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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fvrnressn 7101 | If the value of a function is in the range of the function restricted to the singleton containing the argument, then the value of the function is in the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.) |
⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹)) | ||
Theorem | fvressn 7102 | The value of a function restricted to the singleton containing the argument equals the value of the function for this argument. (Contributed by Alexander van der Vekens, 22-Jul-2018.) |
⊢ (𝑋 ∈ 𝑉 → ((𝐹 ↾ {𝑋})‘𝑋) = (𝐹‘𝑋)) | ||
Theorem | fvn0fvelrnOLD 7103 | Obsolete version of fvn0fvelrn 6868 as of 13-Jan-2025. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹) | ||
Theorem | fvconst 7104 | The value of a constant function. (Contributed by NM, 30-May-1999.) |
⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) = 𝐵) | ||
Theorem | fnsnr 7105 | If a class belongs to a function on a singleton, then that class is the obvious ordered pair. Note that this theorem also holds when 𝐴 is a proper class, but its meaning is then different. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) |
⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 = ⟨𝐴, (𝐹‘𝐴)⟩)) | ||
Theorem | fnsnb 7106 | A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) | ||
Theorem | fmptsn 7107* | Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 28-Feb-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)) | ||
Theorem | fmptsng 7108* | Express a singleton function in maps-to notation. Version of fmptsn 7107 allowing the value 𝐵 to depend on the variable 𝑥. (Contributed by AV, 27-Feb-2019.) |
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)) | ||
Theorem | fmptsnd 7109* | Express a singleton function in maps-to notation. Deduction form of fmptsng 7108. (Contributed by AV, 4-Aug-2019.) |
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)) | ||
Theorem | fmptap 7110* | Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑅 ∪ {𝐴}) = 𝑆 & ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐵) ⇒ ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶) | ||
Theorem | fmptapd 7111* | Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) | ||
Theorem | fmptpr 7112* | Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐸 = 𝐶) & ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐸 = 𝐷) ⇒ ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸)) | ||
Theorem | fvresi 7113 | The value of a restricted identity function. (Contributed by NM, 19-May-2004.) |
⊢ (𝐵 ∈ 𝐴 → (( I ↾ 𝐴)‘𝐵) = 𝐵) | ||
Theorem | fninfp 7114* | Express the class of fixed points of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥}) | ||
Theorem | fnelfp 7115 | Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑋) = 𝑋)) | ||
Theorem | fndifnfp 7116* | Express the class of non-fixed points of a function. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥}) | ||
Theorem | fnelnfp 7117 | Property of a non-fixed point of a function. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑋) ≠ 𝑋)) | ||
Theorem | fnnfpeq0 7118 | A function is the identity iff it moves no points. (Contributed by Stefan O'Rear, 25-Aug-2015.) |
⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ 𝐹 = ( I ↾ 𝐴))) | ||
Theorem | fvunsn 7119 | Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.) |
⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴‘𝐷)) | ||
Theorem | fvsng 7120 | The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.) (Proof shortened by BJ, 25-Feb-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵) | ||
Theorem | fvsn 7121 | The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.) (Proof shortened by BJ, 25-Feb-2023.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵 | ||
Theorem | fvsnun1 7122 | The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7123. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 25-Feb-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ⇒ ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵) | ||
Theorem | fvsnun2 7123 | The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 7122. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 25-Feb-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) & ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∖ {𝐴})) ⇒ ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷)) | ||
Theorem | fnsnsplit 7124 | Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩})) | ||
Theorem | fsnunf 7125 | Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇) | ||
Theorem | fsnunf2 7126 | Adjoining a point to a punctured function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):𝑆⟶𝑇) | ||
Theorem | fsnunfv 7127 | Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌) | ||
Theorem | fsnunres 7128 | Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹) | ||
Theorem | funresdfunsn 7129 | Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself. (Contributed by AV, 2-Dec-2018.) |
⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = 𝐹) | ||
Theorem | fvpr1g 7130 | The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶) | ||
Theorem | fvpr2g 7131 | The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Proof shortened by BJ, 26-Sep-2024.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷) | ||
Theorem | fvpr2gOLD 7132 | Obsolete version of fvpr2g 7131 as of 26-Sep-2024. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷) | ||
Theorem | fvpr1 7133 | The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by BJ, 26-Sep-2024.) |
⊢ 𝐴 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶) | ||
Theorem | fvpr1OLD 7134 | Obsolete version of fvpr1 7133 as of 26-Sep-2024. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶) | ||
Theorem | fvpr2 7135 | The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by BJ, 26-Sep-2024.) |
⊢ 𝐵 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷) | ||
Theorem | fvpr2OLD 7136 | Obsolete version of fvpr2 7135 as of 26-Sep-2024. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐵 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷) | ||
Theorem | fprb 7137* | A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≠ 𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩})) | ||
Theorem | fvtp1 7138 | The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = 𝐷) | ||
Theorem | fvtp2 7139 | The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
⊢ 𝐵 ∈ V & ⊢ 𝐸 ∈ V ⇒ ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐵) = 𝐸) | ||
Theorem | fvtp3 7140 | The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
⊢ 𝐶 ∈ V & ⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹) | ||
Theorem | fvtp1g 7141 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = 𝐷) | ||
Theorem | fvtp2g 7142 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐵) = 𝐸) | ||
Theorem | fvtp3g 7143 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
⊢ (((𝐶 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹) | ||
Theorem | tpres 7144 | An unordered triple of ordered pairs restricted to all but one first components of the pairs is an unordered pair of ordered pairs. (Contributed by AV, 14-Mar-2020.) |
⊢ (𝜑 → 𝑇 = {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ≠ 𝐴) & ⊢ (𝜑 → 𝐶 ≠ 𝐴) ⇒ ⊢ (𝜑 → (𝑇 ↾ (V ∖ {𝐴})) = {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}) | ||
Theorem | fvconst2g 7145 | The value of a constant function. (Contributed by NM, 20-Aug-2005.) |
⊢ ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵) | ||
Theorem | fconst2g 7146 | A constant function expressed as a Cartesian product. (Contributed by NM, 27-Nov-2007.) |
⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))) | ||
Theorem | fvconst2 7147 | The value of a constant function. (Contributed by NM, 16-Apr-2005.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐶 ∈ 𝐴 → ((𝐴 × {𝐵})‘𝐶) = 𝐵) | ||
Theorem | fconst2 7148 | A constant function expressed as a Cartesian product. (Contributed by NM, 20-Aug-1999.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})) | ||
Theorem | fconst5 7149 | Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵})) | ||
Theorem | rnmptc 7150* | Range of a constant function in maps-to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Remove extra hypothesis. (Revised by SN, 17-Apr-2024.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐴 ≠ ∅) ⇒ ⊢ (𝜑 → ran 𝐹 = {𝐵}) | ||
Theorem | rnmptcOLD 7151* | Obsolete version of rnmptc 7150 as of 17-Apr-2024. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → 𝐴 ≠ ∅) ⇒ ⊢ (𝜑 → ran 𝐹 = {𝐵}) | ||
Theorem | fnprb 7152 | A function whose domain has at most two elements can be represented as a set of at most two ordered pairs. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) Eliminate unnecessary antecedent 𝐴 ≠ 𝐵. (Revised by NM, 29-Dec-2018.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) | ||
Theorem | fntpb 7153 | A function whose domain has at most three elements can be represented as a set of at most three ordered pairs. (Contributed by AV, 26-Jan-2021.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) | ||
Theorem | fnpr2g 7154 | A function whose domain has at most two elements can be represented as a set of at most two ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) | ||
Theorem | fpr2g 7155 | A function that maps a pair to a class is a pair of ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))) | ||
Theorem | fconstfv 7156* | A constant function expressed in terms of its functionality, domain, and value. See also fconst2 7148. (Contributed by NM, 27-Aug-2004.) (Proof shortened by OpenAI, 25-Mar-2020.) |
⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)) | ||
Theorem | fconst3 7157 | Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.) |
⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵}))) | ||
Theorem | fconst4 7158 | Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.) |
⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (◡𝐹 “ {𝐵}) = 𝐴)) | ||
Theorem | resfunexg 7159 | The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.) |
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V) | ||
Theorem | resiexd 7160 | The restriction of the identity relation to a set is a set. (Contributed by AV, 15-Feb-2020.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V) | ||
Theorem | fnex 7161 | If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 7159. See fnexALT 7873 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) | ||
Theorem | fnexd 7162 | If the domain of a function is a set, the function is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐹 ∈ V) | ||
Theorem | funex 7163 | If the domain of a function exists, so does the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 7161. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.) |
⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V) | ||
Theorem | opabex 7164* | Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑) ⇒ ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V | ||
Theorem | mptexg 7165* | If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | ||
Theorem | mptexgf 7166 | If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) (Revised by Thierry Arnoux, 17-May-2020.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | ||
Theorem | mptex 7167* | If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg 7165. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V | ||
Theorem | mptexd 7168* | If the domain of a function given by maps-to notation is a set, the function is a set. Deduction version of mptexg 7165. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | ||
Theorem | mptrabex 7169* | If the domain of a function given by maps-to notation is a class abstraction based on a set, the function is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↦ 𝐵) ∈ V | ||
Theorem | fex 7170 | If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝐶) → 𝐹 ∈ V) | ||
Theorem | fexd 7171 | If the domain of a mapping is a set, the function is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐹 ∈ V) | ||
Theorem | mptfvmpt 7172* | A function in maps-to notation as the value of another function in maps-to notation. (Contributed by AV, 20-Aug-2022.) |
⊢ (𝑦 = 𝑌 → 𝑀 = (𝑥 ∈ 𝑉 ↦ 𝐴)) & ⊢ 𝐺 = (𝑦 ∈ 𝑊 ↦ 𝑀) & ⊢ 𝑉 = (𝐹‘𝑋) ⇒ ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴)) | ||
Theorem | eufnfv 7173* | A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) | ||
Theorem | funfvima 7174 | A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.) |
⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) | ||
Theorem | funfvima2 7175 | A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) | ||
Theorem | funfvima2d 7176 | A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by AV, 23-Mar-2024.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴)) | ||
Theorem | fnfvima 7177 | The function value of an operand in a set is contained in the image of that set, using the Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆)) | ||
Theorem | fnfvimad 7178 | A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶)) | ||
Theorem | resfvresima 7179 | The value of the function value of a restriction for a function restricted to the image of the restricting subset. (Contributed by AV, 6-Mar-2021.) |
⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → 𝑆 ⊆ dom 𝐹) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = (𝐻‘(𝐹‘𝑋))) | ||
Theorem | funfvima3 7180 | A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.) |
⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))) | ||
Theorem | rexima 7181* | Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)) | ||
Theorem | ralima 7182* | Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)) | ||
Theorem | fvclss 7183* | Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.) |
⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅}) | ||
Theorem | elabrex 7184* | Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | ||
Theorem | abrexco 7185* | Composition of two image maps 𝐶(𝑦) and 𝐵(𝑤). (Contributed by NM, 27-May-2013.) |
⊢ 𝐵 ∈ V & ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷) ⇒ ⊢ {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤 ∈ 𝐴 𝑥 = 𝐷} | ||
Theorem | imaiun 7186* | The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶) | ||
Theorem | imauni 7187* | The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝐴 “ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝑥) | ||
Theorem | fniunfv 7188* | The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.) |
⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹) | ||
Theorem | funiunfv 7189* |
The indexed union of a function's values is the union of its image under
the index class.
Note: This theorem depends on the fact that our function value is the empty set outside of its domain. If the antecedent is changed to 𝐹 Fn 𝐴, the theorem can be proved without this dependency. (Contributed by NM, 26-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴)) | ||
Theorem | funiunfvf 7190* | The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 7189 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) (Revised by David Abernethy, 15-Apr-2013.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴)) | ||
Theorem | eluniima 7191* | Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.) |
⊢ (Fun 𝐹 → (𝐵 ∈ ∪ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (𝐹‘𝑥))) | ||
Theorem | elunirn 7192* | Membership in the union of the range of a function. See elunirnALT 7193 for a shorter proof which uses ax-pow 5318. See elfvunirn 6869 for a more general version of the reverse direction. (Contributed by NM, 24-Sep-2006.) |
⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) | ||
Theorem | elunirnALT 7193* | Alternate proof of elunirn 7192. It is shorter but requires ax-pow 5318 (through eluniima 7191, funiunfv 7189, ndmfv 6872). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) | ||
Theorem | elunirn2OLD 7194 | Obsolete version of elfvunirn 6869 as of 12-Jan-2025. (Contributed by Thierry Arnoux, 13-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → 𝐵 ∈ ∪ ran 𝐹) | ||
Theorem | fnunirn 7195* | Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝐹 Fn 𝐼 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝐼 𝐴 ∈ (𝐹‘𝑥))) | ||
Theorem | dff13 7196* | A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.) |
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) | ||
Theorem | dff13f 7197* | A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑦𝐹 ⇒ ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) | ||
Theorem | f1veqaeq 7198 | If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷)) | ||
Theorem | f1cofveqaeq 7199 | If the values of a composition of one-to-one functions for two arguments are equal, the arguments themselves must be equal. (Contributed by AV, 3-Feb-2021.) |
⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)) → 𝑋 = 𝑌)) | ||
Theorem | f1cofveqaeqALT 7200 | Alternate proof of f1cofveqaeq 7199, 1 essential step shorter, but having more bytes (305 versus 282). (Contributed by AV, 3-Feb-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)) → 𝑋 = 𝑌)) |
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