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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fun2 6701 | The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.) |
β’ (((πΉ:π΄βΆπΆ β§ πΊ:π΅βΆπΆ) β§ (π΄ β© π΅) = β ) β (πΉ βͺ πΊ):(π΄ βͺ π΅)βΆπΆ) | ||
Theorem | fun2d 6702 | The union of functions with disjoint domains is a function, deduction version of fun2 6701. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
β’ (π β πΉ:π΄βΆπΆ) & β’ (π β πΊ:π΅βΆπΆ) & β’ (π β (π΄ β© π΅) = β ) β β’ (π β (πΉ βͺ πΊ):(π΄ βͺ π΅)βΆπΆ) | ||
Theorem | fnfco 6703 | Composition of two functions. (Contributed by NM, 22-May-2006.) |
β’ ((πΉ Fn π΄ β§ πΊ:π΅βΆπ΄) β (πΉ β πΊ) Fn π΅) | ||
Theorem | fssres 6704 | Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.) |
β’ ((πΉ:π΄βΆπ΅ β§ πΆ β π΄) β (πΉ βΎ πΆ):πΆβΆπ΅) | ||
Theorem | fssresd 6705 | Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β πΉ:π΄βΆπ΅) & β’ (π β πΆ β π΄) β β’ (π β (πΉ βΎ πΆ):πΆβΆπ΅) | ||
Theorem | fssres2 6706 | Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.) |
β’ (((πΉ βΎ π΄):π΄βΆπ΅ β§ πΆ β π΄) β (πΉ βΎ πΆ):πΆβΆπ΅) | ||
Theorem | fresin 6707 | An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.) |
β’ (πΉ:π΄βΆπ΅ β (πΉ βΎ π):(π΄ β© π)βΆπ΅) | ||
Theorem | resasplit 6708 | 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.) |
β’ ((πΉ Fn π΄ β§ πΊ Fn π΅ β§ (πΉ βΎ (π΄ β© π΅)) = (πΊ βΎ (π΄ β© π΅))) β (πΉ βͺ πΊ) = ((πΉ βΎ (π΄ β© π΅)) βͺ ((πΉ βΎ (π΄ β π΅)) βͺ (πΊ βΎ (π΅ β π΄))))) | ||
Theorem | fresaun 6709 | The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
β’ ((πΉ:π΄βΆπΆ β§ πΊ:π΅βΆπΆ β§ (πΉ βΎ (π΄ β© π΅)) = (πΊ βΎ (π΄ β© π΅))) β (πΉ βͺ πΊ):(π΄ βͺ π΅)βΆπΆ) | ||
Theorem | fresaunres2 6710 | 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.) |
β’ ((πΉ:π΄βΆπΆ β§ πΊ:π΅βΆπΆ β§ (πΉ βΎ (π΄ β© π΅)) = (πΊ βΎ (π΄ β© π΅))) β ((πΉ βͺ πΊ) βΎ π΅) = πΊ) | ||
Theorem | fresaunres1 6711 | 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.) |
β’ ((πΉ:π΄βΆπΆ β§ πΊ:π΅βΆπΆ β§ (πΉ βΎ (π΄ β© π΅)) = (πΊ βΎ (π΄ β© π΅))) β ((πΉ βͺ πΊ) βΎ π΄) = πΉ) | ||
Theorem | fcoi1 6712 | Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
β’ (πΉ:π΄βΆπ΅ β (πΉ β ( I βΎ π΄)) = πΉ) | ||
Theorem | fcoi2 6713 | Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
β’ (πΉ:π΄βΆπ΅ β (( I βΎ π΅) β πΉ) = πΉ) | ||
Theorem | feu 6714* | There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.) |
β’ ((πΉ:π΄βΆπ΅ β§ πΆ β π΄) β β!π¦ β π΅ β¨πΆ, π¦β© β πΉ) | ||
Theorem | fcnvres 6715 | The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.) |
β’ (πΉ:π΄βΆπ΅ β β‘(πΉ βΎ π΄) = (β‘πΉ βΎ π΅)) | ||
Theorem | fimacnvdisj 6716 | The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.) |
β’ ((πΉ:π΄βΆπ΅ β§ (π΅ β© πΆ) = β ) β (β‘πΉ β πΆ) = β ) | ||
Theorem | fint 6717* | Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
β’ π΅ β β β β’ (πΉ:π΄βΆβ© π΅ β βπ₯ β π΅ πΉ:π΄βΆπ₯) | ||
Theorem | fin 6718 | Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
β’ (πΉ:π΄βΆ(π΅ β© πΆ) β (πΉ:π΄βΆπ΅ β§ πΉ:π΄βΆπΆ)) | ||
Theorem | f0 6719 | The empty function. (Contributed by NM, 14-Aug-1999.) |
β’ β :β βΆπ΄ | ||
Theorem | f00 6720 | A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.) |
β’ (πΉ:π΄βΆβ β (πΉ = β β§ π΄ = β )) | ||
Theorem | f0bi 6721 | A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.) |
β’ (πΉ:β βΆπ β πΉ = β ) | ||
Theorem | f0dom0 6722 | A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.) |
β’ (πΉ:πβΆπ β (π = β β πΉ = β )) | ||
Theorem | f0rn0 6723* | If there is no element in the range of a function, its domain must be empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.) |
β’ ((πΈ:πβΆπ β§ Β¬ βπ¦ β π π¦ β ran πΈ) β π = β ) | ||
Theorem | fconst 6724 | A Cartesian product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
β’ π΅ β V β β’ (π΄ Γ {π΅}):π΄βΆ{π΅} | ||
Theorem | fconstg 6725 | A Cartesian product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.) |
β’ (π΅ β π β (π΄ Γ {π΅}):π΄βΆ{π΅}) | ||
Theorem | fnconstg 6726 | A Cartesian product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.) |
β’ (π΅ β π β (π΄ Γ {π΅}) Fn π΄) | ||
Theorem | fconst6g 6727 | Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ (π΅ β πΆ β (π΄ Γ {π΅}):π΄βΆπΆ) | ||
Theorem | fconst6 6728 | A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.) |
β’ π΅ β πΆ β β’ (π΄ Γ {π΅}):π΄βΆπΆ | ||
Theorem | f1eq1 6729 | Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.) |
β’ (πΉ = πΊ β (πΉ:π΄β1-1βπ΅ β πΊ:π΄β1-1βπ΅)) | ||
Theorem | f1eq2 6730 | Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.) |
β’ (π΄ = π΅ β (πΉ:π΄β1-1βπΆ β πΉ:π΅β1-1βπΆ)) | ||
Theorem | f1eq3 6731 | Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.) |
β’ (π΄ = π΅ β (πΉ:πΆβ1-1βπ΄ β πΉ:πΆβ1-1βπ΅)) | ||
Theorem | nff1 6732 | Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.) |
β’ β²π₯πΉ & β’ β²π₯π΄ & β’ β²π₯π΅ β β’ β²π₯ πΉ:π΄β1-1βπ΅ | ||
Theorem | dff12 6733* | Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.) |
β’ (πΉ:π΄β1-1βπ΅ β (πΉ:π΄βΆπ΅ β§ βπ¦β*π₯ π₯πΉπ¦)) | ||
Theorem | f1f 6734 | A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.) |
β’ (πΉ:π΄β1-1βπ΅ β πΉ:π΄βΆπ΅) | ||
Theorem | f1fn 6735 | A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.) |
β’ (πΉ:π΄β1-1βπ΅ β πΉ Fn π΄) | ||
Theorem | f1fun 6736 | A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.) |
β’ (πΉ:π΄β1-1βπ΅ β Fun πΉ) | ||
Theorem | f1rel 6737 | A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.) |
β’ (πΉ:π΄β1-1βπ΅ β Rel πΉ) | ||
Theorem | f1dm 6738 | The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.) (Proof shortened by Wolf Lammen, 29-May-2024.) |
β’ (πΉ:π΄β1-1βπ΅ β dom πΉ = π΄) | ||
Theorem | f1dmOLD 6739 | Obsolete version of f1dm 6738 as of 29-May-2024. (Contributed by NM, 8-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (πΉ:π΄β1-1βπ΅ β dom πΉ = π΄) | ||
Theorem | f1ss 6740 | A function that is one-to-one is also one-to-one on some superset of its codomain. (Contributed by Mario Carneiro, 12-Jan-2013.) |
β’ ((πΉ:π΄β1-1βπ΅ β§ π΅ β πΆ) β πΉ:π΄β1-1βπΆ) | ||
Theorem | f1ssr 6741 | A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
β’ ((πΉ:π΄β1-1βπ΅ β§ ran πΉ β πΆ) β πΉ:π΄β1-1βπΆ) | ||
Theorem | f1ssres 6742 | A function that is one-to-one is also one-to-one on any subclass of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.) |
β’ ((πΉ:π΄β1-1βπ΅ β§ πΆ β π΄) β (πΉ βΎ πΆ):πΆβ1-1βπ΅) | ||
Theorem | f1resf1 6743 | The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022.) |
β’ ((πΉ:π΄β1-1βπ΅ β§ πΆ β π΄ β§ (πΉ βΎ πΆ):πΆβΆπ·) β (πΉ βΎ πΆ):πΆβ1-1βπ·) | ||
Theorem | f1cnvcnv 6744 | Two ways to express that a set π΄ (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.) |
β’ (β‘β‘π΄:dom π΄β1-1βV β (Fun β‘π΄ β§ Fun β‘β‘π΄)) | ||
Theorem | f1cof1 6745 | Composition of two one-to-one functions. Generalization of f1co 6746. (Contributed by AV, 18-Sep-2024.) |
β’ ((πΉ:πΆβ1-1βπ· β§ πΊ:π΄β1-1βπ΅) β (πΉ β πΊ):(β‘πΊ β πΆ)β1-1βπ·) | ||
Theorem | f1co 6746 | Composition of one-to-one functions when the codomain of the first matches the domain of the second. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.) (Proof shortened by AV, 20-Sep-2024.) |
β’ ((πΉ:π΅β1-1βπΆ β§ πΊ:π΄β1-1βπ΅) β (πΉ β πΊ):π΄β1-1βπΆ) | ||
Theorem | f1coOLD 6747 | Obsolete version of f1co 6746 as of 20-Sep-2024. (Contributed by NM, 28-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((πΉ:π΅β1-1βπΆ β§ πΊ:π΄β1-1βπ΅) β (πΉ β πΊ):π΄β1-1βπΆ) | ||
Theorem | foeq1 6748 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
β’ (πΉ = πΊ β (πΉ:π΄βontoβπ΅ β πΊ:π΄βontoβπ΅)) | ||
Theorem | foeq2 6749 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
β’ (π΄ = π΅ β (πΉ:π΄βontoβπΆ β πΉ:π΅βontoβπΆ)) | ||
Theorem | foeq3 6750 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
β’ (π΄ = π΅ β (πΉ:πΆβontoβπ΄ β πΉ:πΆβontoβπ΅)) | ||
Theorem | nffo 6751 | Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.) |
β’ β²π₯πΉ & β’ β²π₯π΄ & β’ β²π₯π΅ β β’ β²π₯ πΉ:π΄βontoβπ΅ | ||
Theorem | fof 6752 | An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.) |
β’ (πΉ:π΄βontoβπ΅ β πΉ:π΄βΆπ΅) | ||
Theorem | fofun 6753 | An onto mapping is a function. (Contributed by NM, 29-Mar-2008.) |
β’ (πΉ:π΄βontoβπ΅ β Fun πΉ) | ||
Theorem | fofn 6754 | An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.) |
β’ (πΉ:π΄βontoβπ΅ β πΉ Fn π΄) | ||
Theorem | forn 6755 | The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.) |
β’ (πΉ:π΄βontoβπ΅ β ran πΉ = π΅) | ||
Theorem | dffo2 6756 | Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.) |
β’ (πΉ:π΄βontoβπ΅ β (πΉ:π΄βΆπ΅ β§ ran πΉ = π΅)) | ||
Theorem | foima 6757 | The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.) |
β’ (πΉ:π΄βontoβπ΅ β (πΉ β π΄) = π΅) | ||
Theorem | dffn4 6758 | A function maps onto its range. (Contributed by NM, 10-May-1998.) |
β’ (πΉ Fn π΄ β πΉ:π΄βontoβran πΉ) | ||
Theorem | funforn 6759 | A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.) |
β’ (Fun π΄ β π΄:dom π΄βontoβran π΄) | ||
Theorem | fodmrnu 6760 | An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.) |
β’ ((πΉ:π΄βontoβπ΅ β§ πΉ:πΆβontoβπ·) β (π΄ = πΆ β§ π΅ = π·)) | ||
Theorem | fimadmfo 6761 | A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.) |
β’ (πΉ:π΄βΆπ΅ β πΉ:π΄βontoβ(πΉ β π΄)) | ||
Theorem | fores 6762 | Restriction of an onto function. (Contributed by NM, 4-Mar-1997.) |
β’ ((Fun πΉ β§ π΄ β dom πΉ) β (πΉ βΎ π΄):π΄βontoβ(πΉ β π΄)) | ||
Theorem | fimadmfoALT 6763 | Alternate proof of fimadmfo 6761, based on fores 6762. A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (πΉ:π΄βΆπ΅ β πΉ:π΄βontoβ(πΉ β π΄)) | ||
Theorem | focnvimacdmdm 6764 | The preimage of the codomain of a surjection is its domain. (Contributed by AV, 29-Sep-2024.) |
β’ (πΊ:π΄βontoβπ΅ β (β‘πΊ β π΅) = π΄) | ||
Theorem | focofo 6765 | Composition of onto functions. Generalisation of foco 6766. (Contributed by AV, 29-Sep-2024.) |
β’ ((πΉ:π΄βontoβπ΅ β§ Fun πΊ β§ π΄ β ran πΊ) β (πΉ β πΊ):(β‘πΊ β π΄)βontoβπ΅) | ||
Theorem | foco 6766 | Composition of onto functions. (Contributed by NM, 22-Mar-2006.) (Proof shortened by AV, 29-Sep-2024.) |
β’ ((πΉ:π΅βontoβπΆ β§ πΊ:π΄βontoβπ΅) β (πΉ β πΊ):π΄βontoβπΆ) | ||
Theorem | foconst 6767 | A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.) |
β’ ((πΉ:π΄βΆ{π΅} β§ πΉ β β ) β πΉ:π΄βontoβ{π΅}) | ||
Theorem | f1oeq1 6768 | Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
β’ (πΉ = πΊ β (πΉ:π΄β1-1-ontoβπ΅ β πΊ:π΄β1-1-ontoβπ΅)) | ||
Theorem | f1oeq2 6769 | Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
β’ (π΄ = π΅ β (πΉ:π΄β1-1-ontoβπΆ β πΉ:π΅β1-1-ontoβπΆ)) | ||
Theorem | f1oeq3 6770 | Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
β’ (π΄ = π΅ β (πΉ:πΆβ1-1-ontoβπ΄ β πΉ:πΆβ1-1-ontoβπ΅)) | ||
Theorem | f1oeq23 6771 | Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.) |
β’ ((π΄ = π΅ β§ πΆ = π·) β (πΉ:π΄β1-1-ontoβπΆ β πΉ:π΅β1-1-ontoβπ·)) | ||
Theorem | f1eq123d 6772 | Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.) |
β’ (π β πΉ = πΊ) & β’ (π β π΄ = π΅) & β’ (π β πΆ = π·) β β’ (π β (πΉ:π΄β1-1βπΆ β πΊ:π΅β1-1βπ·)) | ||
Theorem | foeq123d 6773 | Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.) |
β’ (π β πΉ = πΊ) & β’ (π β π΄ = π΅) & β’ (π β πΆ = π·) β β’ (π β (πΉ:π΄βontoβπΆ β πΊ:π΅βontoβπ·)) | ||
Theorem | f1oeq123d 6774 | Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.) |
β’ (π β πΉ = πΊ) & β’ (π β π΄ = π΅) & β’ (π β πΆ = π·) β β’ (π β (πΉ:π΄β1-1-ontoβπΆ β πΊ:π΅β1-1-ontoβπ·)) | ||
Theorem | f1oeq1d 6775 | Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π β πΉ = πΊ) β β’ (π β (πΉ:π΄β1-1-ontoβπ΅ β πΊ:π΄β1-1-ontoβπ΅)) | ||
Theorem | f1oeq2d 6776 | Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π β π΄ = π΅) β β’ (π β (πΉ:π΄β1-1-ontoβπΆ β πΉ:π΅β1-1-ontoβπΆ)) | ||
Theorem | f1oeq3d 6777 | Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π β π΄ = π΅) β β’ (π β (πΉ:πΆβ1-1-ontoβπ΄ β πΉ:πΆβ1-1-ontoβπ΅)) | ||
Theorem | nff1o 6778 | Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.) |
β’ β²π₯πΉ & β’ β²π₯π΄ & β’ β²π₯π΅ β β’ β²π₯ πΉ:π΄β1-1-ontoβπ΅ | ||
Theorem | f1of1 6779 | A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β πΉ:π΄β1-1βπ΅) | ||
Theorem | f1of 6780 | A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β πΉ:π΄βΆπ΅) | ||
Theorem | f1ofn 6781 | A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β πΉ Fn π΄) | ||
Theorem | f1ofun 6782 | A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β Fun πΉ) | ||
Theorem | f1orel 6783 | A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β Rel πΉ) | ||
Theorem | f1odm 6784 | The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β dom πΉ = π΄) | ||
Theorem | dff1o2 6785 | Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β (πΉ Fn π΄ β§ Fun β‘πΉ β§ ran πΉ = π΅)) | ||
Theorem | dff1o3 6786 | Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β (πΉ:π΄βontoβπ΅ β§ Fun β‘πΉ)) | ||
Theorem | f1ofo 6787 | A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β πΉ:π΄βontoβπ΅) | ||
Theorem | dff1o4 6788 | Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β (πΉ Fn π΄ β§ β‘πΉ Fn π΅)) | ||
Theorem | dff1o5 6789 | Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β (πΉ:π΄β1-1βπ΅ β§ ran πΉ = π΅)) | ||
Theorem | f1orn 6790 | A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.) |
β’ (πΉ:π΄β1-1-ontoβran πΉ β (πΉ Fn π΄ β§ Fun β‘πΉ)) | ||
Theorem | f1f1orn 6791 | A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.) |
β’ (πΉ:π΄β1-1βπ΅ β πΉ:π΄β1-1-ontoβran πΉ) | ||
Theorem | f1ocnv 6792 | 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.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β β‘πΉ:π΅β1-1-ontoβπ΄) | ||
Theorem | f1ocnvb 6793 | A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and codomain/range interchanged. (Contributed by NM, 8-Dec-2003.) |
β’ (Rel πΉ β (πΉ:π΄β1-1-ontoβπ΅ β β‘πΉ:π΅β1-1-ontoβπ΄)) | ||
Theorem | f1ores 6794 | The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.) |
β’ ((πΉ:π΄β1-1βπ΅ β§ πΆ β π΄) β (πΉ βΎ πΆ):πΆβ1-1-ontoβ(πΉ β πΆ)) | ||
Theorem | f1orescnv 6795 | The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.) |
β’ ((Fun β‘πΉ β§ (πΉ βΎ π ):π β1-1-ontoβπ) β (β‘πΉ βΎ π):πβ1-1-ontoβπ ) | ||
Theorem | f1imacnv 6796 | Preimage of an image. (Contributed by NM, 30-Sep-2004.) |
β’ ((πΉ:π΄β1-1βπ΅ β§ πΆ β π΄) β (β‘πΉ β (πΉ β πΆ)) = πΆ) | ||
Theorem | foimacnv 6797 | A reverse version of f1imacnv 6796. (Contributed by Jeff Hankins, 16-Jul-2009.) |
β’ ((πΉ:π΄βontoβπ΅ β§ πΆ β π΅) β (πΉ β (β‘πΉ β πΆ)) = πΆ) | ||
Theorem | foun 6798 | The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.) |
β’ (((πΉ:π΄βontoβπ΅ β§ πΊ:πΆβontoβπ·) β§ (π΄ β© πΆ) = β ) β (πΉ βͺ πΊ):(π΄ βͺ πΆ)βontoβ(π΅ βͺ π·)) | ||
Theorem | f1oun 6799 | The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.) |
β’ (((πΉ:π΄β1-1-ontoβπ΅ β§ πΊ:πΆβ1-1-ontoβπ·) β§ ((π΄ β© πΆ) = β β§ (π΅ β© π·) = β )) β (πΉ βͺ πΊ):(π΄ βͺ πΆ)β1-1-ontoβ(π΅ βͺ π·)) | ||
Theorem | f1un 6800 | The union of two one-to-one functions with disjoint domains and codomains. (Contributed by BTernaryTau, 3-Dec-2024.) |
β’ (((πΉ:π΄β1-1βπ΅ β§ πΊ:πΆβ1-1βπ·) β§ ((π΄ β© πΆ) = β β§ (π΅ β© π·) = β )) β (πΉ βͺ πΊ):(π΄ βͺ πΆ)β1-1β(π΅ βͺ π·)) |
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