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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | frn 6601 | The range of a mapping. (Contributed by NM, 3-Aug-1994.) |
⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | ||
Theorem | frnd 6602 | Deduction form of frn 6601. The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵) | ||
Theorem | fdm 6603 | The domain of a mapping. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 29-May-2024.) |
⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | ||
Theorem | fdmOLD 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:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | ||
Theorem | fdmd 6605 | Deduction form of fdm 6603. The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → dom 𝐹 = 𝐴) | ||
Theorem | fdmi 6606 | Inference associated with fdm 6603. The domain of a mapping. (Contributed by NM, 28-Jul-2008.) |
⊢ 𝐹:𝐴⟶𝐵 ⇒ ⊢ dom 𝐹 = 𝐴 | ||
Theorem | dffn3 6607 | A function maps to its range. (Contributed by NM, 1-Sep-1999.) |
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹) | ||
Theorem | ffrn 6608 | A function maps to its range. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹) | ||
Theorem | ffrnb 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:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) | ||
Theorem | ffrnbd 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 𝐹 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹)) | ||
Theorem | fss 6611 | Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) | ||
Theorem | fssd 6612 | Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | ||
Theorem | fssdmd 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.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹) ⇒ ⊢ (𝜑 → 𝐷 ⊆ 𝐴) | ||
Theorem | fssdm 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 & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → 𝐷 ⊆ 𝐴) | ||
Theorem | fimass 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:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵) | ||
Theorem | fimacnv 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:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) | ||
Theorem | fcof 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:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵) | ||
Theorem | fco 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:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | ||
Theorem | fcoOLD 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:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | ||
Theorem | fcod 6620 | Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | ||
Theorem | fco2 6621 | Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.) |
⊢ (((𝐹 ↾ 𝐵):𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | ||
Theorem | fssxp 6622 | A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | ||
Theorem | funssxp 6623 | Two ways of specifying a partial function from 𝐴 to 𝐵. (Contributed by NM, 13-Nov-2007.) |
⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴)) | ||
Theorem | ffdm 6624 | A mapping is a partial function. (Contributed by NM, 25-Nov-2007.) |
⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴)) | ||
Theorem | ffdmd 6625 | The domain of a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → 𝐹:dom 𝐹⟶𝐵) | ||
Theorem | fdmrn 6626 | A different way to write 𝐹 is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.) |
⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹) | ||
Theorem | funcofd 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 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹) | ||
Theorem | fco3OLD 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 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹) | ||
Theorem | opelf 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:𝐴⟶𝐵 ∧ 〈𝐶, 𝐷〉 ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | ||
Theorem | fun 6630 | The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.) |
⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷)) | ||
Theorem | fun2 6631 | The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.) |
⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) | ||
Theorem | fun2d 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.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) | ||
Theorem | fnfco 6633 | Composition of two functions. (Contributed by NM, 22-May-2006.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) | ||
Theorem | fssres 6634 | Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) | ||
Theorem | fssresd 6635 | Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵) | ||
Theorem | fssres2 6636 | Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.) |
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) | ||
Theorem | fresin 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:𝐴⟶𝐵 → (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵) | ||
Theorem | resasplit 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 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))) | ||
Theorem | fresaun 6639 | The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) | ||
Theorem | fresaunres2 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:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺) | ||
Theorem | fresaunres1 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:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹) | ||
Theorem | fcoi1 6642 | Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) | ||
Theorem | fcoi2 6643 | Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ (𝐹:𝐴⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹) | ||
Theorem | feu 6644* | There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 〈𝐶, 𝑦〉 ∈ 𝐹) | ||
Theorem | fcnvres 6645 | The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.) |
⊢ (𝐹:𝐴⟶𝐵 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ 𝐵)) | ||
Theorem | fimacnvdisj 6646 | The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (◡𝐹 “ 𝐶) = ∅) | ||
Theorem | fint 6647* | Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ 𝐵 ≠ ∅ ⇒ ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥) | ||
Theorem | fin 6648 | Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ (𝐹:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴⟶𝐶)) | ||
Theorem | f0 6649 | The empty function. (Contributed by NM, 14-Aug-1999.) |
⊢ ∅:∅⟶𝐴 | ||
Theorem | f00 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 = ∅)) | ||
Theorem | f0bi 6651 | A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.) |
⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅) | ||
Theorem | f0dom0 6652 | A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.) |
⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅)) | ||
Theorem | f0rn0 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:𝑋⟶𝑌 ∧ ¬ ∃𝑦 ∈ 𝑌 𝑦 ∈ ran 𝐸) → 𝑋 = ∅) | ||
Theorem | fconst 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} | ||
Theorem | fconstg 6655 | A Cartesian product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) | ||
Theorem | fnconstg 6656 | A Cartesian product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}) Fn 𝐴) | ||
Theorem | fconst6g 6657 | Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶𝐶) | ||
Theorem | fconst6 6658 | A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.) |
⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ (𝐴 × {𝐵}):𝐴⟶𝐶 | ||
Theorem | f1eq1 6659 | Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.) |
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵)) | ||
Theorem | f1eq2 6660 | Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.) |
⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶)) | ||
Theorem | f1eq3 6661 | Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.) |
⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵)) | ||
Theorem | nff1 6662 | Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵 | ||
Theorem | dff12 6663* | Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.) |
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦)) | ||
Theorem | f1f 6664 | A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.) |
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | ||
Theorem | f1fn 6665 | A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.) |
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) | ||
Theorem | f1fun 6666 | A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.) |
⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹) | ||
Theorem | f1rel 6667 | A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.) |
⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹) | ||
Theorem | f1dm 6668 | 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→𝐵 → dom 𝐹 = 𝐴) | ||
Theorem | f1dmOLD 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:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴) | ||
Theorem | f1ss 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:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶) | ||
Theorem | f1ssr 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:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶) | ||
Theorem | f1ssres 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:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵) | ||
Theorem | f1resf1 6673 | The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷) | ||
Theorem | f1cnvcnv 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 𝐴–1-1→V ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴)) | ||
Theorem | f1cof1 6675 | Composition of two one-to-one functions. Generalization of f1co 6676. (Contributed by AV, 18-Sep-2024.) |
⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)–1-1→𝐷) | ||
Theorem | f1co 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:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) | ||
Theorem | f1coOLD 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:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) | ||
Theorem | foeq1 6678 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵)) | ||
Theorem | foeq2 6679 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) | ||
Theorem | foeq3 6680 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵)) | ||
Theorem | nffo 6681 | Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 | ||
Theorem | fof 6682 | An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.) |
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | ||
Theorem | fofun 6683 | An onto mapping is a function. (Contributed by NM, 29-Mar-2008.) |
⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) | ||
Theorem | fofn 6684 | An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.) |
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | ||
Theorem | forn 6685 | The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.) |
⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | ||
Theorem | dffo2 6686 | Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.) |
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) | ||
Theorem | foima 6687 | The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.) |
⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) | ||
Theorem | dffn4 6688 | A function maps onto its range. (Contributed by NM, 10-May-1998.) |
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | ||
Theorem | funforn 6689 | A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.) |
⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴) | ||
Theorem | fodmrnu 6690 | An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.) |
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | fimadmfo 6691 | A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.) |
⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) | ||
Theorem | fores 6692 | Restriction of an onto function. (Contributed by NM, 4-Mar-1997.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) | ||
Theorem | fimadmfoALT 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:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴)) | ||
Theorem | focnvimacdmdm 6694 | The preimage of the codomain of a surjection is its domain. (Contributed by AV, 29-Sep-2024.) |
⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴) | ||
Theorem | focofo 6695 | Composition of onto functions. Generalisation of foco 6696. (Contributed by AV, 29-Sep-2024.) |
⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵) | ||
Theorem | foco 6696 | Composition of onto functions. (Contributed by NM, 22-Mar-2006.) (Proof shortened by AV, 29-Sep-2024.) |
⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶) | ||
Theorem | foconst 6697 | A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.) |
⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴–onto→{𝐵}) | ||
Theorem | f1oeq1 6698 | Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵)) | ||
Theorem | f1oeq2 6699 | Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶)) | ||
Theorem | f1oeq3 6700 | Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵)) |
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