 5524* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremeqfnfv2f 5525* 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 5521 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)

Theoremeqfunfv 5526* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)

Theoremfvreseq 5527* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)

Theoremfndmdif 5528* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmdifcom 5529 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmdifeq0 5530 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 5531* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfneqeql 5532 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremfneqeql2 5533 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremfnreseql 5534 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremchfnrn 5535* 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 5536 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)

Theoremfunfvbrb 5537 Two ways to say that is in the domain of . (Contributed by Mario Carneiro, 1-May-2014.)

Theoremfvimacnvi 5538 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)

Theoremfvimacnv 5539 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 5229 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass3 5540 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 5539 would be the special case of being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass5 5541* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)

Theoremfunconstss 5542* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)

TheoremfvimacnvALT 5543 Another proof of fvimacnv 5539, based on funimass3 5540. If funimass3 5540 is ever proved directly, as opposed to using funimacnv 5227 pointwise, then the proof of funimacnv 5227 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.)

Theoremelpreima 5544 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfniniseg 5545 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 5546* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfniniseg2 5547* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfnniniseg2 5548* Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremrexsupp 5549* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)

Theoremunpreima 5550 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoreminpreima 5551 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)

Theoremdifpreima 5552 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)

Theoremrespreima 5553 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiinpreima 5554* Preimage of an intersection. (Contributed by FL, 16-Apr-2012.)

Theoremintpreima 5555* Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)

Theoremfimacnv 5556 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)

Theoremsuppss 5557* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.)

Theoremsuppssr 5558 A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.)

Theoremfnopfv 5559 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)

Theoremfvelrn 5560 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)

Theoremfnfvelrn 5561 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)

Theoremffvelrn 5562 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)

Theoremffvelrni 5563 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)

Theoremffvelrnda 5564 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremffvelrnd 5565 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremrexrn 5566* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremralrn 5567* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremralrnmpt 5568* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexrnmpt 5569* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremf0cli 5570 Unconditional closure of a function when the range includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.)

Theoremdff2 5571 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)

Theoremdff3 5572* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdff4 5573* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdffo3 5574* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)

Theoremdffo4 5575* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdffo5 5576* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremexfo 5577* 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 5578* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)

Theoremfoco2 5579 If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremfmpt 5580* Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremf1ompt 5581* Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremfmpti 5582* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremfmptd 5583* Domain and co-domain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremffnfv 5584* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)

Theoremffnfvf 5585 A function maps to a class to which all values belong. This version of ffnfv 5584 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)

Theoremfnfvrnss 5586* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)

Theoremfmpt2d 5587* Domain and co-domain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)

Theoremfmpt2dOLD 5588* Domain and co-domain of the mapping operation; deduction form. (Contributed by NM, 9-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremffvresb 5589* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremfmptco 5590* 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 5591* Version of fmptco 5590 where needn't be distinct from . (Contributed by NM, 27-Dec-2014.)

Theoremfmptcos 5592* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfcompt 5593* 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 5594 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)

Theoremfsn 5595 A function maps a singleton to a singleton iff it is the singleton of a ordered pair. (Contributed by NM, 10-Dec-2003.)

Theoremfsng 5596 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)

Theoremfsn2 5597 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)

Theoremxpsng 5598 The cross product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremxpsn 5599 The cross product of two singletons. (Contributed by NM, 4-Nov-2006.)

Theoremdfmpt 5600 Alternate definition for the "maps to" notation df-mpt 4019 (although it requires that be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)

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