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Theorem List for Metamath Proof Explorer - 5401-5500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtz6.12i 5401 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremfvbr0 5402 Two possibilities for the behavior of a function value. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfvrn0 5403 A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.)

Theoremfvssunirn 5404 The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremndmfv 5405 The value of a class outside its domain is the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremndmfvrcl 5406 Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.)

Theoremelfvdm 5407 If a function value has a member, the argument belongs to the domain. (Contributed by NM, 12-Feb-2007.)

Theoremelfvex 5408 If a function value has a member, the argument is a set. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theoremnfvres 5409 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)

Theoremnfunsn 5410 If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremfv01 5411 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)

Theoremfveqres 5412 Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)

Theoremfunbrfv 5413 The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfunopfv 5414 The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)

Theoremfnbrfvb 5415 Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfnopfvb 5416 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)

Theoremfunbrfvb 5417 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)

Theoremfunopfvb 5418 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)

Theoremfunbrfv2b 5419 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremdffn5 5420* Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfnrnfv 5421* The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfvelrnb 5422* A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)

Theoremdfimafn 5423* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremdfimafn2 5424* Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass4 5425* Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfvelima 5426* Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremfeqmptd 5427* Deduction form of dffn5 5420. (Contributed by Mario Carneiro, 8-Jan-2015.)

Theoremfeqresmpt 5428* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdffn5f 5429* Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.)

Theoremfvelimab 5430* Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)

Theoremfvi 5431 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfviss 5432 The value of the identity function is a subset of the argument. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theoremfniinfv 5433* The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)

Theoremfnsnfv 5434 Singleton of function value. (Contributed by NM, 22-May-1998.)

Theoremfnimapr 5435 The image of a pair under a funtion. (Contributed by Jeff Madsen, 6-Jan-2011.)

Theoremssimaex 5436* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)

Theoremssimaexg 5437* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)

Theoremfunfv 5438 A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.)

Theoremfunfv2 5439* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)

Theoremfunfv2f 5440 The value of a function. Version of funfv2 5439 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)

Theoremfvun 5441 Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.)

Theoremfvun1 5442 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremfvun2 5443 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremdffv2 5444 Alternate definition of function value df-fv 4608 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)

Theoremdmfco 5445 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)

Theoremfvco2 5446 Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)

Theoremfvco 5447 Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.)

Theoremfvco3 5448 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremfvco4i 5449 Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.)

Theoremfvopab3g 5450* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfvopab3ig 5451* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)

Theoremfvmptg 5452* Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpti 5453* Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Theoremfvmpt 5454* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)

Theoremfvmpts 5455* 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 5456* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremfvmpt3i 5457* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptd 5458* Deduction version of fvmpt 5454. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpt2i 5459* Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Theoremfvmpt2 5460* Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)

Theoremfvmptss 5461* 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 5462* 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 5391.) 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 5463* Alternate deduction version of fvmpt 5454, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv 5464* Alternate deduction version of fvmpt 5454, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv2 5465* Alternate deduction version of fvmpt 5454, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremmpteqb 5466* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5474. (Contributed by Mario Carneiro, 14-Nov-2014.)

Theoremfvmptt 5467* Closed theorem form of fvmpt 5454. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptf 5468* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5452 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremfvmptnf 5469* 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 5470 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptn 5470* 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 5452. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)

Theoremfvmptss2 5471* 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 5472* Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)

Theoremfvopab6 5473* 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 5474* 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 5475* 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 5476* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremeqfnfvd 5477* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremeqfnfv2f 5478* 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 5474 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)

Theoremeqfunfv 5479* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)

Theoremfvreseq 5480* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)

Theoremfndmdif 5481* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmdifcom 5482 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmdifeq0 5483 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 5484* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfneqeql 5485 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremfneqeql2 5486 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremfnreseql 5487 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremchfnrn 5488* 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 5489 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)

Theoremfunfvbrb 5490 Two ways to say that is in the domain of . (Contributed by Mario Carneiro, 1-May-2014.)

Theoremfvimacnvi 5491 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)

Theoremfvimacnv 5492 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 5183 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass3 5493 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 5492 would be the special case of being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass5 5494* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)

Theoremfunconstss 5495* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)

TheoremfvimacnvALT 5496 Another proof of fvimacnv 5492, based on funimass3 5493. If funimass3 5493 is ever proved directly, as opposed to using funimacnv 5181 pointwise, then the proof of funimacnv 5181 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.)

Theoremelpreima 5497 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfniniseg 5498 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 5499* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfniniseg2 5500* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

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