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Theorem List for Metamath Proof Explorer - 5601-5700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfco2 5601 Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)

Theoremfssxp 5602 A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfex2 5603 A function with bounded domain and range is a set. This version of fex 5969 is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)

Theoremfunssxp 5604 Two ways of specifying a partial function from to . (Contributed by NM, 13-Nov-2007.)

Theoremffdm 5605 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)

Theoremopelf 5606 The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfun 5607 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)

Theoremfun2 5608 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)

Theoremfnfco 5609 Composition of two functions. (Contributed by NM, 22-May-2006.)

Theoremfssres 5610 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)

Theoremfssres2 5611 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)

Theoremfresin 5612 An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Theoremresasplit 5613 If two functions agree on their common domain, express their union as a union of three functions with pairwise disjoint domains. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremfresaun 5614 The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremfresaunres2 5615 From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremfresaunres1 5616 From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.)

Theoremfcoi1 5617 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfcoi2 5618 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfeu 5619* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)

Theoremfcnvres 5620 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)

Theoremfimacnvdisj 5621 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)

Theoremfint 5622* Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfin 5623 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfabexg 5624* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)

Theoremfabex 5625* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)

Theoremdmfex 5626 If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremf0 5627 The empty function. (Contributed by NM, 14-Aug-1999.)

Theoremf00 5628 A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)

Theoremfconst 5629 A cross product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfconstg 5630 A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)

Theoremfnconstg 5631 A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)

Theoremfconst6g 5632 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfconst6 5633 A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)

Theoremf1eq1 5634 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1eq2 5635 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1eq3 5636 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremnff1 5637 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)

Theoremdff12 5638* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)

Theoremf1f 5639 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)

Theoremf1fn 5640 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)

Theoremf1fun 5641 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)

Theoremf1rel 5642 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremf1dm 5643 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)

Theoremf1ss 5644 A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)

Theoremf1ssr 5645 Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremf1ssres 5646 A function that is one-to-one is also one-to-one on some aubset of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)

Theoremf1cnvcnv 5647 Two ways to express that a set (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.)

Theoremf1co 5648 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)

Theoremfoeq1 5649 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremfoeq2 5650 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremfoeq3 5651 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremnffo 5652 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)

Theoremfof 5653 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)

Theoremfofun 5654 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)

Theoremfofn 5655 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)

Theoremforn 5656 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)

Theoremdffo2 5657 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)

Theoremfoima 5658 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)

Theoremdffn4 5659 A function maps onto its range. (Contributed by NM, 10-May-1998.)

Theoremfunforn 5660 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)

Theoremfodmrnu 5661 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)

Theoremfores 5662 Restriction of a function. (Contributed by NM, 4-Mar-1997.)

Theoremfoco 5663 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)

Theoremfoconst 5664 A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.)

Theoremf1oeq1 5665 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq2 5666 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq3 5667 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq23 5668 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)

Theoremf1eq123d 5669 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremfoeq123d 5670 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremf1oeq123d 5671 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremnff1o 5672 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)

Theoremf1of1 5673 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)

Theoremf1of 5674 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)

Theoremf1ofn 5675 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)

Theoremf1ofun 5676 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)

Theoremf1orel 5677 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)

Theoremf1odm 5678 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)

Theoremdff1o2 5679 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremdff1o3 5680 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1ofo 5681 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)

Theoremdff1o4 5682 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremdff1o5 5683 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1orn 5684 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)

Theoremf1f1orn 5685 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)

Theoremf1oabexg 5686* The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)

Theoremf1ocnv 5687 The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1ocnvb 5688 A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)

Theoremf1ores 5689 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)

Theoremf1orescnv 5690 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremf1imacnv 5691 Preimage of an image. (Contributed by NM, 30-Sep-2004.)

Theoremfoimacnv 5692 A reverse version of f1imacnv 5691. (Contributed by Jeffrey Hankins, 16-Jul-2009.)

Theoremfoun 5693 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremf1oun 5694 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)

Theoremfun11iun 5695* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.)

Theoremresdif 5696 The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremresin 5697 The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremf1oco 5698 Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)

Theoremf1cnv 5699 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)

Theoremfuncocnv2 5700 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)

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