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

Theoremfunsn 5501 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)

Theoremfunprg 5502 A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)

Theoremfuntpg 5503 A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.)

Theoremfunpr 5504 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfuntp 5505 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfnsn 5506 Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremfnprg 5507 Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfntpg 5508 Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.)

Theoremfntp 5509 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfun0 5510 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)

Theoremfuncnvcnv 5511 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)

Theoremfuncnv2 5512* A simpler equivalence for single-rooted (see funcnv 5513). (Contributed by NM, 9-Aug-2004.)

Theoremfuncnv 5513* 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 5512 for a simpler version. (Contributed by NM, 13-Aug-2004.)

Theoremfuncnv3 5514* A condition showing a class is single-rooted. (See funcnv 5513). (Contributed by NM, 26-May-2006.)

Theoremfun2cnv 5515* 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 5516 A single-valued relation is a function. (See fun2cnv 5515 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)

Theoremfncnv 5517* Single-rootedness (see funcnv 5513) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)

Theoremfun11 5518* 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 5519* The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)

Theoremfuncnvuni 5520* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5513 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)

Theoremfun11uni 5521* 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 5522 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 5523 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)

Theoremfuncnvres 5524 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)

Theoremcnvresid 5525 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)

Theoremfuncnvres2 5526 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 5527 The image of the preimage of a function. (Contributed by NM, 25-May-2004.)

Theoremfunimass1 5528 A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)

Theoremfunimass2 5529 A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)

Theoremimadif 5530 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)

Theoremimain 5531 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremfunimaexg 5532 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 5533 The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 4322. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.)

Theoremisarep1 5534* 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 5535* 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 5533. (Contributed by NM, 26-Oct-2006.)

Theoremfneq1 5536 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)

Theoremfneq2 5537 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)

Theoremfneq1d 5538 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfneq2d 5539 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfneq12d 5540 Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)

Theoremfneq1i 5541 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfneq2i 5542 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)

Theoremnffn 5543 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)

Theoremfnfun 5544 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)

Theoremfnrel 5545 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)

Theoremfndm 5546 The domain of a function. (Contributed by NM, 2-Aug-1994.)

Theoremfunfni 5547 Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)

Theoremfndmu 5548 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)

Theoremfnbr 5549 The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)

Theoremfnop 5550 The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)

Theoremfneu 5551* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfneu2 5552* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)

Theoremfnun 5553 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)

Theoremfnunsn 5554 Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremfnco 5555 Composition of two functions. (Contributed by NM, 22-May-2006.)

Theoremfnresdm 5556 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)

Theoremfnresdisj 5557 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)

Theorem2elresin 5558 Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)

Theoremfnssresb 5559 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)

Theoremfnssres 5560 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)

Theoremfnresin1 5561 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)

Theoremfnresin2 5562 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)

Theoremfnres 5563* An equivalence for functionality of a restriction. Compare dffun8 5482. (Contributed by Mario Carneiro, 20-May-2015.)

Theoremfnresi 5564 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)

Theoremfnima 5565 The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfn0 5566 A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfnimadisj 5567 A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)

Theoremfnimaeq0 5568 Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 27128. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremdfmpt3 5569 Alternate definition for the "maps to" notation df-mpt 4270. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremfnopabg 5570* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Theoremfnopab 5571* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)

Theoremmptfng 5572* The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)

Theoremfnmpt 5573* The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)

Theoremmpt0 5574 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremfnmpti 5575* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdmmpti 5576* Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmptun 5577 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremfeq1 5578 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Theoremfeq2 5579 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Theoremfeq3 5580 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Theoremfeq23 5581 Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfeq1d 5582 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)

Theoremfeq2d 5583 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq12d 5584 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq123d 5585 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq123 5586 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)

Theoremfeq1i 5587 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq2i 5588 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)

Theoremfeq23i 5589 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq23d 5590 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)

Theoremnff 5591 Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremelimf 5592 Eliminate a mapping hypothesis for the weak deduction theorem dedth 3782, when a special case is provable, in order to convert from a hypothesis to an antecedent. (Contributed by NM, 24-Aug-2006.)

Theoremffn 5593 A mapping is a function. (Contributed by NM, 2-Aug-1994.)

Theoremdffn2 5594 Any function is a mapping into . (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremffun 5595 A mapping is a function. (Contributed by NM, 3-Aug-1994.)

Theoremfrel 5596 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)

Theoremfdm 5597 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)

Theoremfdmi 5598 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)

Theoremfrn 5599 The range of a mapping. (Contributed by NM, 3-Aug-1994.)

Theoremdffn3 5600 A function maps to its range. (Contributed by NM, 1-Sep-1999.)

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