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Theorem List for Metamath Proof Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfun2 6701 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
(((𝐹:𝐴⟢𝐢 ∧ 𝐺:𝐡⟢𝐢) ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐡)⟢𝐢)
 
Theoremfun2d 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.)
(πœ‘ β†’ 𝐹:𝐴⟢𝐢)    &   (πœ‘ β†’ 𝐺:𝐡⟢𝐢)    &   (πœ‘ β†’ (𝐴 ∩ 𝐡) = βˆ…)    β‡’   (πœ‘ β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐡)⟢𝐢)
 
Theoremfnfco 6703 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn 𝐴 ∧ 𝐺:𝐡⟢𝐴) β†’ (𝐹 ∘ 𝐺) Fn 𝐡)
 
Theoremfssres 6704 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
((𝐹:𝐴⟢𝐡 ∧ 𝐢 βŠ† 𝐴) β†’ (𝐹 β†Ύ 𝐢):𝐢⟢𝐡)
 
Theoremfssresd 6705 Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ 𝐢 βŠ† 𝐴)    β‡’   (πœ‘ β†’ (𝐹 β†Ύ 𝐢):𝐢⟢𝐡)
 
Theoremfssres2 6706 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
(((𝐹 β†Ύ 𝐴):𝐴⟢𝐡 ∧ 𝐢 βŠ† 𝐴) β†’ (𝐹 β†Ύ 𝐢):𝐢⟢𝐡)
 
Theoremfresin 6707 An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝐹:𝐴⟢𝐡 β†’ (𝐹 β†Ύ 𝑋):(𝐴 ∩ 𝑋)⟢𝐡)
 
Theoremresasplit 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.)
((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐡 ∧ (𝐹 β†Ύ (𝐴 ∩ 𝐡)) = (𝐺 β†Ύ (𝐴 ∩ 𝐡))) β†’ (𝐹 βˆͺ 𝐺) = ((𝐹 β†Ύ (𝐴 ∩ 𝐡)) βˆͺ ((𝐹 β†Ύ (𝐴 βˆ– 𝐡)) βˆͺ (𝐺 β†Ύ (𝐡 βˆ– 𝐴)))))
 
Theoremfresaun 6709 The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.)
((𝐹:𝐴⟢𝐢 ∧ 𝐺:𝐡⟢𝐢 ∧ (𝐹 β†Ύ (𝐴 ∩ 𝐡)) = (𝐺 β†Ύ (𝐴 ∩ 𝐡))) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐡)⟢𝐢)
 
Theoremfresaunres2 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.)
((𝐹:𝐴⟢𝐢 ∧ 𝐺:𝐡⟢𝐢 ∧ (𝐹 β†Ύ (𝐴 ∩ 𝐡)) = (𝐺 β†Ύ (𝐴 ∩ 𝐡))) β†’ ((𝐹 βˆͺ 𝐺) β†Ύ 𝐡) = 𝐺)
 
Theoremfresaunres1 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.)
((𝐹:𝐴⟢𝐢 ∧ 𝐺:𝐡⟢𝐢 ∧ (𝐹 β†Ύ (𝐴 ∩ 𝐡)) = (𝐺 β†Ύ (𝐴 ∩ 𝐡))) β†’ ((𝐹 βˆͺ 𝐺) β†Ύ 𝐴) = 𝐹)
 
Theoremfcoi1 6712 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴⟢𝐡 β†’ (𝐹 ∘ ( I β†Ύ 𝐴)) = 𝐹)
 
Theoremfcoi2 6713 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴⟢𝐡 β†’ (( I β†Ύ 𝐡) ∘ 𝐹) = 𝐹)
 
Theoremfeu 6714* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
((𝐹:𝐴⟢𝐡 ∧ 𝐢 ∈ 𝐴) β†’ βˆƒ!𝑦 ∈ 𝐡 ⟨𝐢, π‘¦βŸ© ∈ 𝐹)
 
Theoremfcnvres 6715 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
(𝐹:𝐴⟢𝐡 β†’ β—‘(𝐹 β†Ύ 𝐴) = (◑𝐹 β†Ύ 𝐡))
 
Theoremfimacnvdisj 6716 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
((𝐹:𝐴⟢𝐡 ∧ (𝐡 ∩ 𝐢) = βˆ…) β†’ (◑𝐹 β€œ 𝐢) = βˆ…)
 
Theoremfint 6717* Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
𝐡 β‰  βˆ…    β‡’   (𝐹:𝐴⟢∩ 𝐡 ↔ βˆ€π‘₯ ∈ 𝐡 𝐹:𝐴⟢π‘₯)
 
Theoremfin 6718 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴⟢(𝐡 ∩ 𝐢) ↔ (𝐹:𝐴⟢𝐡 ∧ 𝐹:𝐴⟢𝐢))
 
Theoremf0 6719 The empty function. (Contributed by NM, 14-Aug-1999.)
βˆ…:βˆ…βŸΆπ΄
 
Theoremf00 6720 A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)
(𝐹:π΄βŸΆβˆ… ↔ (𝐹 = βˆ… ∧ 𝐴 = βˆ…))
 
Theoremf0bi 6721 A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
(𝐹:βˆ…βŸΆπ‘‹ ↔ 𝐹 = βˆ…)
 
Theoremf0dom0 6722 A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
(𝐹:π‘‹βŸΆπ‘Œ β†’ (𝑋 = βˆ… ↔ 𝐹 = βˆ…))
 
Theoremf0rn0 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.)
((𝐸:π‘‹βŸΆπ‘Œ ∧ Β¬ βˆƒπ‘¦ ∈ π‘Œ 𝑦 ∈ ran 𝐸) β†’ 𝑋 = βˆ…)
 
Theoremfconst 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.)
𝐡 ∈ V    β‡’   (𝐴 Γ— {𝐡}):𝐴⟢{𝐡}
 
Theoremfconstg 6725 A Cartesian product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
(𝐡 ∈ 𝑉 β†’ (𝐴 Γ— {𝐡}):𝐴⟢{𝐡})
 
Theoremfnconstg 6726 A Cartesian product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
(𝐡 ∈ 𝑉 β†’ (𝐴 Γ— {𝐡}) Fn 𝐴)
 
Theoremfconst6g 6727 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐡 ∈ 𝐢 β†’ (𝐴 Γ— {𝐡}):𝐴⟢𝐢)
 
Theoremfconst6 6728 A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
𝐡 ∈ 𝐢    β‡’   (𝐴 Γ— {𝐡}):𝐴⟢𝐢
 
Theoremf1eq1 6729 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 β†’ (𝐹:𝐴–1-1→𝐡 ↔ 𝐺:𝐴–1-1→𝐡))
 
Theoremf1eq2 6730 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐡 β†’ (𝐹:𝐴–1-1→𝐢 ↔ 𝐹:𝐡–1-1→𝐢))
 
Theoremf1eq3 6731 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐡 β†’ (𝐹:𝐢–1-1→𝐴 ↔ 𝐹:𝐢–1-1→𝐡))
 
Theoremnff1 6732 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
β„²π‘₯𝐹    &   β„²π‘₯𝐴    &   β„²π‘₯𝐡    β‡’   β„²π‘₯ 𝐹:𝐴–1-1→𝐡
 
Theoremdff12 6733* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
(𝐹:𝐴–1-1→𝐡 ↔ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘¦βˆƒ*π‘₯ π‘₯𝐹𝑦))
 
Theoremf1f 6734 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
(𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴⟢𝐡)
 
Theoremf1fn 6735 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴–1-1→𝐡 β†’ 𝐹 Fn 𝐴)
 
Theoremf1fun 6736 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴–1-1→𝐡 β†’ Fun 𝐹)
 
Theoremf1rel 6737 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴–1-1→𝐡 β†’ Rel 𝐹)
 
Theoremf1dm 6738 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.) (Proof shortened by Wolf Lammen, 29-May-2024.)
(𝐹:𝐴–1-1→𝐡 β†’ dom 𝐹 = 𝐴)
 
Theoremf1dmOLD 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.)
(𝐹:𝐴–1-1→𝐡 β†’ dom 𝐹 = 𝐴)
 
Theoremf1ss 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.)
((𝐹:𝐴–1-1→𝐡 ∧ 𝐡 βŠ† 𝐢) β†’ 𝐹:𝐴–1-1→𝐢)
 
Theoremf1ssr 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.)
((𝐹:𝐴–1-1→𝐡 ∧ ran 𝐹 βŠ† 𝐢) β†’ 𝐹:𝐴–1-1→𝐢)
 
Theoremf1ssres 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.)
((𝐹:𝐴–1-1→𝐡 ∧ 𝐢 βŠ† 𝐴) β†’ (𝐹 β†Ύ 𝐢):𝐢–1-1→𝐡)
 
Theoremf1resf1 6743 The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022.)
((𝐹:𝐴–1-1→𝐡 ∧ 𝐢 βŠ† 𝐴 ∧ (𝐹 β†Ύ 𝐢):𝐢⟢𝐷) β†’ (𝐹 β†Ύ 𝐢):𝐢–1-1→𝐷)
 
Theoremf1cnvcnv 6744 Two ways to express that a set 𝐴 (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.)
(◑◑𝐴:dom 𝐴–1-1β†’V ↔ (Fun ◑𝐴 ∧ Fun ◑◑𝐴))
 
Theoremf1cof1 6745 Composition of two one-to-one functions. Generalization of f1co 6746. (Contributed by AV, 18-Sep-2024.)
((𝐹:𝐢–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐡) β†’ (𝐹 ∘ 𝐺):(◑𝐺 β€œ 𝐢)–1-1→𝐷)
 
Theoremf1co 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.)
((𝐹:𝐡–1-1→𝐢 ∧ 𝐺:𝐴–1-1→𝐡) β†’ (𝐹 ∘ 𝐺):𝐴–1-1→𝐢)
 
Theoremf1coOLD 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.)
((𝐹:𝐡–1-1→𝐢 ∧ 𝐺:𝐴–1-1→𝐡) β†’ (𝐹 ∘ 𝐺):𝐴–1-1→𝐢)
 
Theoremfoeq1 6748 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 β†’ (𝐹:𝐴–onto→𝐡 ↔ 𝐺:𝐴–onto→𝐡))
 
Theoremfoeq2 6749 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐡 β†’ (𝐹:𝐴–onto→𝐢 ↔ 𝐹:𝐡–onto→𝐢))
 
Theoremfoeq3 6750 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐡 β†’ (𝐹:𝐢–onto→𝐴 ↔ 𝐹:𝐢–onto→𝐡))
 
Theoremnffo 6751 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
β„²π‘₯𝐹    &   β„²π‘₯𝐴    &   β„²π‘₯𝐡    β‡’   β„²π‘₯ 𝐹:𝐴–onto→𝐡
 
Theoremfof 6752 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴–onto→𝐡 β†’ 𝐹:𝐴⟢𝐡)
 
Theoremfofun 6753 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
(𝐹:𝐴–onto→𝐡 β†’ Fun 𝐹)
 
Theoremfofn 6754 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
(𝐹:𝐴–onto→𝐡 β†’ 𝐹 Fn 𝐴)
 
Theoremforn 6755 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴–onto→𝐡 β†’ ran 𝐹 = 𝐡)
 
Theoremdffo2 6756 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
(𝐹:𝐴–onto→𝐡 ↔ (𝐹:𝐴⟢𝐡 ∧ ran 𝐹 = 𝐡))
 
Theoremfoima 6757 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
(𝐹:𝐴–onto→𝐡 β†’ (𝐹 β€œ 𝐴) = 𝐡)
 
Theoremdffn4 6758 A function maps onto its range. (Contributed by NM, 10-May-1998.)
(𝐹 Fn 𝐴 ↔ 𝐹:𝐴–ontoβ†’ran 𝐹)
 
Theoremfunforn 6759 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
(Fun 𝐴 ↔ 𝐴:dom 𝐴–ontoβ†’ran 𝐴)
 
Theoremfodmrnu 6760 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
((𝐹:𝐴–onto→𝐡 ∧ 𝐹:𝐢–onto→𝐷) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))
 
Theoremfimadmfo 6761 A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.)
(𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴–ontoβ†’(𝐹 β€œ 𝐴))
 
Theoremfores 6762 Restriction of an onto function. (Contributed by NM, 4-Mar-1997.)
((Fun 𝐹 ∧ 𝐴 βŠ† dom 𝐹) β†’ (𝐹 β†Ύ 𝐴):𝐴–ontoβ†’(𝐹 β€œ 𝐴))
 
TheoremfimadmfoALT 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.)
(𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴–ontoβ†’(𝐹 β€œ 𝐴))
 
Theoremfocnvimacdmdm 6764 The preimage of the codomain of a surjection is its domain. (Contributed by AV, 29-Sep-2024.)
(𝐺:𝐴–onto→𝐡 β†’ (◑𝐺 β€œ 𝐡) = 𝐴)
 
Theoremfocofo 6765 Composition of onto functions. Generalisation of foco 6766. (Contributed by AV, 29-Sep-2024.)
((𝐹:𝐴–onto→𝐡 ∧ Fun 𝐺 ∧ 𝐴 βŠ† ran 𝐺) β†’ (𝐹 ∘ 𝐺):(◑𝐺 β€œ 𝐴)–onto→𝐡)
 
Theoremfoco 6766 Composition of onto functions. (Contributed by NM, 22-Mar-2006.) (Proof shortened by AV, 29-Sep-2024.)
((𝐹:𝐡–onto→𝐢 ∧ 𝐺:𝐴–onto→𝐡) β†’ (𝐹 ∘ 𝐺):𝐴–onto→𝐢)
 
Theoremfoconst 6767 A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.)
((𝐹:𝐴⟢{𝐡} ∧ 𝐹 β‰  βˆ…) β†’ 𝐹:𝐴–ontoβ†’{𝐡})
 
Theoremf1oeq1 6768 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 β†’ (𝐹:𝐴–1-1-onto→𝐡 ↔ 𝐺:𝐴–1-1-onto→𝐡))
 
Theoremf1oeq2 6769 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐡 β†’ (𝐹:𝐴–1-1-onto→𝐢 ↔ 𝐹:𝐡–1-1-onto→𝐢))
 
Theoremf1oeq3 6770 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐡 β†’ (𝐹:𝐢–1-1-onto→𝐴 ↔ 𝐹:𝐢–1-1-onto→𝐡))
 
Theoremf1oeq23 6771 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
((𝐴 = 𝐡 ∧ 𝐢 = 𝐷) β†’ (𝐹:𝐴–1-1-onto→𝐢 ↔ 𝐹:𝐡–1-1-onto→𝐷))
 
Theoremf1eq123d 6772 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(πœ‘ β†’ 𝐹 = 𝐺)    &   (πœ‘ β†’ 𝐴 = 𝐡)    &   (πœ‘ β†’ 𝐢 = 𝐷)    β‡’   (πœ‘ β†’ (𝐹:𝐴–1-1→𝐢 ↔ 𝐺:𝐡–1-1→𝐷))
 
Theoremfoeq123d 6773 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(πœ‘ β†’ 𝐹 = 𝐺)    &   (πœ‘ β†’ 𝐴 = 𝐡)    &   (πœ‘ β†’ 𝐢 = 𝐷)    β‡’   (πœ‘ β†’ (𝐹:𝐴–onto→𝐢 ↔ 𝐺:𝐡–onto→𝐷))
 
Theoremf1oeq123d 6774 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(πœ‘ β†’ 𝐹 = 𝐺)    &   (πœ‘ β†’ 𝐴 = 𝐡)    &   (πœ‘ β†’ 𝐢 = 𝐷)    β‡’   (πœ‘ β†’ (𝐹:𝐴–1-1-onto→𝐢 ↔ 𝐺:𝐡–1-1-onto→𝐷))
 
Theoremf1oeq1d 6775 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(πœ‘ β†’ 𝐹 = 𝐺)    β‡’   (πœ‘ β†’ (𝐹:𝐴–1-1-onto→𝐡 ↔ 𝐺:𝐴–1-1-onto→𝐡))
 
Theoremf1oeq2d 6776 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ (𝐹:𝐴–1-1-onto→𝐢 ↔ 𝐹:𝐡–1-1-onto→𝐢))
 
Theoremf1oeq3d 6777 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ (𝐹:𝐢–1-1-onto→𝐴 ↔ 𝐹:𝐢–1-1-onto→𝐡))
 
Theoremnff1o 6778 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
β„²π‘₯𝐹    &   β„²π‘₯𝐴    &   β„²π‘₯𝐡    β‡’   β„²π‘₯ 𝐹:𝐴–1-1-onto→𝐡
 
Theoremf1of1 6779 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴–1-1-onto→𝐡 β†’ 𝐹:𝐴–1-1→𝐡)
 
Theoremf1of 6780 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴–1-1-onto→𝐡 β†’ 𝐹:𝐴⟢𝐡)
 
Theoremf1ofn 6781 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴–1-1-onto→𝐡 β†’ 𝐹 Fn 𝐴)
 
Theoremf1ofun 6782 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴–1-1-onto→𝐡 β†’ Fun 𝐹)
 
Theoremf1orel 6783 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
(𝐹:𝐴–1-1-onto→𝐡 β†’ Rel 𝐹)
 
Theoremf1odm 6784 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴–1-1-onto→𝐡 β†’ dom 𝐹 = 𝐴)
 
Theoremdff1o2 6785 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴–1-1-onto→𝐡 ↔ (𝐹 Fn 𝐴 ∧ Fun ◑𝐹 ∧ ran 𝐹 = 𝐡))
 
Theoremdff1o3 6786 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴–1-1-onto→𝐡 ↔ (𝐹:𝐴–onto→𝐡 ∧ Fun ◑𝐹))
 
Theoremf1ofo 6787 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
(𝐹:𝐴–1-1-onto→𝐡 β†’ 𝐹:𝐴–onto→𝐡)
 
Theoremdff1o4 6788 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴–1-1-onto→𝐡 ↔ (𝐹 Fn 𝐴 ∧ ◑𝐹 Fn 𝐡))
 
Theoremdff1o5 6789 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴–1-1-onto→𝐡 ↔ (𝐹:𝐴–1-1→𝐡 ∧ ran 𝐹 = 𝐡))
 
Theoremf1orn 6790 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
(𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◑𝐹))
 
Theoremf1f1orn 6791 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
(𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
 
Theoremf1ocnv 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.)
(𝐹:𝐴–1-1-onto→𝐡 β†’ ◑𝐹:𝐡–1-1-onto→𝐴)
 
Theoremf1ocnvb 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.)
(Rel 𝐹 β†’ (𝐹:𝐴–1-1-onto→𝐡 ↔ ◑𝐹:𝐡–1-1-onto→𝐴))
 
Theoremf1ores 6794 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)
((𝐹:𝐴–1-1→𝐡 ∧ 𝐢 βŠ† 𝐴) β†’ (𝐹 β†Ύ 𝐢):𝐢–1-1-ontoβ†’(𝐹 β€œ 𝐢))
 
Theoremf1orescnv 6795 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)
((Fun ◑𝐹 ∧ (𝐹 β†Ύ 𝑅):𝑅–1-1-onto→𝑃) β†’ (◑𝐹 β†Ύ 𝑃):𝑃–1-1-onto→𝑅)
 
Theoremf1imacnv 6796 Preimage of an image. (Contributed by NM, 30-Sep-2004.)
((𝐹:𝐴–1-1→𝐡 ∧ 𝐢 βŠ† 𝐴) β†’ (◑𝐹 β€œ (𝐹 β€œ 𝐢)) = 𝐢)
 
Theoremfoimacnv 6797 A reverse version of f1imacnv 6796. (Contributed by Jeff Hankins, 16-Jul-2009.)
((𝐹:𝐴–onto→𝐡 ∧ 𝐢 βŠ† 𝐡) β†’ (𝐹 β€œ (◑𝐹 β€œ 𝐢)) = 𝐢)
 
Theoremfoun 6798 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
(((𝐹:𝐴–onto→𝐡 ∧ 𝐺:𝐢–onto→𝐷) ∧ (𝐴 ∩ 𝐢) = βˆ…) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–ontoβ†’(𝐡 βˆͺ 𝐷))
 
Theoremf1oun 6799 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
(((𝐹:𝐴–1-1-onto→𝐡 ∧ 𝐺:𝐢–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐢) = βˆ… ∧ (𝐡 ∩ 𝐷) = βˆ…)) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1-ontoβ†’(𝐡 βˆͺ 𝐷))
 
Theoremf1un 6800 The union of two one-to-one functions with disjoint domains and codomains. (Contributed by BTernaryTau, 3-Dec-2024.)
(((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) ∧ ((𝐴 ∩ 𝐢) = βˆ… ∧ (𝐡 ∩ 𝐷) = βˆ…)) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1β†’(𝐡 βˆͺ 𝐷))
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