HomeTheorem List (p. 328 of 503)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | br8d 32701* | Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by Thierry Arnoux, 21-Mar-2019.) |
| ⊢ (𝑎 = 𝐴 → (𝜓 ↔ 𝜒)) & ⊢ (𝑏 = 𝐵 → (𝜒 ↔ 𝜃)) & ⊢ (𝑐 = 𝐶 → (𝜃 ↔ 𝜏)) & ⊢ (𝑑 = 𝐷 → (𝜏 ↔ 𝜂)) & ⊢ (𝑒 = 𝐸 → (𝜂 ↔ 𝜁)) & ⊢ (𝑓 = 𝐹 → (𝜁 ↔ 𝜎)) & ⊢ (𝑔 = 𝐺 → (𝜎 ↔ 𝜌)) & ⊢ (ℎ = 𝐻 → (𝜌 ↔ 𝜇)) & ⊢ (𝜑 → 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜓)}) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ 𝑃) & ⊢ (𝜑 → 𝐻 ∈ 𝑃) ⇒ ⊢ (𝜑 → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜇)) | ||
| Theorem | fnfvor 32702 | Relation between two functions implies the same relation for the function value at a given 𝑋. See also fnfvof 7639. (Contributed by Thierry Arnoux, 15-Jan-2026.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∘r 𝑅𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹‘𝑋)𝑅(𝐺‘𝑋)) | ||
| Theorem | ofrco 32703 | Function relation between function compositions. (Contributed by Thierry Arnoux, 15-Jan-2026.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐴) & ⊢ (𝜑 → 𝐻:𝐶⟶𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∘r 𝑅𝐺) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∘r 𝑅(𝐺 ∘ 𝐻)) | ||
| Theorem | opabdm 32704* | Domain of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.) |
| ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑}) | ||
| Theorem | opabrn 32705* | Range of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.) |
| ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑}) | ||
| Theorem | opabssi 32706* | Sufficient condition for a collection of ordered pairs to be a subclass of a relation. (Contributed by Peter Mazsa, 21-Oct-2019.) (Revised by Thierry Arnoux, 18-Feb-2022.) |
| ⊢ (𝜑 → ⟨𝑥, 𝑦⟩ ∈ 𝐴) ⇒ ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ 𝐴 | ||
| Theorem | opabid2ss 32707* | One direction of opabid2 5775 which holds without a Rel 𝐴 requirement. (Contributed by Thierry Arnoux, 18-Feb-2022.) |
| ⊢ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ⊆ 𝐴 | ||
| Theorem | ssrelf 32708* | A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Thierry Arnoux, 6-Nov-2017.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))) | ||
| Theorem | eqrelrd2 32709* | A version of eqrelrdv2 5742 with explicit nonfree declarations. (Contributed by Thierry Arnoux, 28-Aug-2017.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑦𝐵 & ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) ⇒ ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) | ||
| Theorem | erbr3b 32710 | Biconditional for equivalent elements. (Contributed by Thierry Arnoux, 6-Jan-2020.) |
| ⊢ ((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | ||
| Theorem | iunsnima 32711 | Image of a singleton by an indexed union involving that singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵) | ||
| Theorem | iunsnima2 32712* | Version of iunsnima 32711 with different variables. (Contributed by Thierry Arnoux, 22-Jun-2024.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) = 𝐶) | ||
| Theorem | fconst7v 32713* | An alternative way to express a constant function. (Contributed by Glauco Siliprandi, 5-Feb-2022.) Removed hyphotheses as suggested by SN (Revised by Thierry Arnoux, 10-Jan-2026.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) ⇒ ⊢ (𝜑 → 𝐹 = (𝐴 × {𝐵})) | ||
| Theorem | constcof 32714 | Composition with a constant function. See also fcoconst 7079. (Contributed by Thierry Arnoux, 11-Jan-2026.) |
| ⊢ (𝜑 → 𝐹:𝑋⟶𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) = (𝑋 × {𝑌})) | ||
| Theorem | ac6sf2 32715* | Alternate version of ac6 10391 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.) (Revised by Thierry Arnoux, 17-May-2020.) |
| ⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑦𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | ||
| Theorem | ac6mapd 32716* | Axiom of choice equivalent, deduction form. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ (𝑦 = (𝑓‘𝑥) → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝜓) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ (𝐵 ↑m 𝐴)∀𝑥 ∈ 𝐴 𝜒) | ||
| Theorem | fnresin 32717 | Restriction of a function with a subclass of its domain. (Contributed by Thierry Arnoux, 10-Oct-2017.) |
| ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐵) Fn (𝐴 ∩ 𝐵)) | ||
| Theorem | fresunsn 32718 | Recover the original function from a point-added function. See also funresdfunsn 7135 and fsnunres 7134. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}) = 𝐹) | ||
| Theorem | f1o3d 32719* | Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.) |
| ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) ⇒ ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) | ||
| Theorem | eldmne0 32720 | A function of nonempty domain is not empty. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
| ⊢ (𝑋 ∈ dom 𝐹 → 𝐹 ≠ ∅) | ||
| Theorem | f1rnen 32721 | Equinumerosity of the range of an injective function. (Contributed by Thierry Arnoux, 7-Jul-2023.) |
| ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉) → ran 𝐹 ≈ 𝐴) | ||
| Theorem | f1oeq3dd 32722 | Equality deduction for one-to-one onto functions. (Contributed by Thierry Arnoux, 10-Jan-2026.) |
| ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵) | ||
| Theorem | rinvf1o 32723 | Sufficient conditions for the restriction of an involution to be a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.) |
| ⊢ Fun 𝐹 & ⊢ ◡𝐹 = 𝐹 & ⊢ (𝐹 “ 𝐴) ⊆ 𝐵 & ⊢ (𝐹 “ 𝐵) ⊆ 𝐴 & ⊢ 𝐴 ⊆ dom 𝐹 & ⊢ 𝐵 ⊆ dom 𝐹 ⇒ ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵 | ||
| Theorem | fresf1o 32724 | Conditions for a restriction to be a one-to-one onto function. (Contributed by Thierry Arnoux, 7-Dec-2016.) |
| ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1-onto→𝐶) | ||
| Theorem | nfpconfp 32725 | The set of fixed points of 𝐹 is the complement of the set of points moved by 𝐹. (Contributed by Thierry Arnoux, 17-Nov-2023.) |
| ⊢ (𝐹 Fn 𝐴 → (𝐴 ∖ dom (𝐹 ∖ I )) = dom (𝐹 ∩ I )) | ||
| Theorem | fmptco1f1o 32726* | The action of composing (to the right) with a bijection is itself a bijection of functions. (Contributed by Thierry Arnoux, 3-Jan-2021.) |
| ⊢ 𝐴 = (𝑅 ↑m 𝐸) & ⊢ 𝐵 = (𝑅 ↑m 𝐷) & ⊢ 𝐹 = (𝑓 ∈ 𝐴 ↦ (𝑓 ∘ 𝑇)) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝑋) & ⊢ (𝜑 → 𝑇:𝐷–1-1-onto→𝐸) ⇒ ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | ||
| Theorem | cofmpt2 32727* | Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 15-Jul-2023.) |
| ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐸) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐷)) | ||
| Theorem | f1mptrn 32728* | Express injection for a mapping operation. (Contributed by Thierry Arnoux, 3-May-2020.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵) ⇒ ⊢ (𝜑 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵)) | ||
| Theorem | dfimafnf 32729* | Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.) |
| ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 ⇒ ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) | ||
| Theorem | funimass4f 32730 | Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.) |
| ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝐹 ⇒ ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | ||
| Theorem | suppss2f 32731* | Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.) (Revised by AV, 1-Sep-2020.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐴 & ⊢ Ⅎ𝑘𝑊 & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊) | ||
| Theorem | ofrn 32732 | The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ 𝐶) | ||
| Theorem | ofrn2 32733 | The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺))) | ||
| Theorem | off2 32734* | The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.) |
| ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈) | ||
| Theorem | ofresid 32735 | Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺)) | ||
| Theorem | unipreima 32736* | Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.) |
| ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) | ||
| Theorem | opfv 32737 | Value of a function producing ordered pairs. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
| ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩) | ||
| Theorem | xppreima 32738 | The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.) |
| ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (◡𝐹 “ (𝑌 × 𝑍)) = ((◡(1st ∘ 𝐹) “ 𝑌) ∩ (◡(2nd ∘ 𝐹) “ 𝑍))) | ||
| Theorem | 2ndimaxp 32739 | Image of a cartesian product by 2nd. (Contributed by Thierry Arnoux, 23-Jun-2024.) |
| ⊢ (𝐴 ≠ ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵) | ||
| Theorem | dmdju 32740* | Domain of a disjoint union of non-empty sets. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅) ⇒ ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = 𝐴) | ||
| Theorem | djussxp2 32741* | Stronger version of djussxp 5792. (Contributed by Thierry Arnoux, 23-Jun-2024.) |
| ⊢ ∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵) | ||
| Theorem | 2ndresdju 32742* | The 2nd function restricted to a disjoint union is injective. (Contributed by Thierry Arnoux, 23-Jun-2024.) |
| ⊢ 𝑈 = ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑊) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝐶) & ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴) ⇒ ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1→𝐴) | ||
| Theorem | 2ndresdjuf1o 32743* | The 2nd function restricted to a disjoint union is a bijection. See also e.g. 2ndconst 8042. (Contributed by Thierry Arnoux, 23-Jun-2024.) |
| ⊢ 𝑈 = ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑊) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝐶) & ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴) ⇒ ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1-onto→𝐴) | ||
| Theorem | xppreima2 32744* | The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐶) & ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ⇒ ⊢ (𝜑 → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍))) | ||
| Theorem | abfmpunirn 32745* | Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑}) & ⊢ {𝑦 ∣ 𝜑} ∈ V & ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 𝜓)) | ||
| Theorem | rabfmpunirn 32746* | Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑊 ∣ 𝜑}) & ⊢ 𝑊 ∈ V & ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓)) | ||
| Theorem | abfmpeld 32747* | Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜓}) & ⊢ (𝜑 → {𝑦 ∣ 𝜓} ∈ V) & ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒))) | ||
| Theorem | abfmpel 32748* | Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑}) & ⊢ {𝑦 ∣ 𝜑} ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜓)) | ||
| Theorem | fmptdF 32749 | Domain and codomain of the mapping operation; deduction form. This version of fmptd 7058 uses bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | ||
| Theorem | fmptcof2 32750* | Composition of two functions expressed as ordered-pair class abstractions. (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.) (Revised by Thierry Arnoux, 10-May-2017.) |
| ⊢ Ⅎ𝑥𝑆 & ⊢ Ⅎ𝑦𝑇 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝜑 & ⊢ (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 | fcomptf 32751* | Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 7078. (Contributed by Thierry Arnoux, 30-Jun-2017.) |
| ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥)))) | ||
| Theorem | acunirnmpt 32752* | Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 6-Nov-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅) & ⊢ 𝐶 = ran (𝑗 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵)) | ||
| Theorem | acunirnmpt2 32753* | Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅) & ⊢ 𝐶 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)) | ||
| Theorem | acunirnmpt2f 32754* | Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅) & ⊢ Ⅎ𝑗𝐴 & ⊢ Ⅎ𝑗𝐶 & ⊢ Ⅎ𝑗𝐷 & ⊢ 𝐶 = ∪ 𝑗 ∈ 𝐴 𝐵 & ⊢ (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)) | ||
| Theorem | aciunf1lem 32755* | Choice in an index union. (Contributed by Thierry Arnoux, 8-Nov-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅) & ⊢ Ⅎ𝑗𝐴 & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑥)) = 𝑥)) | ||
| Theorem | aciunf1 32756* | Choice in an index union. (Contributed by Thierry Arnoux, 4-May-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘)) | ||
| Theorem | ofoprabco 32757* | Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
| ⊢ Ⅎ𝑎𝑀 & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 = (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)) & ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦))) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑁 ∘ 𝑀)) | ||
| Theorem | ofpreima 32758* | Express the preimage of a function operation as a union of preimages. (Contributed by Thierry Arnoux, 8-Mar-2018.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 Fn (𝐵 × 𝐶)) ⇒ ⊢ (𝜑 → (◡(𝐹 ∘f 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) | ||
| Theorem | ofpreima2 32759* | Express the preimage of a function operation as a union of preimages. This version of ofpreima 32758 iterates the union over a smaller set. (Contributed by Thierry Arnoux, 8-Mar-2018.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 Fn (𝐵 × 𝐶)) ⇒ ⊢ (𝜑 → (◡(𝐹 ∘f 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) | ||
| Theorem | funcnv5mpt 32760* | Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-Mar-2017.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶))) | ||
| Theorem | funcnv4mpt 32761* | Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))) | ||
| Theorem | preimane 32762 | Different elements have different preimages. (Contributed by Thierry Arnoux, 7-May-2023.) |
| ⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐹) & ⊢ (𝜑 → 𝑌 ∈ ran 𝐹) ⇒ ⊢ (𝜑 → (◡𝐹 “ {𝑋}) ≠ (◡𝐹 “ {𝑌})) | ||
| Theorem | fnpreimac 32763* | Choose a set 𝑥 containing a preimage of each element of a given set 𝐵. (Contributed by Thierry Arnoux, 7-May-2023.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∃𝑥 ∈ 𝒫 𝐴(𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵)) | ||
| Theorem | fgreu 32764* | Exactly one point of a function's graph has a given first element. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∃!𝑝 ∈ 𝐹 𝑋 = (1st ‘𝑝)) | ||
| Theorem | fcnvgreu 32765* | If the converse of a relation 𝐴 is a function, exactly one point of its graph has a given second element (that is, function value). (Contributed by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑝 ∈ 𝐴 𝑌 = (2nd ‘𝑝)) | ||
| Theorem | rnmposs 32766* | The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 23-May-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → ran 𝐹 ⊆ 𝐷) | ||
| Theorem | mptssALT 32767* | Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5999. (Contributed by Thierry Arnoux, 30-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
| Theorem | dfcnv2 32768* | Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.) |
| ⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))) | ||
| Theorem | partfun2 32769* | Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. See also partfun 6637 and ifmpt2v 7460. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ 𝜑} ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ if(𝜑, 𝐵, 𝐶)) = ((𝑥 ∈ 𝐷 ↦ 𝐵) ∪ (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶)) | ||
| Theorem | rnressnsn 32770 | The range of a restriction to a singleton is a singleton. See dmressnsn 5980. (Contributed by Thierry Arnoux, 25-Jan-2026.) |
| ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ran (𝐹 ↾ {𝐴}) = {(𝐹‘𝐴)}) | ||
| Theorem | mpomptxf 32771* | Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐵(𝑥) is not assumed to be constant w.r.t 𝑥. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Thierry Arnoux, 31-Mar-2018.) |
| ⊢ Ⅎ𝑥𝐶 & ⊢ Ⅎ𝑦𝐶 & ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷) ⇒ ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) | ||
| Theorem | of0r 32772 | Function operation with the empty function. (Contributed by Thierry Arnoux, 27-May-2025.) |
| ⊢ (𝐹 ∘f 𝑅∅) = ∅ | ||
| Theorem | elmaprd 32773 | Deduction associated with elmapd 8778. Reverse direction of elmapdd 8779. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m 𝐴)) ⇒ ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | ||
| Theorem | suppovss 32774* | A bound for the support of an operation. (Contributed by Thierry Arnoux, 19-Jul-2023.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ 𝐶)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑍 ∈ 𝐷) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍))) | ||
| Theorem | elsuppfnd 32775 | Deduce membership in the support of a function. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → (𝐹‘𝑋) ≠ 𝑍) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐹 supp 𝑍)) | ||
| Theorem | fisuppov1 32776* | Formula building theorem for finite support: operator with left annihilator. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 0 ∈ 𝑋) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) & ⊢ (𝜑 → 𝐷 ⊆ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐸) & ⊢ (𝜑 → 𝐹 finSupp 0 ) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ( 0 𝑂𝑦) = 𝑍) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) finSupp 𝑍) | ||
| Theorem | suppun2 32777 | The support of a union is the union of the supports. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝑍 ∈ 𝑋) ⇒ ⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) = ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) | ||
| Theorem | fdifsupp 32778 | Express the support of a function 𝐹 outside of 𝐵 in two different ways. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 Fn 𝐴) ⇒ ⊢ (𝜑 → ((𝐹 ↾ (𝐴 ∖ 𝐵)) supp 𝑍) = ((𝐹 supp 𝑍) ∖ 𝐵)) | ||
| Theorem | suppiniseg 32779 | Relation between the support (𝐹 supp 𝑍) and the initial segment (◡𝐹 “ {𝑍}). (Contributed by Thierry Arnoux, 25-Jun-2024.) |
| ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍})) | ||
| Theorem | fsuppinisegfi 32780 | The initial segment (◡𝐹 “ {𝑌}) of a nonzero 𝑌 is finite if 𝐹 has finite support. (Contributed by Thierry Arnoux, 21-Jun-2024.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 0 ∈ 𝑊) & ⊢ (𝜑 → 𝑌 ∈ (V ∖ { 0 })) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ∈ Fin) | ||
| Theorem | fressupp 32781 | The restriction of a function to its support. (Contributed by Thierry Arnoux, 25-Jun-2024.) |
| ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ (𝐹 supp 𝑍)) = (𝐹 ∖ (V × {𝑍}))) | ||
| Theorem | fdifsuppconst 32782 | A function is a zero constant outside of its support. (Contributed by Thierry Arnoux, 22-Jun-2024.) |
| ⊢ 𝐴 = (dom 𝐹 ∖ (𝐹 supp 𝑍)) ⇒ ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ 𝐴) = (𝐴 × {𝑍})) | ||
| Theorem | ressupprn 32783 | The range of a function restricted to its support. (Contributed by Thierry Arnoux, 25-Jun-2024.) |
| ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 })) | ||
| Theorem | supppreima 32784 | Express the support of a function as the preimage of its range except zero. (Contributed by Thierry Arnoux, 24-Jun-2024.) |
| ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (ran 𝐹 ∖ {𝑍}))) | ||
| Theorem | fsupprnfi 32785 | Finite support implies finite range. (Contributed by Thierry Arnoux, 24-Jun-2024.) |
| ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → ran 𝐹 ∈ Fin) | ||
| Theorem | mptiffisupp 32786* | Conditions for a mapping function defined with a conditional to have finite support. (Contributed by Thierry Arnoux, 20-Feb-2025.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍)) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐹 finSupp 𝑍) | ||
| Theorem | cosnopne 32787 | Composition of two ordered pair singletons with non-matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐴 ≠ 𝐷) ⇒ ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅) | ||
| Theorem | cosnop 32788 | Composition of two ordered pair singletons with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) ⇒ ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}) = {⟨𝐶, 𝐵⟩}) | ||
| Theorem | cnvprop 32789 | Converse of a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩}) | ||
| Theorem | brprop 32790 | Binary relation for a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)))) | ||
| Theorem | mptprop 32791* | Rewrite pairs of ordered pairs as mapping to functions. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷))) | ||
| Theorem | coprprop 32792 | Composition of two pairs of ordered pairs with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) & ⊢ (𝜑 → 𝐸 ∈ 𝑋) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) & ⊢ (𝜑 → 𝐸 ≠ 𝐹) ⇒ ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∘ {⟨𝐸, 𝐴⟩, ⟨𝐹, 𝐶⟩}) = {⟨𝐸, 𝐵⟩, ⟨𝐹, 𝐷⟩}) | ||
| Theorem | fmptunsnop 32793* | Two ways to express a function with a value replaced. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩})) | ||
| Theorem | gtiso 32794 | Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.) |
| ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) → (𝐹 Isom < , ◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ◡ ≤ (𝐴, 𝐵))) | ||
| Theorem | isoun 32795* | Infer an isomorphism from a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.) |
| ⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥𝑅𝑦) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) → 𝑧𝑆𝑤) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑥𝑅𝑦) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐵) → ¬ 𝑧𝑆𝑤) & ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅) & ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅) ⇒ ⊢ (𝜑 → (𝐻 ∪ 𝐺) Isom 𝑅, 𝑆 ((𝐴 ∪ 𝐶), (𝐵 ∪ 𝐷))) | ||
| Theorem | disjdsct 32796* | A disjoint collection is distinct, i.e. each set in this collection is different of all others, provided that it does not contain the empty set This can be expressed as "the converse of the mapping function is a function", or "the mapping function is single-rooted". (Cf. funcnv 6559) (Contributed by Thierry Arnoux, 28-Feb-2017.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (𝑉 ∖ {∅})) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) ⇒ ⊢ (𝜑 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵)) | ||
| Theorem | df1stres 32797* | Definition for a restriction of the 1st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
| ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑥) | ||
| Theorem | df2ndres 32798* | Definition for a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
| ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑦) | ||
| Theorem | 1stpreimas 32799 | The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.) |
| ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (◡(1st ↾ 𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋}))) | ||
| Theorem | 1stpreima 32800 | The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.) |
| ⊢ (𝐴 ⊆ 𝐵 → (◡(1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶)) | ||
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