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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | disjxpin 29401* | Derive a disjunction over a Cartesian product from the disjunctions over its first and second elements. (Contributed by Thierry Arnoux, 9-Mar-2018.) |
⊢ (𝑥 = (1st ‘𝑝) → 𝐶 = 𝐸) & ⊢ (𝑦 = (2nd ‘𝑝) → 𝐷 = 𝐹) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐶) & ⊢ (𝜑 → Disj 𝑦 ∈ 𝐵 𝐷) ⇒ ⊢ (𝜑 → Disj 𝑝 ∈ (𝐴 × 𝐵)(𝐸 ∩ 𝐹)) | ||
Theorem | iundisjf 29402* | Rewrite a countable union as a disjoint union. Cf. iundisj 23316. (Contributed by Thierry Arnoux, 31-Dec-2016.) |
⊢ Ⅎ𝑘𝐴 & ⊢ Ⅎ𝑛𝐵 & ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) ⇒ ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) | ||
Theorem | iundisj2f 29403* | A disjoint union is disjoint. Cf. iundisj2 23317. (Contributed by Thierry Arnoux, 30-Dec-2016.) |
⊢ Ⅎ𝑘𝐴 & ⊢ Ⅎ𝑛𝐵 & ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) ⇒ ⊢ Disj 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) | ||
Theorem | disjrdx 29404* | Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.) |
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶) & ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐶 𝐷)) | ||
Theorem | disjex 29405* | Two ways to say that two classes are disjoint (or equal). (Contributed by Thierry Arnoux, 4-Oct-2016.) |
⊢ ((∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝐴 = 𝐵) ↔ (𝐴 = 𝐵 ∨ (𝐴 ∩ 𝐵) = ∅)) | ||
Theorem | disjexc 29406* | A variant of disjex 29405, applicable for more generic families. (Contributed by Thierry Arnoux, 4-Oct-2016.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ ((∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝑥 = 𝑦) → (𝐴 = 𝐵 ∨ (𝐴 ∩ 𝐵) = ∅)) | ||
Theorem | disjunsn 29407* | Append an element to a disjoint collection. Similar to ralunsn 4422, gsumunsn 18359, etc. (Contributed by Thierry Arnoux, 28-Mar-2018.) |
⊢ (𝑥 = 𝑀 → 𝐵 = 𝐶) ⇒ ⊢ ((𝑀 ∈ 𝑉 ∧ ¬ 𝑀 ∈ 𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {𝑀})𝐵 ↔ (Disj 𝑥 ∈ 𝐴 𝐵 ∧ (∪ 𝑥 ∈ 𝐴 𝐵 ∩ 𝐶) = ∅))) | ||
Theorem | disjun0 29408* | Adding the empty element preserves disjointness. (Contributed by Thierry Arnoux, 30-May-2020.) |
⊢ (Disj 𝑥 ∈ 𝐴 𝑥 → Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥) | ||
Theorem | disjiunel 29409* | A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.) |
⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) & ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) & ⊢ (𝜑 → 𝐸 ⊆ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 𝐸)) ⇒ ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅) | ||
Theorem | disjuniel 29410* | A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.) |
⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵)) ⇒ ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅) | ||
Theorem | xpdisjres 29411 | Restriction of a constant function (or other Cartesian product) outside of its domain. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) ↾ 𝐶) = ∅) | ||
Theorem | opeldifid 29412 | Ordered pair elementhood outside of the diagonal. (Contributed by Thierry Arnoux, 1-Jan-2020.) |
⊢ (Rel 𝐴 → (〈𝑋, 𝑌〉 ∈ (𝐴 ∖ I ) ↔ (〈𝑋, 𝑌〉 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌))) | ||
Theorem | difres 29413 | Case when class difference in unaffected by restriction. (Contributed by Thierry Arnoux, 1-Jan-2020.) |
⊢ (𝐴 ⊆ (𝐵 × V) → (𝐴 ∖ (𝐶 ↾ 𝐵)) = (𝐴 ∖ 𝐶)) | ||
Theorem | imadifxp 29414 | Image of the difference with a Cartesian product. (Contributed by Thierry Arnoux, 13-Dec-2017.) |
⊢ (𝐶 ⊆ 𝐴 → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅 “ 𝐶) ∖ 𝐵)) | ||
Theorem | relfi 29415 | A relation (set) is finite if and only if both its domain and range are finite. (Contributed by Thierry Arnoux, 27-Aug-2017.) |
⊢ (Rel 𝐴 → (𝐴 ∈ Fin ↔ (dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin))) | ||
Theorem | funresdm1 29416 | Restriction of a disjoint union to the domain of the first term. (Contributed by Thierry Arnoux, 9-Dec-2021.) |
⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = 𝐴) | ||
Theorem | fnunres1 29417 | Restriction of a disjoint union to the domain of the first function. (Contributed by Thierry Arnoux, 9-Dec-2021.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹) | ||
Theorem | fcoinver 29418 | Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 29419. (Contributed by Thierry Arnoux, 3-Jan-2020.) |
⊢ (𝐹 Fn 𝑋 → (◡𝐹 ∘ 𝐹) Er 𝑋) | ||
Theorem | fcoinvbr 29419 | Binary relation for the equivalence relation from fcoinver 29418. (Contributed by Thierry Arnoux, 3-Jan-2020.) |
⊢ ∼ = (◡𝐹 ∘ 𝐹) ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ (𝐹‘𝑋) = (𝐹‘𝑌))) | ||
Theorem | brabgaf 29420* | The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) (Revised by Thierry Arnoux, 17-May-2020.) |
⊢ Ⅎ𝑥𝜓 & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝑅𝐵 ↔ 𝜓)) | ||
Theorem | brelg 29421 | Two things in a binary relation belong to the relation's domain. (Contributed by Thierry Arnoux, 29-Aug-2017.) |
⊢ ((𝑅 ⊆ (𝐶 × 𝐷) ∧ 𝐴𝑅𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | ||
Theorem | br8d 29422* | 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 | opabdm 29423* | Domain of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.) |
⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑}) | ||
Theorem | opabrn 29424* | Range of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.) |
⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑}) | ||
Theorem | ssrelf 29425* | 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 29426* | A version of eqrelrdv2 5219 with explicit non-free 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 29427 | Biconditional for equivalent elements. (Contributed by Thierry Arnoux, 6-Jan-2020.) |
⊢ ((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | ||
Theorem | iunsnima 29428 | 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 | ac6sf2 29429* | Alternate version of ac6 9302 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 | fnresin 29430 | 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 | f1o3d 29431* | 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 | rinvf1o 29432 | 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 29433 | 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 | fmptco1f1o 29434* | 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 | f1mptrn 29435* | 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 29436* | 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 29437 | 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 | elimampt 29438* | Membership in the image of a mapping. (Contributed by Thierry Arnoux, 3-Jan-2022.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝐷 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵)) | ||
Theorem | suppss2f 29439* | 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 | fovcld 29440 | Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.) |
⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) | ||
Theorem | ofrn 29441 | 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 29442 | 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 29443* | 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 29444 | 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 | fimarab 29445* | Expressing the image of a set as a restricted abstract builder. (Contributed by Thierry Arnoux, 27-Jan-2020.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦}) | ||
Theorem | unipreima 29446* | Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.) |
⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) | ||
Theorem | sspreima 29447 | 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 | opfv 29448 | 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 29449 | 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 | xppreima2 29450* | 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 | elunirn2 29451 | Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.) |
⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → 𝐵 ∈ ∪ ran 𝐹) | ||
Theorem | abfmpunirn 29452* | 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 29453* | 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 29454* | 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 29455* | 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 29456 | Domain and co-domain of the mapping operation; deduction form. This version of fmptd 6385 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 29457* | 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 29458* | Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 6400. (Contributed by Thierry Arnoux, 30-Jun-2017.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥)))) | ||
Theorem | acunirnmpt 29459* | 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 29460* | 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 29461* | 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 29462* | 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 29463* | 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 29464* | 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 29465* | 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 29466* | Express the preimage of a function operation as a union of preimages. This version of ofpreima 29465 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 | funcnvmptOLD 29467* | Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))) | ||
Theorem | funcnvmpt 29468* | Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵)) | ||
Theorem | funcnv5mpt 29469* | 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 29470* | 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 | fgreu 29471* | 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 29472* | 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 | rnmpt2ss 29473* | 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 29474* | Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5454. (Contributed by Thierry Arnoux, 30-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
Theorem | partfun 29475 | Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. (Contributed by Thierry Arnoux, 30-Mar-2017.) |
⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝐷)) = ((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝐷)) | ||
Theorem | dfcnv2 29476* | 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 | mpt2mptxf 29477* | 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 | gtiso 29478 | 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 29479* | 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 29480* | 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 5958) (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 29481* | 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 29482* | 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 29483 | The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.) |
⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (◡(1st ↾ 𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋}))) | ||
Theorem | 1stpreima 29484 | The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.) |
⊢ (𝐴 ⊆ 𝐵 → (◡(1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶)) | ||
Theorem | 2ndpreima 29485 | The preimage by 2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.) |
⊢ (𝐴 ⊆ 𝐶 → (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴)) | ||
Theorem | curry2ima 29486* | The image of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 25-Sep-2017.) |
⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) ⇒ ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)}) | ||
Theorem | supssd 29487* | Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅)) | ||
Theorem | infssd 29488* | Inequality deduction for infimum of a subset. (Contributed by AV, 4-Oct-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐶 ⊆ 𝐵) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅)) | ||
Theorem | imafi2 29489 | The image by a finite set is finite. See also imafi 8259. (Contributed by Thierry Arnoux, 25-Apr-2020.) |
⊢ (𝐴 ∈ Fin → (𝐴 “ 𝐵) ∈ Fin) | ||
Theorem | unifi3 29490 | If a union is finite, then all its elements are finite. See unifi 8255. (Contributed by Thierry Arnoux, 27-Aug-2017.) |
⊢ (∪ 𝐴 ∈ Fin → 𝐴 ⊆ Fin) | ||
Theorem | snct 29491 | A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.) |
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≼ ω) | ||
Theorem | prct 29492 | An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ≼ ω) | ||
Theorem | mpt2cti 29493* | An operation is countable if both its domains are countable. (Contributed by Thierry Arnoux, 17-Sep-2017.) |
⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 ⇒ ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ≼ ω) | ||
Theorem | abrexct 29494* | An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≼ ω) | ||
Theorem | mptctf 29495 | A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ≼ ω → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) | ||
Theorem | abrexctf 29496* | An image set of a countable set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≼ ω) | ||
Theorem | padct 29497* | Index a countable set with integers and pad with 𝑍. (Contributed by Thierry Arnoux, 1-Jun-2020.) |
⊢ ((𝐴 ≼ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) | ||
Theorem | cnvoprab 29498* | The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜓 & ⊢ (𝑎 = 〈𝑥, 𝑦〉 → (𝜓 ↔ 𝜑)) & ⊢ (𝜓 → 𝑎 ∈ (V × V)) ⇒ ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑧, 𝑎〉 ∣ 𝜓} | ||
Theorem | f1od2 29499* | Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌)) & ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))) ⇒ ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷) | ||
Theorem | fcobij 29500* | Composing functions with a bijection yields a bijection between sets of functions. (Contributed by Thierry Arnoux, 25-Aug-2017.) |
⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑇 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ↦ (𝐺 ∘ 𝑓)):(𝑆 ↑𝑚 𝑅)–1-1-onto→(𝑇 ↑𝑚 𝑅)) |
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