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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ffun 6201 | A mapping is a function. (Contributed by NM, 3-Aug-1994.) |
⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | ||
Theorem | ffund 6202 | A mapping is a function, deduction version. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
Theorem | frel 6203 | A mapping is a relation. (Contributed by NM, 3-Aug-1994.) |
⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹) | ||
Theorem | fdm 6204 | The domain of a mapping. (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | ||
Theorem | fdmi 6205 | The domain of a mapping. (Contributed by NM, 28-Jul-2008.) |
⊢ 𝐹:𝐴⟶𝐵 ⇒ ⊢ dom 𝐹 = 𝐴 | ||
Theorem | frn 6206 | The range of a mapping. (Contributed by NM, 3-Aug-1994.) |
⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | ||
Theorem | dffn3 6207 | A function maps to its range. (Contributed by NM, 1-Sep-1999.) |
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹) | ||
Theorem | ffrn 6208 | A function maps to its range. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹) | ||
Theorem | fss 6209 | 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 6210 | 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 | fco 6211 | Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | ||
Theorem | fco2 6212 | 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 6213 | 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 6214 | 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 6215 | A mapping is a partial function. (Contributed by NM, 25-Nov-2007.) |
⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴)) | ||
Theorem | ffdmd 6216 | The domain of a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → 𝐹:dom 𝐹⟶𝐵) | ||
Theorem | fdmrn 6217 | A different way to write 𝐹 is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.) |
⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹) | ||
Theorem | opelf 6218 | 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 6219 | 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 6220 | 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 6221 | The union of functions with disjoint domains is a function, deduction version of fun2 6220. (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 6222 | Composition of two functions. (Contributed by NM, 22-May-2006.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) | ||
Theorem | fssres 6223 | 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 6224 | 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 6225 | 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 6226 | 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 6227 | 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 6228 | 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 6229 | 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 6230 | 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 6231 | 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 6232 | 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 6233* | 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 | fimass 6234 | The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵) | ||
Theorem | fcnvres 6235 | The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.) |
⊢ (𝐹:𝐴⟶𝐵 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ 𝐵)) | ||
Theorem | fimacnvdisj 6236 | 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 6237* | 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 6238 | 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 6239 | The empty function. (Contributed by NM, 14-Aug-1999.) |
⊢ ∅:∅⟶𝐴 | ||
Theorem | f00 6240 | 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 6241 | A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.) |
⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅) | ||
Theorem | f0dom0 6242 | A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.) |
⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅)) | ||
Theorem | f0rn0 6243* | 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 6244 | 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 6245 | A Cartesian product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) | ||
Theorem | fnconstg 6246 | A Cartesian product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}) Fn 𝐴) | ||
Theorem | fconst6g 6247 | Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶𝐶) | ||
Theorem | fconst6 6248 | 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 6249 | 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 6250 | 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 6251 | 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 6252 | 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 6253* | 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 6254 | A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.) |
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | ||
Theorem | f1fn 6255 | 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 6256 | A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.) |
⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹) | ||
Theorem | f1rel 6257 | A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.) |
⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹) | ||
Theorem | f1dm 6258 | The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.) |
⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴) | ||
Theorem | f1ss 6259 | 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 6260 | 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 6261 | A function that is one-to-one is also one-to-one on some subset of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵) | ||
Theorem | f1cnvcnv 6262 | 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 | f1co 6263 | Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.) |
⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) | ||
Theorem | foeq1 6264 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵)) | ||
Theorem | foeq2 6265 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) | ||
Theorem | foeq3 6266 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵)) | ||
Theorem | nffo 6267 | Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 | ||
Theorem | fof 6268 | An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.) |
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | ||
Theorem | fofun 6269 | An onto mapping is a function. (Contributed by NM, 29-Mar-2008.) |
⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) | ||
Theorem | fofn 6270 | An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.) |
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | ||
Theorem | forn 6271 | The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.) |
⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | ||
Theorem | dffo2 6272 | Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.) |
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) | ||
Theorem | foima 6273 | The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.) |
⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) | ||
Theorem | dffn4 6274 | A function maps onto its range. (Contributed by NM, 10-May-1998.) |
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | ||
Theorem | funforn 6275 | A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.) |
⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴) | ||
Theorem | fodmrnu 6276 | 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 | fores 6277 | Restriction of an onto function. (Contributed by NM, 4-Mar-1997.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) | ||
Theorem | foco 6278 | Composition of onto functions. (Contributed by NM, 22-Mar-2006.) |
⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶) | ||
Theorem | foconst 6279 | A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.) |
⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴–onto→{𝐵}) | ||
Theorem | f1oeq1 6280 | 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 6281 | 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 6282 | 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)) | ||
Theorem | f1oeq23 6283 | Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷)) | ||
Theorem | f1eq123d 6284 | Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷)) | ||
Theorem | foeq123d 6285 | Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐹:𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐷)) | ||
Theorem | f1oeq123d 6286 | Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷)) | ||
Theorem | f1oeq3d 6287 | Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵)) | ||
Theorem | nff1o 6288 | Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐹:𝐴–1-1-onto→𝐵 | ||
Theorem | f1of1 6289 | A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵) | ||
Theorem | f1of 6290 | A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵) | ||
Theorem | f1ofn 6291 | A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴) | ||
Theorem | f1ofun 6292 | A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) | ||
Theorem | f1orel 6293 | A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) | ||
Theorem | f1odm 6294 | The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴) | ||
Theorem | dff1o2 6295 | 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→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵)) | ||
Theorem | dff1o3 6296 | 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→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹)) | ||
Theorem | f1ofo 6297 | A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵) | ||
Theorem | dff1o4 6298 | 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→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) | ||
Theorem | dff1o5 6299 | 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→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵)) | ||
Theorem | f1orn 6300 | A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.) |
⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) |
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