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Theorem List for Metamath Proof Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrn 6601 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → ran 𝐹𝐵)
 
Theoremfrnd 6602 Deduction form of frn 6601. The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → ran 𝐹𝐵)
 
Theoremfdm 6603 The domain of a mapping. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 29-May-2024.)
(𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
 
TheoremfdmOLD 6604 Obsolete version of fdm 6603 as of 29-May-2024. (Contributed by NM, 2-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
 
Theoremfdmd 6605 Deduction form of fdm 6603. The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → dom 𝐹 = 𝐴)
 
Theoremfdmi 6606 Inference associated with fdm 6603. The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
𝐹:𝐴𝐵       dom 𝐹 = 𝐴
 
Theoremdffn3 6607 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
(𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
 
Theoremffrn 6608 A function maps to its range. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹)
 
Theoremffrnb 6609 Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6608. (Contributed by BJ, 21-Sep-2024.)
(𝐹:𝐴𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵))
 
Theoremffrnbd 6610 A function maps to its range iff the the range is a subset of its codomain. Generalization of ffrn 6608. (Contributed by AV, 20-Sep-2024.)
(𝜑 → ran 𝐹𝐵)       (𝜑 → (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹))
 
Theoremfss 6611 Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹:𝐴𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
 
Theoremfssd 6612 Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐹:𝐴𝐶)
 
Theoremfssdmd 6613 Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, deduction form. (Contributed by AV, 21-Aug-2022.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐷 ⊆ dom 𝐹)       (𝜑𝐷𝐴)
 
Theoremfssdm 6614 Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, semi-deduction form. (Contributed by AV, 21-Aug-2022.)
𝐷 ⊆ dom 𝐹    &   (𝜑𝐹:𝐴𝐵)       (𝜑𝐷𝐴)
 
Theoremfimass 6615 The image of a class under a function with domain and codomain is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐹:𝐴𝐵 → (𝐹𝑋) ⊆ 𝐵)
 
Theoremfimacnv 6616 The preimage of the codomain of a function is the function's domain. (Contributed by FL, 25-Jan-2007.) (Proof shortened by AV, 20-Sep-2024.)
(𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐴)
 
Theoremfcof 6617 Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 6618. (Contributed by AV, 18-Sep-2024.)
((𝐹:𝐴𝐵 ∧ Fun 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)
 
Theoremfco 6618 Composition of two functions with domain and codomain as a function with domain and codomain. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Proof shortened by AV, 20-Sep-2024.)
((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
 
TheoremfcoOLD 6619 Obsolete version of fco 6618 as of 20-Sep-2024. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
 
Theoremfcod 6620 Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐵𝐶)    &   (𝜑𝐺:𝐴𝐵)       (𝜑 → (𝐹𝐺):𝐴𝐶)
 
Theoremfco2 6621 Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
(((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
 
Theoremfssxp 6622 A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
 
Theoremfunssxp 6623 Two ways of specifying a partial function from 𝐴 to 𝐵. (Contributed by NM, 13-Nov-2007.)
((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
 
Theoremffdm 6624 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
(𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
 
Theoremffdmd 6625 The domain of a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑𝐹:dom 𝐹𝐵)
 
Theoremfdmrn 6626 A different way to write 𝐹 is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
(Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
 
Theoremfuncofd 6627 Composition of two functions as a function with domain and codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof shortened by AV, 20-Sep-2024.)
(𝜑 → Fun 𝐹)    &   (𝜑 → Fun 𝐺)       (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)
 
Theoremfco3OLD 6628 Obsolete version of funcofd 6627 as 20-Sep-2024. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Fun 𝐹)    &   (𝜑 → Fun 𝐺)       (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)
 
Theoremopelf 6629 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.)
((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶𝐴𝐷𝐵))
 
Theoremfun 6630 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹:𝐴𝐶𝐺:𝐵𝐷) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷))
 
Theoremfun2 6631 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
(((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
 
Theoremfun2d 6632 The union of functions with disjoint domains is a function, deduction version of fun2 6631. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐹:𝐴𝐶)    &   (𝜑𝐺:𝐵𝐶)    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑 → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
 
Theoremfnfco 6633 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn 𝐴𝐺:𝐵𝐴) → (𝐹𝐺) Fn 𝐵)
 
Theoremfssres 6634 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
 
Theoremfssresd 6635 Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶):𝐶𝐵)
 
Theoremfssres2 6636 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
(((𝐹𝐴):𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
 
Theoremfresin 6637 An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝐹:𝐴𝐵 → (𝐹𝑋):(𝐴𝑋)⟶𝐵)
 
Theoremresasplit 6638 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 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
 
Theoremfresaun 6639 The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.)
((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
 
Theoremfresaunres2 6640 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:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)
 
Theoremfresaunres1 6641 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:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)
 
Theoremfcoi1 6642 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
 
Theoremfcoi2 6643 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
 
Theoremfeu 6644* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹)
 
Theoremfcnvres 6645 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
(𝐹:𝐴𝐵(𝐹𝐴) = (𝐹𝐵))
 
Theoremfimacnvdisj 6646 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐹𝐶) = ∅)
 
Theoremfint 6647* Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
𝐵 ≠ ∅       (𝐹:𝐴 𝐵 ↔ ∀𝑥𝐵 𝐹:𝐴𝑥)
 
Theoremfin 6648 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹:𝐴𝐵𝐹:𝐴𝐶))
 
Theoremf0 6649 The empty function. (Contributed by NM, 14-Aug-1999.)
∅:∅⟶𝐴
 
Theoremf00 6650 A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)
(𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 
Theoremf0bi 6651 A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
(𝐹:∅⟶𝑋𝐹 = ∅)
 
Theoremf0dom0 6652 A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
(𝐹:𝑋𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))
 
Theoremf0rn0 6653* 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:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸) → 𝑋 = ∅)
 
Theoremfconst 6654 A Cartesian product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
𝐵 ∈ V       (𝐴 × {𝐵}):𝐴⟶{𝐵}
 
Theoremfconstg 6655 A Cartesian product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
(𝐵𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
 
Theoremfnconstg 6656 A Cartesian product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
(𝐵𝑉 → (𝐴 × {𝐵}) Fn 𝐴)
 
Theoremfconst6g 6657 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐵𝐶 → (𝐴 × {𝐵}):𝐴𝐶)
 
Theoremfconst6 6658 A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
𝐵𝐶       (𝐴 × {𝐵}):𝐴𝐶
 
Theoremf1eq1 6659 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 → (𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵))
 
Theoremf1eq2 6660 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐴1-1𝐶𝐹:𝐵1-1𝐶))
 
Theoremf1eq3 6661 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐶1-1𝐴𝐹:𝐶1-1𝐵))
 
Theoremnff1 6662 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴1-1𝐵
 
Theoremdff12 6663* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
 
Theoremf1f 6664 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
(𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
 
Theoremf1fn 6665 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
 
Theoremf1fun 6666 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1𝐵 → Fun 𝐹)
 
Theoremf1rel 6667 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1𝐵 → Rel 𝐹)
 
Theoremf1dm 6668 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.) (Proof shortened by Wolf Lammen, 29-May-2024.)
(𝐹:𝐴1-1𝐵 → dom 𝐹 = 𝐴)
 
Theoremf1dmOLD 6669 Obsolete version of f1dm 6668 as of 29-May-2024. (Contributed by NM, 8-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐹:𝐴1-1𝐵 → dom 𝐹 = 𝐴)
 
Theoremf1ss 6670 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:𝐴1-1𝐵𝐵𝐶) → 𝐹:𝐴1-1𝐶)
 
Theoremf1ssr 6671 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:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴1-1𝐶)
 
Theoremf1ssres 6672 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:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1𝐵)
 
Theoremf1resf1 6673 The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022.)
((𝐹:𝐴1-1𝐵𝐶𝐴 ∧ (𝐹𝐶):𝐶𝐷) → (𝐹𝐶):𝐶1-1𝐷)
 
Theoremf1cnvcnv 6674 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 𝐴1-1→V ↔ (Fun 𝐴 ∧ Fun 𝐴))
 
Theoremf1cof1 6675 Composition of two one-to-one functions. Generalization of f1co 6676. (Contributed by AV, 18-Sep-2024.)
((𝐹:𝐶1-1𝐷𝐺:𝐴1-1𝐵) → (𝐹𝐺):(𝐺𝐶)–1-1𝐷)
 
Theoremf1co 6676 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:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → (𝐹𝐺):𝐴1-1𝐶)
 
Theoremf1coOLD 6677 Obsolete version of f1co 6676 as of 20-Sep-2024. (Contributed by NM, 28-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → (𝐹𝐺):𝐴1-1𝐶)
 
Theoremfoeq1 6678 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵))
 
Theoremfoeq2 6679 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐴onto𝐶𝐹:𝐵onto𝐶))
 
Theoremfoeq3 6680 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐶onto𝐴𝐹:𝐶onto𝐵))
 
Theoremnffo 6681 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴onto𝐵
 
Theoremfof 6682 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
 
Theoremfofun 6683 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
(𝐹:𝐴onto𝐵 → Fun 𝐹)
 
Theoremfofn 6684 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
(𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
 
Theoremforn 6685 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
 
Theoremdffo2 6686 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
 
Theoremfoima 6687 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
(𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
 
Theoremdffn4 6688 A function maps onto its range. (Contributed by NM, 10-May-1998.)
(𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
 
Theoremfunforn 6689 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
(Fun 𝐴𝐴:dom 𝐴onto→ran 𝐴)
 
Theoremfodmrnu 6690 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremfimadmfo 6691 A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.)
(𝐹:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))
 
Theoremfores 6692 Restriction of an onto function. (Contributed by NM, 4-Mar-1997.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
 
TheoremfimadmfoALT 6693 Alternate proof of fimadmfo 6691, based on fores 6692. 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:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))
 
Theoremfocnvimacdmdm 6694 The preimage of the codomain of a surjection is its domain. (Contributed by AV, 29-Sep-2024.)
(𝐺:𝐴onto𝐵 → (𝐺𝐵) = 𝐴)
 
Theoremfocofo 6695 Composition of onto functions. Generalisation of foco 6696. (Contributed by AV, 29-Sep-2024.)
((𝐹:𝐴onto𝐵 ∧ Fun 𝐺𝐴 ⊆ ran 𝐺) → (𝐹𝐺):(𝐺𝐴)–onto𝐵)
 
Theoremfoco 6696 Composition of onto functions. (Contributed by NM, 22-Mar-2006.) (Proof shortened by AV, 29-Sep-2024.)
((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)
 
Theoremfoconst 6697 A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.)
((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴onto→{𝐵})
 
Theoremf1oeq1 6698 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 → (𝐹:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))
 
Theoremf1oeq2 6699 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
 
Theoremf1oeq3 6700 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1-onto𝐵))
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