Home | Metamath
Proof Explorer Theorem List (p. 69 of 462) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28971) |
Hilbert Space Explorer
(28972-30494) |
Users' Mathboxes
(30495-46134) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fvco4i 6801 | 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 6802* | 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 6803* | 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 | brfvopabrbr 6804* | The binary relation of a function value which is an ordered-pair class abstraction of a restricted binary relation is the restricted binary relation. The first hypothesis can often be obtained by using fvmptopab 7255. (Contributed by AV, 29-Oct-2021.) |
⊢ (𝐴‘𝑍) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐵‘𝑍)𝑦 ∧ 𝜑)} & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓)) & ⊢ Rel (𝐵‘𝑍) ⇒ ⊢ (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓)) | ||
Theorem | fvmptg 6805* | 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 6806* | 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 6807* | 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 6808 | 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 6809* | 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 6810* | 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 6811* | 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 6812* | 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 | fvmptdf 6813* | Deduction version of fvmptd 6814 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by AV, 29-Mar-2024.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmptd 6814* | Deduction version of fvmpt 6807. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof shortened by AV, 29-Mar-2024.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmptd2 6815* | Deduction version of fvmpt 6807 (where the definition of the mapping does not depend on the common antecedent 𝜑). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | mptrcl 6816* | 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 6817* | 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 6818* | 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) | ||
Theorem | fvmptss 6819* | If all the values of the mapping are subsets of a class 𝐶, then so is any evaluation of the mapping, even if 𝐷 is not in the base set 𝐴. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶) | ||
Theorem | fvmpt2d 6820* | Deduction version of fvmpt2 6818. (Contributed by Thierry Arnoux, 8-Dec-2016.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | ||
Theorem | fvmptex 6821* | Express a function 𝐹 whose value 𝐵 may not always be a set in terms of another function 𝐺 for which sethood is guaranteed. (Note that ( I ‘𝐵) is just shorthand for if(𝐵 ∈ V, 𝐵, ∅), and it is always a set by fvex 6719.) Note also that these functions are not the same; wherever 𝐵(𝐶) is not a set, 𝐶 is not in the domain of 𝐹 (so it evaluates to the empty set), but 𝐶 is in the domain of 𝐺, and 𝐺(𝐶) is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵)) ⇒ ⊢ (𝐹‘𝐶) = (𝐺‘𝐶) | ||
Theorem | fvmptd3f 6822* | Alternate deduction version of fvmpt 6807 with three nonfreeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) | ||
Theorem | fvmptd2f 6823* | Alternate deduction version of fvmpt 6807, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) (Proof shortened by AV, 19-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) | ||
Theorem | fvmptdv 6824* | Alternate deduction version of fvmpt 6807, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) | ||
Theorem | fvmptdv2 6825* | Alternate deduction version of fvmpt 6807, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = 𝐶)) | ||
Theorem | mpteqb 6826* | Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 6841. (Contributed by Mario Carneiro, 14-Nov-2014.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)) | ||
Theorem | fvmptt 6827* | Closed theorem form of fvmpt 6807. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmptf 6828* | Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6805 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmptnf 6829* | The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 6831 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ⇒ ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅) | ||
Theorem | fvmptd3 6830* | Deduction version of fvmpt 6807. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmptn 6831* | This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class 𝐶 it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvmptg 6805. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.) |
⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐷) = ∅) | ||
Theorem | fvmptss2 6832* | A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝐹‘𝐷) ⊆ 𝐶 | ||
Theorem | elfvmptrab1w 6833* | Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. Version of elfvmptrab1 6834 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) & ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) ⇒ ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)) | ||
Theorem | elfvmptrab1 6834* | Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker elfvmptrab1w 6833 when possible. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) & ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) ⇒ ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)) | ||
Theorem | elfvmptrab 6835* | Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.) |
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}) & ⊢ (𝑋 ∈ 𝑉 → 𝑀 ∈ V) ⇒ ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑀)) | ||
Theorem | fvopab4ndm 6836* | Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.) |
⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⇒ ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅) | ||
Theorem | fvmptndm 6837* | Value of a function given by the maps-to notation, outside of its domain. (Contributed by AV, 31-Dec-2020.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (¬ 𝑋 ∈ 𝐴 → (𝐹‘𝑋) = ∅) | ||
Theorem | fvmptrabfv 6838* | Value of a function mapping a set to a class abstraction restricting the value of another function. (Contributed by AV, 18-Feb-2022.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑}) & ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} | ||
Theorem | fvopab5 6839* | The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝜑} & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑦𝜓)) | ||
Theorem | fvopab6 6840* | Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐵)} & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶) | ||
Theorem | eqfnfv 6841* | Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | ||
Theorem | eqfnfv2 6842* | Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))) | ||
Theorem | eqfnfv3 6843* | Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))))) | ||
Theorem | eqfnfvd 6844* | Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | eqfnfv2f 6845* | Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 6841 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐺 ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | ||
Theorem | eqfunfv 6846* | Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.) |
⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = (𝐺‘𝑥)))) | ||
Theorem | fvreseq0 6847* | Equality of restricted functions is determined by their values (for functions with different domains). (Contributed by AV, 6-Jan-2019.) |
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) | ||
Theorem | fvreseq1 6848* | Equality of a function restricted to the domain of another function. (Contributed by AV, 6-Jan-2019.) |
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) | ||
Theorem | fvreseq 6849* | Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.) (Proof shortened by AV, 4-Mar-2019.) |
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) | ||
Theorem | fnmptfvd 6850* | A function with a given domain is a mapping defined by its function values. (Contributed by AV, 1-Mar-2019.) |
⊢ (𝜑 → 𝑀 Fn 𝐴) & ⊢ (𝑖 = 𝑎 → 𝐷 = 𝐶) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐷 ∈ 𝑈) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷)) | ||
Theorem | fndmdif 6851* | Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)}) | ||
Theorem | fndmdifcom 6852 | The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = dom (𝐺 ∖ 𝐹)) | ||
Theorem | fndmdifeq0 6853 | The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom (𝐹 ∖ 𝐺) = ∅ ↔ 𝐹 = 𝐺)) | ||
Theorem | fndmin 6854* | Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)}) | ||
Theorem | fneqeql 6855 | Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹 ∩ 𝐺) = 𝐴)) | ||
Theorem | fneqeql2 6856 | Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ 𝐴 ⊆ dom (𝐹 ∩ 𝐺))) | ||
Theorem | fnreseql 6857 | Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺))) | ||
Theorem | chfnrn 6858* | The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.) |
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → ran 𝐹 ⊆ ∪ 𝐴) | ||
Theorem | funfvop 6859 | Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | ||
Theorem | funfvbrb 6860 | Two ways to say that 𝐴 is in the domain of 𝐹. (Contributed by Mario Carneiro, 1-May-2014.) |
⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ 𝐴𝐹(𝐹‘𝐴))) | ||
Theorem | fvimacnvi 6861 | A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵) | ||
Theorem | fvimacnv 6862 | The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 6452 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵))) | ||
Theorem | funimass3 6863 | A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 6862 would be the special case of 𝐴 being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵))) | ||
Theorem | funimass5 6864* | A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐴 ⊆ (◡𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | ||
Theorem | funconstss 6865* | Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵}))) | ||
Theorem | fvimacnvALT 6866 | Alternate proof of fvimacnv 6862, based on funimass3 6863. If funimass3 6863 is ever proved directly, as opposed to using funimacnv 6450 pointwise, then the proof of funimacnv 6450 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵))) | ||
Theorem | elpreima 6867 | Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ (◡𝐹 “ 𝐶) ↔ (𝐵 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ 𝐶))) | ||
Theorem | elpreimad 6868 | Membership in the preimage of a set under a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ∈ (◡𝐹 “ 𝐶)) | ||
Theorem | fniniseg 6869 | Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro , 28-Apr-2015.) |
⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵))) | ||
Theorem | fncnvima2 6870* | Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ 𝐵) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝐵}) | ||
Theorem | fniniseg2 6871* | Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵}) | ||
Theorem | unpreima 6872 | Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵))) | ||
Theorem | inpreima 6873 | Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.) |
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵))) | ||
Theorem | difpreima 6874 | Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.) |
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∖ 𝐵)) = ((◡𝐹 “ 𝐴) ∖ (◡𝐹 “ 𝐵))) | ||
Theorem | respreima 6875 | The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ 𝐵)) | ||
Theorem | cnvimainrn 6876 | The preimage of the intersection of the range of a class and a class 𝐴 is the preimage of the class 𝐴. (Contributed by AV, 17-Sep-2024.) |
⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴)) | ||
Theorem | sspreima 6877 | The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵)) | ||
Theorem | iinpreima 6878* | Preimage of an intersection. (Contributed by FL, 16-Apr-2012.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵)) | ||
Theorem | intpreima 6879* | Preimage of an intersection. (Contributed by FL, 28-Apr-2012.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ ∩ 𝐴) = ∩ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) | ||
Theorem | fimacnvOLD 6880 | Obsolete version of fimacnv 6556 as of 20-Sep-2024. (Contributed by FL, 25-Jan-2007.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) | ||
Theorem | fimacnvinrn 6881 | Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) | ||
Theorem | fimacnvinrn2 6882 | Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 17-Feb-2017.) |
⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ 𝐵))) | ||
Theorem | rescnvimafod 6883 | The restriction of a function to a preimage of a class is a function onto the intersection of this class and the range of the function. (Contributed by AV, 13-Sep-2024.) (Revised by AV, 29-Sep-2024.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵)) & ⊢ (𝜑 → 𝐷 = (◡𝐹 “ 𝐵)) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸) | ||
Theorem | fvn0ssdmfun 6884* | If a class' function values for certain arguments is not the empty set, the arguments are contained in the domain of the class, and the class restricted to the arguments is a function, analogous to fvfundmfvn0 6744. (Contributed by AV, 27-Jan-2020.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (∀𝑎 ∈ 𝐷 (𝐹‘𝑎) ≠ ∅ → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ 𝐷))) | ||
Theorem | fnopfv 6885 | Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 〈𝐵, (𝐹‘𝐵)〉 ∈ 𝐹) | ||
Theorem | fvelrn 6886 | A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) | ||
Theorem | nelrnfvne 6887 | A function value cannot be any element not contained in the range of the function. (Contributed by AV, 28-Jan-2020.) |
⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹 ∧ 𝑌 ∉ ran 𝐹) → (𝐹‘𝑋) ≠ 𝑌) | ||
Theorem | fveqdmss 6888* | If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the domain of the function is contained in the domain of the class. (Contributed by AV, 28-Jan-2020.) |
⊢ 𝐷 = dom 𝐵 ⇒ ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴) | ||
Theorem | fveqressseq 6889* | If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the class restricted to the domain of the function is the function itself. (Contributed by AV, 28-Jan-2020.) |
⊢ 𝐷 = dom 𝐵 ⇒ ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴 ↾ 𝐷) = 𝐵) | ||
Theorem | fnfvelrn 6890 | A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ∈ ran 𝐹) | ||
Theorem | ffvelrn 6891 | A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵) | ||
Theorem | ffvelrni 6892 | A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.) |
⊢ 𝐹:𝐴⟶𝐵 ⇒ ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵) | ||
Theorem | ffvelrnda 6893 | A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵) | ||
Theorem | ffvelrnd 6894 | A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝐵) | ||
Theorem | rexrn 6895* | Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.) |
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)) | ||
Theorem | ralrn 6896* | Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.) |
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐹 Fn 𝐴 → (∀𝑥 ∈ ran 𝐹𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)) | ||
Theorem | elrnrexdm 6897* | For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.) |
⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥))) | ||
Theorem | elrnrexdmb 6898* | For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.) |
⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥))) | ||
Theorem | eldmrexrn 6899* | For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.) |
⊢ (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))) | ||
Theorem | eldmrexrnb 6900* | For any element in the domain of a function, there is an element in the range of the function which is the value of the function at that element. Because of the definition df-fv 6377 of the value of a function, the theorem is only valid in general if the empty set is not contained in the range of the function (the implication "to the right" is always valid). Indeed, with the definition df-fv 6377 of the value of a function, (𝐹‘𝑌) = ∅ may mean that the value of 𝐹 at 𝑌 is the empty set or that 𝐹 is not defined at 𝑌. (Contributed by Alexander van der Vekens, 17-Dec-2017.) |
⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |