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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ffnfv 7001* | A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.) |
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | ||
Theorem | ffnfvf 7002 | A function maps to a class to which all values belong. This version of ffnfv 7001 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝐹 ⇒ ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | ||
Theorem | fnfvrnss 7003* | An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.) |
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵) | ||
Theorem | frnssb 7004* | A function is a function into a subset of its codomain if all of its values are elements of this subset. (Contributed by AV, 7-Feb-2021.) |
⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → (𝐹:𝐴⟶𝑊 ↔ 𝐹:𝐴⟶𝑉)) | ||
Theorem | rnmptss 7005* | The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶) | ||
Theorem | fmpt2d 7006* | Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | ||
Theorem | ffvresb 7007* | A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ (Fun 𝐹 → ((𝐹 ↾ 𝐴):𝐴⟶𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵))) | ||
Theorem | f1oresrab 7008* | Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝜒 ↔ 𝜓)) ⇒ ⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | f1ossf1o 7009* | Restricting a bijection, which is a mapping from a restricted class abstraction, to a subset is a bijection. (Contributed by AV, 7-Aug-2022.) |
⊢ 𝑋 = {𝑤 ∈ 𝐴 ∣ (𝜓 ∧ 𝜒)} & ⊢ 𝑌 = {𝑤 ∈ 𝐴 ∣ 𝜓} & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝑌 ↦ 𝐵) & ⊢ (𝜑 → 𝐺:𝑌–1-1-onto→𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 = 𝐵) → (𝜏 ↔ [𝑥 / 𝑤]𝜒)) ⇒ ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→{𝑦 ∈ 𝐶 ∣ 𝜏}) | ||
Theorem | fmptco 7010* | Composition of two functions expressed as ordered-pair class abstractions. If 𝐹 has the equation (𝑥 + 2) and 𝐺 the equation (3∗𝑧) then (𝐺 ∘ 𝐹) has the equation (3∗(𝑥 + 2)). (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅)) & ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆)) & ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇)) | ||
Theorem | fmptcof 7011* | Version of fmptco 7010 where 𝜑 needn't be distinct from 𝑥. (Contributed by NM, 27-Dec-2014.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅)) & ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆)) & ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇)) | ||
Theorem | fmptcos 7012* | Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅)) & ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆)) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆)) | ||
Theorem | cofmpt 7013* | Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.) |
⊢ (𝜑 → 𝐹:𝐶⟶𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))) | ||
Theorem | fcompt 7014* | Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.) |
⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥)))) | ||
Theorem | fcoconst 7015 | Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.) |
⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹‘𝑌)})) | ||
Theorem | fsn 7016 | A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {〈𝐴, 𝐵〉}) | ||
Theorem | fsn2 7017 | A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) | ||
Theorem | fsng 7018 | A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {〈𝐴, 𝐵〉})) | ||
Theorem | fsn2g 7019 | A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by Thierry Arnoux, 11-Jul-2020.) |
⊢ (𝐴 ∈ 𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}))) | ||
Theorem | xpsng 7020 | The Cartesian product of two singletons is the singleton consisting in the associated ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) | ||
Theorem | xpprsng 7021 | The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐴, 𝐵} × {𝐶}) = {〈𝐴, 𝐶〉, 〈𝐵, 𝐶〉}) | ||
Theorem | xpsn 7022 | The Cartesian product of two singletons is the singleton consisting in the associated ordered pair. (Contributed by NM, 4-Nov-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉} | ||
Theorem | f1o2sn 7023 | A singleton consisting in a nested ordered pair is a one-to-one function from the cartesian product of two singletons onto a singleton (case where the two singletons are equal). (Contributed by AV, 15-Aug-2019.) |
⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {〈〈𝐸, 𝐸〉, 𝑋〉}:({𝐸} × {𝐸})–1-1-onto→{𝑋}) | ||
Theorem | residpr 7024 | Restriction of the identity to a pair. (Contributed by AV, 11-Dec-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ( I ↾ {𝐴, 𝐵}) = {〈𝐴, 𝐴〉, 〈𝐵, 𝐵〉}) | ||
Theorem | dfmpt 7025 | Alternate definition for the maps-to notation df-mpt 5159 (although it requires that 𝐵 be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} | ||
Theorem | fnasrn 7026 | A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) | ||
Theorem | idref 7027* | Two ways to state that a relation is reflexive on a class. (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Revised by NM, 30-Mar-2016.) |
⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) | ||
Theorem | funiun 7028* | A function is a union of singletons of ordered pairs indexed by its domain. (Contributed by AV, 18-Sep-2020.) |
⊢ (Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{〈𝑥, (𝐹‘𝑥)〉}) | ||
Theorem | funopsn 7029* | If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 15-Jul-2021.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6492, as relsnopg 5715 is to relop 5762. (New usage is discouraged.) |
⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V ⇒ ⊢ ((Fun 𝐹 ∧ 𝐹 = 〈𝑋, 𝑌〉) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {〈𝑎, 𝑎〉})) | ||
Theorem | funop 7030* | An ordered pair is a function iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6492, as relsnopg 5715 is to relop 5762. (New usage is discouraged.) |
⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V ⇒ ⊢ (Fun 〈𝑋, 𝑌〉 ↔ ∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉})) | ||
Theorem | funopdmsn 7031 | The domain of a function which is an ordered pair is a singleton. (Contributed by AV, 15-Nov-2021.) (Avoid depending on this detail.) |
⊢ 𝐺 = 〈𝑋, 𝑌〉 & ⊢ 𝑋 ∈ 𝑉 & ⊢ 𝑌 ∈ 𝑊 ⇒ ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵) | ||
Theorem | funsndifnop 7032 | A singleton of an ordered pair is not an ordered pair if the components are different. (Contributed by AV, 23-Sep-2020.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐺 = {〈𝐴, 𝐵〉} ⇒ ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐺 ∈ (V × V)) | ||
Theorem | funsneqopb 7033 | A singleton of an ordered pair is an ordered pair iff the components are equal. (Contributed by AV, 24-Sep-2020.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐺 = {〈𝐴, 𝐵〉} ⇒ ⊢ (𝐴 = 𝐵 ↔ 𝐺 ∈ (V × V)) | ||
Theorem | ressnop0 7034 | If 𝐴 is not in 𝐶, then the restriction of a singleton of 〈𝐴, 𝐵〉 to 𝐶 is null. (Contributed by Scott Fenton, 15-Apr-2011.) |
⊢ (¬ 𝐴 ∈ 𝐶 → ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅) | ||
Theorem | fpr 7035 | A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (𝐴 ≠ 𝐵 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶{𝐶, 𝐷}) | ||
Theorem | fprg 7036 | A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.) |
⊢ (((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐹) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐻) ∧ 𝐴 ≠ 𝐵) → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶{𝐶, 𝐷}) | ||
Theorem | ftpg 7037 | A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → {〈𝑋, 𝐴〉, 〈𝑌, 𝐵〉, 〈𝑍, 𝐶〉}:{𝑋, 𝑌, 𝑍}⟶{𝐴, 𝐵, 𝐶}) | ||
Theorem | ftp 7038 | A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by Alexander van der Vekens, 23-Jan-2018.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V & ⊢ 𝑍 ∈ V & ⊢ 𝐴 ≠ 𝐵 & ⊢ 𝐴 ≠ 𝐶 & ⊢ 𝐵 ≠ 𝐶 ⇒ ⊢ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉, 〈𝐶, 𝑍〉}:{𝐴, 𝐵, 𝐶}⟶{𝑋, 𝑌, 𝑍} | ||
Theorem | fnressn 7039 | A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 ↾ {𝐵}) = {〈𝐵, (𝐹‘𝐵)〉}) | ||
Theorem | funressn 7040 | A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ (Fun 𝐹 → (𝐹 ↾ {𝐵}) ⊆ {〈𝐵, (𝐹‘𝐵)〉}) | ||
Theorem | fressnfv 7041 | The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶)) | ||
Theorem | fvrnressn 7042 | 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 7043 | 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 | fvn0fvelrn 7044 | If the value of a function is not null, the value is an element of the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.) |
⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹) | ||
Theorem | fvconst 7045 | The value of a constant function. (Contributed by NM, 30-May-1999.) |
⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) = 𝐵) | ||
Theorem | fnsnr 7046 | 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 7047 | 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 7048* | 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 7049* | Express a singleton function in maps-to notation. Version of fmptsn 7048 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 7050* | Express a singleton function in maps-to notation. Deduction form of fmptsng 7049. (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 7051* | 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 7052* | 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 7053* | 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 7054 | The value of a restricted identity function. (Contributed by NM, 19-May-2004.) |
⊢ (𝐵 ∈ 𝐴 → (( I ↾ 𝐴)‘𝐵) = 𝐵) | ||
Theorem | fninfp 7055* | 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 7056 | 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 7057* | 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 7058 | 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 7059 | 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 7060 | 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 7061 | 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 7062 | 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 7063 | The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7064. (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 7064 | The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 7063. (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 7065 | 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 7066 | 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 7067 | 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 7068 | 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 7069 | 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 7070 | 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 7071 | 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 7072 | 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 7073 | Obsolete version of fvpr2g 7072 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 7074 | 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 7075 | Obsolete version of fvpr1 7074 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 7076 | 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 7077 | Obsolete version of fvpr2 7076 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 7078* | 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 7079 | 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 7080 | 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 7081 | 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 7082 | 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 7083 | 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 7084 | 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 7085 | 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 7086 | The value of a constant function. (Contributed by NM, 20-Aug-2005.) |
⊢ ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵) | ||
Theorem | fconst2g 7087 | A constant function expressed as a Cartesian product. (Contributed by NM, 27-Nov-2007.) |
⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))) | ||
Theorem | fvconst2 7088 | The value of a constant function. (Contributed by NM, 16-Apr-2005.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐶 ∈ 𝐴 → ((𝐴 × {𝐵})‘𝐶) = 𝐵) | ||
Theorem | fconst2 7089 | A constant function expressed as a Cartesian product. (Contributed by NM, 20-Aug-1999.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})) | ||
Theorem | fconst5 7090 | 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 7091* | 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 7092* | Obsolete version of rnmptc 7091 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 7093 | 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 7094 | 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 7095 | 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 7096 | 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 7097* | A constant function expressed in terms of its functionality, domain, and value. See also fconst2 7089. (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 7098 | Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.) |
⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵}))) | ||
Theorem | fconst4 7099 | Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.) |
⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (◡𝐹 “ {𝐵}) = 𝐴)) | ||
Theorem | resfunexg 7100 | 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) |
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