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

Theoremunidmrn 5401 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)

Theoremrelcnvfld 5402 if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)

Theoremdfdm2 5403 Alternate definition of domain df-dm 4890 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)

Theoremunixp 5404 The double class union of a non-empty cross product is the union of it members. (Contributed by NM, 17-Sep-2006.)

Theoremunixp0 5405 A cross product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)

Theoremunixpid 5406 Field of a square cross product. (Contributed by FL, 10-Oct-2009.)

Theoremcnvexg 5407 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)

Theoremcnvex 5408 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)

Theoremrelcnvexb 5409 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)

Theoremressn 5410 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremcnviin 5411* The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)

Theoremcnvpo 5412 The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)

Theoremcnvso 5413 The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)

Theoremcoexg 5414 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)

Theoremcoex 5415 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)

Theoremxpco 5416 Composition of two cross products. (Contributed by Thierry Arnoux, 17-Nov-2017.)

Theoremxpcoid 5417 Composition of two square cross products. (Contributed by Thierry Arnoux, 14-Jan-2018.)

2.4.8  Definite description binder (inverted iota)

Syntaxcio 5418 Extend class notation with Russell's definition description binder (inverted iota).

Theoremiotajust 5419* Soundness justification theorem for df-iota 5420. (Contributed by Andrew Salmon, 29-Jun-2011.)

Definitiondf-iota 5420* Define Russell's definition description binder, which can be read as "the unique such that ," where ordinarily contains as a free variable. Our definition is meaningful only when there is exactly one such that is true (see iotaval 5431); otherwise, it evaluates to the empty set (see iotanul 5435). Russell used the inverted iota symbol to represent the binder.

Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use riotacl2 6565 (or iotacl 5443 for unbounded iota), as demonstrated in the proof of supub 7466. This can be easier than applying riotasbc 6567 or a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF.

(Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremdfiota2 5421* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremnfiota1 5422 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfiotad 5423 Deduction version of nfiota 5424. (Contributed by NM, 18-Feb-2013.)

Theoremnfiota 5424 Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.)

Theoremcbviota 5425 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Theoremcbviotav 5426* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Theoremsb8iota 5427 Variable substitution in description binder. Compare sb8eu 2301. (Contributed by NM, 18-Mar-2013.)

Theoremiotaeq 5428 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremiotabi 5429 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremuniabio 5430* Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotaval 5431* Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotauni 5432 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiotaint 5433 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremiota1 5434 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremiotanul 5435 Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one that satisfies . (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotassuni 5436 The class is a subset of the union of all elements satisfying . (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremiotaex 5437 Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiota4 5438 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiota4an 5439 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiota5 5440* A method for computing iota. (Contributed by NM, 17-Sep-2013.)

Theoremiotabidv 5441* Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)

Theoremiotabii 5442 Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremiotacl 5443 Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5420). If you have a bounded iota-based definition, riotacl2 6565 may be useful.

(Contributed by Andrew Salmon, 1-Aug-2011.)

Theoremiota2df 5444 A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Theoremiota2d 5445* A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Theoremiota2 5446* The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremsniota 5447 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremdfiota4 5448 The operation using the operator. (Contributed by Scott Fenton, 6-Oct-2017.)

Theoremcsbiotag 5449* Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)

2.4.9  Functions

Syntaxwfun 5450 Extend the definition of a wff to include the function predicate. (Read: is a function.)

Syntaxwfn 5451 Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .)

Syntaxwf 5452 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .)

Syntaxwf1 5453 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 5454 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 5455 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 5456 Extend the definition of a class to include the value of a function. (Read: The value of at , or " of .")

Syntaxwiso 5457 Extend the definition of a wff to include the isomorphism property. (Read: is an , isomorphism of onto .)

Definitiondf-fun 5458 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 12675). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4268 with the maps-to notation (see df-mpt 4270 and df-mpt2 6088). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5459), a function with a given domain and codomain (df-f 5460), a one-to-one function (df-f1 5461), an onto function (df-fo 5462), or a one-to-one onto function (df-f1o 5463). For alternate definitions, see dffun2 5466, dffun3 5467, dffun4 5468, dffun5 5469, dffun6 5471, dffun7 5481, dffun8 5482, and dffun9 5483. (Contributed by NM, 1-Aug-1994.)

Definitiondf-fn 5459 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 5594, dffn3 5600, dffn4 5661, and dffn5 5774. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f 5460 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. For alternate definitions, see dff2 5883, dff3 5884, and dff4 5885. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f1 5461 Define a one-to-one function. For equivalent definitions see dff12 5640 and dff13 6006. 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 5462 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 5659, dffo3 5886, dffo4 5887, and dffo5 5888. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f1o 5463 Define a one-to-one onto function. For equivalent definitions see dff1o2 5681, dff1o3 5682, dff1o4 5684, and dff1o5 5685. 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 5464* Define the value of a function, , also known as function application. For example, (we prove this in cos0 12753 after we define cosine in df-cos 12675). Typically, function is defined using maps-to notation (see df-mpt 4270 and df-mpt2 6088), but this is not required. For example, (ex-fv 21753). Note that df-ov 6086 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 5757 and fvprc 5724). 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 5798, dffv3 5726, fv2 5725, and fv3 5746 (the latter two previously required to be a set.) Restricted equivalents that require to be a function are shown in funfv 5792 and funfv2 5793. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 5772. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 5727. (Revised by Scott Fenton, 6-Oct-2017.)

Definitiondf-isom 5465* 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 5466* Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun3 5467* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun4 5468* Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.)

Theoremdffun5 5469* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun6f 5470* 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 5471* Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)

Theoremfunmo 5472* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)

Theoremfunrel 5473 A function is a relation. (Contributed by NM, 1-Aug-1994.)

Theoremfunss 5474 Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)

Theoremfuneq 5475 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)

Theoremfuneqi 5476 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremfuneqd 5477 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)

Theoremnffun 5478 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)

Theoremfuneu 5479* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfuneu2 5480* There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)

Theoremdffun7 5481* 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 5482 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)

Theoremdffun8 5482* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5481. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremdffun9 5483* Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Theoremfunfn 5484 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)

Theoremfuni 5485 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)

Theoremnfunv 5486 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)

Theoremfunopg 5487 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 5488* 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 5489* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)

Theoremfunopab4 5490* A class of ordered pairs of values in the form used by df-mpt 4270 is a function. (Contributed by NM, 17-Feb-2013.)

Theoremfunmpt 5491 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremfunmpt2 5492 Functionality of a class given by a "maps to" notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)

Theoremfunco 5493 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 5494 A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)

Theoremfunssres 5495 The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.)

Theoremfun2ssres 5496 Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)

Theoremfunun 5497 The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.)

Theoremfuncnvsn 5498 The converse singleton of an ordered pair is a function. This is equivalent to funsn 5501 via cnvsn 5354, but stating it this way allows us to skip the sethood assumptions on and . (Contributed by NM, 30-Apr-2015.)

Theoremfunsng 5499 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)

Theoremfnsng 5500 Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)

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