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Theorem List for Metamath Proof Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrelcnvfld 5201 if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)

Theoremdfdm2 5202 Alternate definition of domain df-dm 4698 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)

Theoremunixp 5203 The double class union of a non-empty cross product is the union of it members. (Contributed by NM, 17-Sep-2006.)

Theoremunixp0 5204 A cross product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)

Theoremunixpid 5205 Field of a square cross product. (Contributed by FL, 10-Oct-2009.)

Theoremcnvexg 5206 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)

Theoremcnvex 5207 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)

Theoremrelcnvexb 5208 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)

Theoremressn 5209 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremcnviin 5210* The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)

Theoremcnvpo 5211 The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)

Theoremcnvso 5212 The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)

Theoremcoexg 5213 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)

Theoremcoex 5214 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)

2.4.8  Functions

Syntaxwfun 5215 Extend the definition of a wff to include the function predicate. (Read: is a function.)

Syntaxwfn 5216 Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .)

Syntaxwf 5217 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .)

Syntaxwf1 5218 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 5219 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 5220 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 5221 Extend the definition of a class to include the value of a function. (Read: The value of at , or " of .")

Syntaxwiso 5222 Extend the definition of a wff to include the isomorphism property. (Read: is an , isomorphism of onto .)

Definitiondf-fun 5223 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 12348). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4078 with the maps-to notation (see df-mpt 4080 and df-mpt2 5825). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5224), a function with a given domain and codomain (df-f 5225), a one-to-one function (df-f1 5226), an onto function (df-fo 5227), or a one-to-one onto function (df-f1o 5228). For alternate definitions, see dffun2 5231, dffun3 5232, dffun4 5233, dffun5 5234, dffun6 5236, dffun7 5246, dffun8 5247, and dffun9 5248. (Contributed by NM, 1-Aug-1994.)

Definitiondf-fn 5224 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 5356, dffn3 5362, dffn4 5423, and dffn5 5530. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f 5225 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. For alternate definitions, see dff2 5634, dff3 5635, and dff4 5636. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f1 5226 Define a one-to-one function. For equivalent definitions see dff12 5402 and dff13 5745. 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 5227 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 5421, dffo3 5637, dffo4 5638, and dffo5 5639. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f1o 5228 Define a one-to-one onto function. For equivalent definitions see dff1o2 5443, dff1o3 5444, dff1o4 5446, and dff1o5 5447. 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 5229* Define the value of a function, , also known as function application. For example, (we prove this in cos0 12426 after we define cosine in df-cos 12348). Typically function is defined using maps-to notation (see df-mpt 4080 and df-mpt2 5825), but this is not required. For example, (ex-fv 20807). Note that df-ov 5823 will define two-argument functions using ordered pairs as . Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 5040), our definition apparently does not appear in the literature. However, it 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 5514 and fvprc 5483). 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 5554 and dffv3 6281. For other alternate definitions (that fail to evaluate to the empty set for proper classes), see fv2 5482, fv3 5502, and fv4 6282. Restricted equivalents that require to be a function are shown in funfv 5548 and funfv2 5549. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 5528. (Contributed by NM, 1-Aug-1994.)

Definitiondf-isom 5230* 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 5231* Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun3 5232* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun4 5233* Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.)

Theoremdffun5 5234* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun6f 5235* 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 5236* Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)

Theoremfunmo 5237* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)

Theoremfunrel 5238 A function is a relation. (Contributed by NM, 1-Aug-1994.)

Theoremfunss 5239 Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)

Theoremfuneq 5240 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)

Theoremfuneqi 5241 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremfuneqd 5242 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)

Theoremnffun 5243 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)

Theoremfuneu 5244* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfuneu2 5245* There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)

Theoremdffun7 5246* 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 5247 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)

Theoremdffun8 5247* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5246. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremdffun9 5248* Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Theoremfunfn 5249 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)

Theoremfuni 5250 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)

Theoremnfunv 5251 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)

Theoremfunopg 5252 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 5253* 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 5254* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)

Theoremfunopab4 5255* A class of ordered pairs of values in the form used by df-mpt 4080 is a function. (Contributed by NM, 17-Feb-2013.)

Theoremfunmpt 5256 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremfunmpt2 5257 Functionality of a class given by a "maps to" notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)

Theoremfunco 5258 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 5259 A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)

Theoremfunssres 5260 The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.)

Theoremfun2ssres 5261 Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)

Theoremfunun 5262 The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.)

Theoremfuncnvsn 5263 The converse singleton of an ordered pair is a function. This is equivalent to funsn 5266 via cnvsn 5153, but stating it this way allows us to skip the sethood assumptions on and . (Contributed by NM, 30-Apr-2015.)

Theoremfunsng 5264 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)

Theoremfnsng 5265 Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremfunsn 5266 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)

Theoremfunprg 5267 A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)

Theoremfunpr 5268 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfuntp 5269 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfnsn 5270 Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremfnprg 5271 Domain of a function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfntp 5272 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfun0 5273 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)

Theoremfuncnvcnv 5274 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)

Theoremfuncnv2 5275* A simpler equivalence for single-rooted (see funcnv 5276). (Contributed by NM, 9-Aug-2004.)

Theoremfuncnv 5276* 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 5275 for a simpler version. (Contributed by NM, 13-Aug-2004.)

Theoremfuncnv3 5277* A condition showing a class is single-rooted. (See funcnv 5276). (Contributed by NM, 26-May-2006.)

Theoremfun2cnv 5278* 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 5279 A single-valued relation is a function. (See fun2cnv 5278 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)

Theoremfncnv 5280* Single-rootedness (see funcnv 5276) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)

Theoremfun11 5281* 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 5282* The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)

Theoremfuncnvuni 5283* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5276 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)

Theoremfun11uni 5284* 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 5285 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 5286 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)

Theoremfuncnvres 5287 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)

Theoremcnvresid 5288 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)

Theoremfuncnvres2 5289 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 5290 The image of the preimage of a function. (Contributed by NM, 25-May-2004.)

Theoremfunimass1 5291 A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)

Theoremfunimass2 5292 A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)

Theoremimadif 5293 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)

Theoremimain 5294 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremfunimaexg 5295 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 5296 The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 4132. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.)

Theoremisarep1 5297* 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 5298* 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 5296. (Contributed by NM, 26-Oct-2006.)

Theoremfneq1 5299 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)

Theoremfneq2 5300 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)

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