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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | f0bi 6001 | A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.) |
⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅) | ||
Theorem | f0dom0 6002 | A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.) |
⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅)) | ||
Theorem | f0rn0 6003* | 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 6004 | 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 6005 | A Cartesian product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) | ||
Theorem | fnconstg 6006 | A Cartesian product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}) Fn 𝐴) | ||
Theorem | fconst6g 6007 | Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶𝐶) | ||
Theorem | fconst6 6008 | 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 6009 | 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 6010 | 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 6011 | 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 6012 | 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 6013* | 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 6014 | A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.) |
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | ||
Theorem | f1fn 6015 | 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 6016 | A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.) |
⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹) | ||
Theorem | f1rel 6017 | A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.) |
⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹) | ||
Theorem | f1dm 6018 | The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.) |
⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴) | ||
Theorem | f1ss 6019 | 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 6020 | 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 6021 | 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 6022 | 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 6023 | 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 6024 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵)) | ||
Theorem | foeq2 6025 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) | ||
Theorem | foeq3 6026 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵)) | ||
Theorem | nffo 6027 | Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 | ||
Theorem | fof 6028 | An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.) |
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | ||
Theorem | fofun 6029 | An onto mapping is a function. (Contributed by NM, 29-Mar-2008.) |
⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) | ||
Theorem | fofn 6030 | An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.) |
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | ||
Theorem | forn 6031 | The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.) |
⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | ||
Theorem | dffo2 6032 | Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.) |
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) | ||
Theorem | foima 6033 | The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.) |
⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) | ||
Theorem | dffn4 6034 | A function maps onto its range. (Contributed by NM, 10-May-1998.) |
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | ||
Theorem | funforn 6035 | A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.) |
⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴) | ||
Theorem | fodmrnu 6036 | 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 6037 | Restriction of an onto function. (Contributed by NM, 4-Mar-1997.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) | ||
Theorem | foco 6038 | 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 6039 | A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.) |
⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴–onto→{𝐵}) | ||
Theorem | f1oeq1 6040 | 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 6041 | 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 6042 | 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 6043 | 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 6044 | 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 6045 | 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 6046 | 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 6047 | 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 6048 | 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 6049 | 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 6050 | 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 6051 | 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 6052 | 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 6053 | 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 6054 | 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 6055 | 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 6056 | 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 6057 | 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 6058 | 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 6059 | 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 6060 | 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)) | ||
Theorem | f1f1orn 6061 | A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.) |
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹) | ||
Theorem | f1ocnv 6062 | 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→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | ||
Theorem | f1ocnvb 6063 | A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.) |
⊢ (Rel 𝐹 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹:𝐵–1-1-onto→𝐴)) | ||
Theorem | f1ores 6064 | 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→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) | ||
Theorem | f1orescnv 6065 | The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.) |
⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → (◡𝐹 ↾ 𝑃):𝑃–1-1-onto→𝑅) | ||
Theorem | f1imacnv 6066 | Preimage of an image. (Contributed by NM, 30-Sep-2004.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝐶)) = 𝐶) | ||
Theorem | foimacnv 6067 | A reverse version of f1imacnv 6066. (Contributed by Jeff Hankins, 16-Jul-2009.) |
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝐶)) = 𝐶) | ||
Theorem | foun 6068 | The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.) |
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–onto→(𝐵 ∪ 𝐷)) | ||
Theorem | f1oun 6069 | 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→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷)) | ||
Theorem | resdif 6070 | The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.) |
⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐶 ∖ 𝐷)) | ||
Theorem | resin 6071 | The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.) |
⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∩ 𝐷)) | ||
Theorem | f1oco 6072 | Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.) |
⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝐺:𝐴–1-1-onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1-onto→𝐶) | ||
Theorem | f1cnv 6073 | The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.) |
⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴) | ||
Theorem | funcocnv2 6074 | Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) | ||
Theorem | fococnv2 6075 | The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) | ||
Theorem | f1ococnv2 6076 | The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) | ||
Theorem | f1cocnv2 6077 | Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.) |
⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) | ||
Theorem | f1ococnv1 6078 | The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) | ||
Theorem | f1cocnv1 6079 | Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.) |
⊢ (𝐹:𝐴–1-1→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) | ||
Theorem | funcoeqres 6080 | Re-express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ ((Fun 𝐺 ∧ (𝐹 ∘ 𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻 ∘ ◡𝐺)) | ||
Theorem | f10 6081 | The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.) |
⊢ ∅:∅–1-1→𝐴 | ||
Theorem | f10d 6082 | The empty set maps one-to-one into any class, deduction version. (Contributed by AV, 25-Nov-2020.) |
⊢ (𝜑 → 𝐹 = ∅) ⇒ ⊢ (𝜑 → 𝐹:dom 𝐹–1-1→𝐴) | ||
Theorem | f1o00 6083 | One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.) |
⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) | ||
Theorem | fo00 6084 | Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.) |
⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) | ||
Theorem | f1o0 6085 | One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.) |
⊢ ∅:∅–1-1-onto→∅ | ||
Theorem | f1oi 6086 | A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | ||
Theorem | f1ovi 6087 | The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.) |
⊢ I :V–1-1-onto→V | ||
Theorem | f1osn 6088 | A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} | ||
Theorem | f1osng 6089 | A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) | ||
Theorem | f1sng 6090 | A singleton of an ordered pair is a one-to-one function. (Contributed by AV, 17-Apr-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) | ||
Theorem | fsnd 6091 | A singleton of an ordered pair is a function. (Contributed by AV, 17-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → {〈𝐴, 𝐵〉}:{𝐴}⟶𝑊) | ||
Theorem | f1oprswap 6092 | A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉, 〈𝐵, 𝐴〉}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵}) | ||
Theorem | f1oprg 6093 | An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 6092. (Contributed by Alexander van der Vekens, 14-Aug-2017.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) → {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉}:{𝐴, 𝐶}–1-1-onto→{𝐵, 𝐷})) | ||
Theorem | tz6.12-2 6094* | Function value when 𝐹 is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅) | ||
Theorem | fveu 6095* | The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.) |
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) | ||
Theorem | brprcneu 6096* | If 𝐴 is a proper class, then there is no unique binary relationship with 𝐴 as the first element. (Contributed by Scott Fenton, 7-Oct-2017.) |
⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥) | ||
Theorem | fvprc 6097 | A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.) |
⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | ||
Theorem | fv2 6098* | Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥)} | ||
Theorem | dffv3 6099* | A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.) |
⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) | ||
Theorem | dffv4 6100* | The previous definition of function value, from before the ℩ operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 5412), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} |
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