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

Theoremiota2df 5401 A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Theoremiota2d 5402* A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Theoremiota2 5403* The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremsniota 5404 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremdfiota4 5405 The operation using the operator. (Contributed by Scott Fenton, 6-Oct-2017.)

Theoremcsbiotag 5406* Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)

2.4.9  Functions

Syntaxwfun 5407 Extend the definition of a wff to include the function predicate. (Read: is a function.)

Syntaxwfn 5408 Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .)

Syntaxwf 5409 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .)

Syntaxwf1 5410 Extend the definition of a wff to include one-to-one functions. (Read: maps one-to-one into .) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27.

Syntaxwfo 5411 Extend the definition of a wff to include onto functions. (Read: maps onto .) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27.

Syntaxwf1o 5412 Extend the definition of a wff to include one-to-one onto functions. (Read: maps one-to-one onto .) The notation ("1-1" above the arrow and "onto" below the arrow) is from Definition 6.15(6) of [TakeutiZaring] p. 27.

Syntaxcfv 5413 Extend the definition of a class to include the value of a function. (Read: The value of at , or " of .")

Syntaxwiso 5414 Extend the definition of a wff to include the isomorphism property. (Read: is an , isomorphism of onto .)

Definitiondf-fun 5415 Define predicate that determines if some class is a function. Definition 10.1 of [Quine] p. 65. For example, the expression is true once we define cosine (df-cos 12628). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4226 with the maps-to notation (see df-mpt 4228 and df-mpt2 6045). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5416), a function with a given domain and codomain (df-f 5417), a one-to-one function (df-f1 5418), an onto function (df-fo 5419), or a one-to-one onto function (df-f1o 5420). For alternate definitions, see dffun2 5423, dffun3 5424, dffun4 5425, dffun5 5426, dffun6 5428, dffun7 5438, dffun8 5439, and dffun9 5440. (Contributed by NM, 1-Aug-1994.)

Definitiondf-fn 5416 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 5551, dffn3 5557, dffn4 5618, and dffn5 5731. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f 5417 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. For alternate definitions, see dff2 5840, dff3 5841, and dff4 5842. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f1 5418 Define a one-to-one function. For equivalent definitions see dff12 5597 and dff13 5963. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow). (Contributed by NM, 1-Aug-1994.)

Definitiondf-fo 5419 Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). For alternate definitions, see dffo2 5616, dffo3 5843, dffo4 5844, and dffo5 5845. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f1o 5420 Define a one-to-one onto function. For equivalent definitions see dff1o2 5638, dff1o3 5639, dff1o4 5641, and dff1o5 5642. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow). (Contributed by NM, 1-Aug-1994.)

Definitiondf-fv 5421* Define the value of a function, , also known as function application. For example, (we prove this in cos0 12706 after we define cosine in df-cos 12628). Typically, function is defined using maps-to notation (see df-mpt 4228 and df-mpt2 6045), but this is not required. For example, (ex-fv 21704). Note that df-ov 6043 will define two-argument functions using ordered pairs as . This particular definition is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful (as shown by ndmfv 5714 and fvprc 5681). The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar notation for a function's value at , i.e. " of ," but without context-dependent notational ambiguity. Alternate definitions are dffv2 5755, dffv3 5683, fv2 5682, and fv3 5703 (the latter two previously required to be a set.) Restricted equivalents that require to be a function are shown in funfv 5749 and funfv2 5750. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 5729. (Contributed by Scott Fenton, 6-Oct-2017.)

Definitiondf-isom 5422* Define the isomorphism predicate. We read this as " is an , isomorphism of onto ." Normally, and are ordering relations on and respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that and are subscripts. (Contributed by NM, 4-Mar-1997.)

Theoremdffun2 5423* Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun3 5424* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun4 5425* Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.)

Theoremdffun5 5426* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun6f 5427* Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremdffun6 5428* Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)

Theoremfunmo 5429* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)

Theoremfunrel 5430 A function is a relation. (Contributed by NM, 1-Aug-1994.)

Theoremfunss 5431 Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)

Theoremfuneq 5432 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)

Theoremfuneqi 5433 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremfuneqd 5434 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)

Theoremnffun 5435 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)

Theoremfuneu 5436* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfuneu2 5437* There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)

Theoremdffun7 5438* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one." However, dffun8 5439 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)

Theoremdffun8 5439* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5438. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremdffun9 5440* Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Theoremfunfn 5441 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)

Theoremfuni 5442 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)

Theoremnfunv 5443 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)

Theoremfunopg 5444 A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfunopab 5445* A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.)

Theoremfunopabeq 5446* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)

Theoremfunopab4 5447* A class of ordered pairs of values in the form used by df-mpt 4228 is a function. (Contributed by NM, 17-Feb-2013.)

Theoremfunmpt 5448 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremfunmpt2 5449 Functionality of a class given by a "maps to" notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)

Theoremfunco 5450 The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfunres 5451 A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)

Theoremfunssres 5452 The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.)

Theoremfun2ssres 5453 Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)

Theoremfunun 5454 The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.)

Theoremfuncnvsn 5455 The converse singleton of an ordered pair is a function. This is equivalent to funsn 5458 via cnvsn 5311, but stating it this way allows us to skip the sethood assumptions on and . (Contributed by NM, 30-Apr-2015.)

Theoremfunsng 5456 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)

Theoremfnsng 5457 Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremfunsn 5458 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)

Theoremfunprg 5459 A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)

Theoremfuntpg 5460 A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.)

Theoremfunpr 5461 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfuntp 5462 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfnsn 5463 Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremfnprg 5464 Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfntpg 5465 Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.)

Theoremfntp 5466 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfun0 5467 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)

Theoremfuncnvcnv 5468 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)

Theoremfuncnv2 5469* A simpler equivalence for single-rooted (see funcnv 5470). (Contributed by NM, 9-Aug-2004.)

Theoremfuncnv 5470* The converse of a class is a function iff the class is single-rooted, which means that for any in the range of there is at most one such that . Definition of single-rooted in [Enderton] p. 43. See funcnv2 5469 for a simpler version. (Contributed by NM, 13-Aug-2004.)

Theoremfuncnv3 5471* A condition showing a class is single-rooted. (See funcnv 5470). (Contributed by NM, 26-May-2006.)

Theoremfun2cnv 5472* The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that is not necessarily a function. (Contributed by NM, 13-Aug-2004.)

Theoremsvrelfun 5473 A single-valued relation is a function. (See fun2cnv 5472 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)

Theoremfncnv 5474* Single-rootedness (see funcnv 5470) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)

Theoremfun11 5475* Two ways of stating that is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.)

Theoremfununi 5476* The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)

Theoremfuncnvuni 5477* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5470 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)

Theoremfun11uni 5478* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)

Theoremfunin 5479 The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfunres11 5480 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)

Theoremfuncnvres 5481 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)

Theoremcnvresid 5482 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)

Theoremfuncnvres2 5483 The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)

Theoremfunimacnv 5484 The image of the preimage of a function. (Contributed by NM, 25-May-2004.)

Theoremfunimass1 5485 A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)

Theoremfunimass2 5486 A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)

Theoremimadif 5487 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)

Theoremimain 5488 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremfunimaexg 5489 Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.)

Theoremfunimaex 5490 The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 4280. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.)

Theoremisarep1 5491* Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by i.e. the class . If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Theoremisarep2 5492* Part of a study of the Axiom of Replacement used by the Isabelle prover. In Isabelle, the sethood of PrimReplace is apparently postulated implicitly by its type signature " i, i, i => o => i", which automatically asserts that it is a set without using any axioms. To prove that it is a set in Metamath, we need the hypotheses of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement in the form funimaex 5490. (Contributed by NM, 26-Oct-2006.)

Theoremfneq1 5493 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)

Theoremfneq2 5494 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)

Theoremfneq1d 5495 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfneq2d 5496 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfneq12d 5497 Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)

Theoremfneq1i 5498 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfneq2i 5499 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)

Theoremnffn 5500 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)

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