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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elfv 6101* | Membership in a function value. (Contributed by NM, 30-Apr-2004.) |
⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥))) | ||
Theorem | fveq1 6102 | Equality theorem for function value. (Contributed by NM, 29-Dec-1996.) |
⊢ (𝐹 = 𝐺 → (𝐹‘𝐴) = (𝐺‘𝐴)) | ||
Theorem | fveq2 6103 | Equality theorem for function value. (Contributed by NM, 29-Dec-1996.) |
⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵)) | ||
Theorem | fveq1i 6104 | Equality inference for function value. (Contributed by NM, 2-Sep-2003.) |
⊢ 𝐹 = 𝐺 ⇒ ⊢ (𝐹‘𝐴) = (𝐺‘𝐴) | ||
Theorem | fveq1d 6105 | Equality deduction for function value. (Contributed by NM, 2-Sep-2003.) |
⊢ (𝜑 → 𝐹 = 𝐺) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) | ||
Theorem | fveq2i 6106 | Equality inference for function value. (Contributed by NM, 28-Jul-1999.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐹‘𝐴) = (𝐹‘𝐵) | ||
Theorem | fveq2d 6107 | Equality deduction for function value. (Contributed by NM, 29-May-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵)) | ||
Theorem | fveq12i 6108 | Equality deduction for function value. (Contributed by FL, 27-Jun-2014.) |
⊢ 𝐹 = 𝐺 & ⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐹‘𝐴) = (𝐺‘𝐵) | ||
Theorem | fveq12d 6109 | Equality deduction for function value. (Contributed by FL, 22-Dec-2008.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) | ||
Theorem | nffv 6110 | Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥(𝐹‘𝐴) | ||
Theorem | nffvmpt1 6111* | Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.) |
⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐶) | ||
Theorem | nffvd 6112 | Deduction version of bound-variable hypothesis builder nffv 6110. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐹) & ⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝐴)) | ||
Theorem | fvex 6113 | The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by NM, 30-Dec-1996.) |
⊢ (𝐹‘𝐴) ∈ V | ||
Theorem | fvif 6114 | Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵)) | ||
Theorem | iffv 6115 | Move a conditional outside of a function. (Contributed by Thierry Arnoux, 28-Sep-2018.) |
⊢ (if(𝜑, 𝐹, 𝐺)‘𝐴) = if(𝜑, (𝐹‘𝐴), (𝐺‘𝐴)) | ||
Theorem | fv3 6116* | Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐹‘𝐴) = {𝑥 ∣ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)} | ||
Theorem | fvres 6117 | The value of a restricted function. (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴)) | ||
Theorem | fvresd 6118 | The value of a restricted function, deduction version of fvres 6117. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴)) | ||
Theorem | funssfv 6119 | The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.) |
⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴)) | ||
Theorem | tz6.12-1 6120* | Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) |
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) | ||
Theorem | tz6.12 6121* | Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.) |
⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦) | ||
Theorem | tz6.12f 6122* | Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.) |
⊢ Ⅎ𝑦𝐹 ⇒ ⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦) | ||
Theorem | tz6.12c 6123* | Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) |
⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) | ||
Theorem | tz6.12i 6124 | Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐵 ≠ ∅ → ((𝐹‘𝐴) = 𝐵 → 𝐴𝐹𝐵)) | ||
Theorem | fvbr0 6125 | Two possibilities for the behavior of a function value. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅) | ||
Theorem | fvrn0 6126 | A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.) |
⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) | ||
Theorem | fvssunirn 6127 | The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹 | ||
Theorem | ndmfv 6128 | The value of a class outside its domain is the empty set. (Contributed by NM, 24-Aug-1995.) |
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | ||
Theorem | ndmfvrcl 6129 | Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.) |
⊢ dom 𝐹 = 𝑆 & ⊢ ¬ ∅ ∈ 𝑆 ⇒ ⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆) | ||
Theorem | elfvdm 6130 | If a function value has a member, the argument belongs to the domain. (Contributed by NM, 12-Feb-2007.) |
⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹) | ||
Theorem | elfvex 6131 | If a function value has a member, the argument is a set. (Contributed by Mario Carneiro, 6-Nov-2015.) |
⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ V) | ||
Theorem | elfvexd 6132 | If a function value is nonempty, its argument is a set. Deduction form of elfvex 6131. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (𝐵‘𝐶)) ⇒ ⊢ (𝜑 → 𝐶 ∈ V) | ||
Theorem | eliman0 6133 | A non-nul function value is an element of the image of the function. (Contributed by Thierry Arnoux, 25-Jun-2019.) |
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ (𝐹‘𝐴) = ∅) → (𝐹‘𝐴) ∈ (𝐹 “ 𝐵)) | ||
Theorem | nfvres 6134 | The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.) |
⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅) | ||
Theorem | nfunsn 6135 | If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅) | ||
Theorem | fvfundmfvn0 6136 | If a class' value at an argument is not the empty set, the argument is contained in the domain of the class, and the class restricted to the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | ||
Theorem | 0fv 6137 | Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.) |
⊢ (∅‘𝐴) = ∅ | ||
Theorem | fv2prc 6138 | A function's value at a function's value at a proper class is the empty set. (Contributed by AV, 8-Apr-2021.) |
⊢ (¬ 𝐴 ∈ V → ((𝐹‘𝐴)‘𝐵) = ∅) | ||
Theorem | elfv2ex 6139 | If a function value of a function value has a member, the first argument is a set. (Contributed by AV, 8-Apr-2021.) |
⊢ (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V) | ||
Theorem | fveqres 6140 | Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.) |
⊢ ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴)) | ||
Theorem | csbfv12 6141 | Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 20-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) | ||
Theorem | csbfv2g 6142* | Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.) |
⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (𝐹‘⦋𝐴 / 𝑥⦌𝐵)) | ||
Theorem | csbfv 6143* | Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴) | ||
Theorem | funbrfv 6144 | The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵)) | ||
Theorem | funopfv 6145 | The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.) |
⊢ (Fun 𝐹 → (〈𝐴, 𝐵〉 ∈ 𝐹 → (𝐹‘𝐴) = 𝐵)) | ||
Theorem | fnbrfvb 6146 | Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) | ||
Theorem | fnopfvb 6147 | Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐹)) | ||
Theorem | funbrfvb 6148 | Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | ||
Theorem | funopfvb 6149 | Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | ||
Theorem | funbrfv2b 6150 | Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.) |
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹‘𝐴) = 𝐵))) | ||
Theorem | dffn5 6151* | Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | ||
Theorem | fnrnfv 6152* | The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) | ||
Theorem | fvelrnb 6153* | A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.) |
⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)) | ||
Theorem | foelrni 6154* | A member of a surjective function's codomain is a value of the function. (Contributed by Thierry Arnoux, 23-Jan-2020.) |
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌) | ||
Theorem | dfimafn 6155* | Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦}) | ||
Theorem | dfimafn2 6156* | Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)}) | ||
Theorem | funimass4 6157* | Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | ||
Theorem | fvelima 6158* | Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴) | ||
Theorem | feqmptd 6159* | Deduction form of dffn5 6151. (Contributed by Mario Carneiro, 8-Jan-2015.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | ||
Theorem | feqresmpt 6160* | Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) | ||
Theorem | feqmptdf 6161 | Deduction form of dffn5f 6162. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | ||
Theorem | dffn5f 6162* | Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | ||
Theorem | fvelimab 6163* | Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) | ||
Theorem | fvelimabd 6164* | Deduction form of fvelimab 6163. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) | ||
Theorem | fvi 6165 | The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴) | ||
Theorem | fviss 6166 | The value of the identity function is a subset of the argument. (Contributed by Mario Carneiro, 27-Feb-2016.) |
⊢ ( I ‘𝐴) ⊆ 𝐴 | ||
Theorem | fniinfv 6167* | The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.) |
⊢ (𝐹 Fn 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ ran 𝐹) | ||
Theorem | fnsnfv 6168 | Singleton of function value. (Contributed by NM, 22-May-1998.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵})) | ||
Theorem | opabiotafun 6169* | Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 19-May-2015.) |
⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} ⇒ ⊢ Fun 𝐹 | ||
Theorem | opabiotadm 6170* | Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.) |
⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} ⇒ ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑} | ||
Theorem | opabiota 6171* | Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.) |
⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = (℩𝑦𝜓)) | ||
Theorem | fnimapr 6172 | The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹‘𝐵), (𝐹‘𝐶)}) | ||
Theorem | ssimaex 6173* | The existence of a subimage. (Contributed by NM, 8-Apr-2007.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))) | ||
Theorem | ssimaexg 6174* | The existence of a subimage. (Contributed by FL, 15-Apr-2007.) |
⊢ ((𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))) | ||
Theorem | funfv 6175 | A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.) |
⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) | ||
Theorem | funfv2 6176* | The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.) |
⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) | ||
Theorem | funfv2f 6177 | The value of a function. Version of funfv2 6176 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐹 ⇒ ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) | ||
Theorem | fvun 6178 | Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.) |
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝐴) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴))) | ||
Theorem | fvun1 6179 | The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋)) | ||
Theorem | fvun2 6180 | The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋)) | ||
Theorem | dffv2 6181 | Alternate definition of function value df-fv 5812 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.) |
⊢ (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) | ||
Theorem | dmfco 6182 | Domains of a function composition. (Contributed by NM, 27-Jan-1997.) |
⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝐴) ∈ dom 𝐹)) | ||
Theorem | fvco2 6183 | Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.) |
⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) | ||
Theorem | fvco 6184 | Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.) |
⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝐴) = (𝐹‘(𝐺‘𝐴))) | ||
Theorem | fvco3 6185 | Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.) |
⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) | ||
Theorem | fvco4i 6186 | Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ ∅ = (𝐹‘∅) & ⊢ Fun 𝐺 ⇒ ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)) | ||
Theorem | fvopab3g 6187* | Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦𝜑) & ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 𝜒)) | ||
Theorem | fvopab3ig 6188* | Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 ∈ 𝐶 → ∃*𝑦𝜑) & ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝜒 → (𝐹‘𝐴) = 𝐵)) | ||
Theorem | fvmptg 6189* | Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmpti 6190* | Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.) |
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ⇒ ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶)) | ||
Theorem | fvmpt 6191* | Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.) |
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmpt2f 6192 | Value of a function given by the "maps to" notation. (Contributed by Thierry Arnoux, 9-Mar-2017.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) | ||
Theorem | fvtresfn 6193* | Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⇒ ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = (𝑋 ↾ 𝑉)) | ||
Theorem | fvmpts 6194* | Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ 𝐵) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) | ||
Theorem | fvmpt3 6195* | Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) & ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmpt3i 6196* | Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmptd 6197* | Deduction version of fvmpt 6191. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | mptrcl 6198* | Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴) | ||
Theorem | fvmpt2i 6199* | Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = ( I ‘𝐵)) | ||
Theorem | fvmpt2 6200* | Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝑥) = 𝐵) |
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