Home | Metamath
Proof Explorer Theorem List (p. 289 of 424) | < 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
(1-27159) |
Hilbert Space Explorer
(27160-28684) |
Users' Mathboxes
(28685-42360) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | brelg 28801 | 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 28802* | 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 28803* | Domain of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.) |
⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑}) | ||
Theorem | opabrn 28804* | Range of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.) |
⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑}) | ||
Theorem | ssrelf 28805* | 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 28806* | A version of eqrelrdv2 5142 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 28807 | Biconditional for equivalent elements. (Contributed by Thierry Arnoux, 6-Jan-2020.) |
⊢ ((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | ||
Theorem | iunsnima 28808 | 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 | mptexgf 28809 | If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) (Revised by Thierry Arnoux, 17-May-2020.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | ||
Theorem | ac6sf2 28810* | Alternate version of ac6 9185 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 | idssxp 28811 | A diagonal set as a subset of a Cartesian product. (Contributed by Thierry Arnoux, 29-Dec-2019.) |
⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) | ||
Theorem | fnresin 28812 | 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 28813* | 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 28814 | 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 28815 | 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 | f1mptrn 28816* | 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 28817* | 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 28818 | 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 28819* | 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 28820 | 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 28821 | 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 28822 | 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 28823* | 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 28824 | 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 28825* | 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 28826* | Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.) |
⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) | ||
Theorem | sspreima 28827 | 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 28828 | 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 28829 | 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 28830* | 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 28831 | 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 28832* | 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 28833* | 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 28834* | 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 28835* | 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 28836 | Domain and co-domain of the mapping operation; deduction form. This version of fmptd 6292 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 | mpteq12df 28837 | An equality theorem for the maps to notation. (Contributed by Thierry Arnoux, 30-May-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | ||
Theorem | resmptf 28838 | Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
Theorem | fmptcof2 28839* | 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 28840* | Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 6306. (Contributed by Thierry Arnoux, 30-Jun-2017.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥)))) | ||
Theorem | acunirnmpt 28841* | 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 28842* | 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 28843* | 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 28844* | 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 28845* | 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 | cofmpt 28846* | Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.) |
⊢ (𝜑 → 𝐹:𝐶⟶𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))) | ||
Theorem | ofoprabco 28847* | 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 28848* | 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 28849* | Express the preimage of a function operation as a union of preimages. This version of ofpreima 28848 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 28850* | 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 28851* | 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 28852* | 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 28853* | 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 28854* | 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 28855* | 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 28856* | 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 28857* | Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5373. (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 28858 | 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 28859* | 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 28860* | 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 28861 | 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 28862* | 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 28863* | 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 5872) (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 28864* | 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 28865* | 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 28866 | The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.) |
⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (◡(1st ↾ 𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋}))) | ||
Theorem | 1stpreima 28867 | 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 28868 | 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 28869* | 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 28870* | 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 28871* | 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 28872 | The image by a finite set is finite. See also imafi 8142. (Contributed by Thierry Arnoux, 25-Apr-2020.) |
⊢ (𝐴 ∈ Fin → (𝐴 “ 𝐵) ∈ Fin) | ||
Theorem | unifi3 28873 | If a union is finite, then all its elements are finite. See unifi 8138. (Contributed by Thierry Arnoux, 27-Aug-2017.) |
⊢ (∪ 𝐴 ∈ Fin → 𝐴 ⊆ Fin) | ||
Theorem | snct 28874 | A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.) |
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≼ ω) | ||
Theorem | prct 28875 | An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ≼ ω) | ||
Theorem | fnct 28876 | If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ ω) | ||
Theorem | dmct 28877 | The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω) | ||
Theorem | cnvct 28878 | If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω) | ||
Theorem | rnct 28879 | The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → ran 𝐴 ≼ ω) | ||
Theorem | mptct 28880* | A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) | ||
Theorem | mpt2cti 28881* | 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 28882* | 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 28883 | 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 28884* | 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 28885* | 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 28886* | 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 28887* | 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 28888* | 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)) | ||
Theorem | fcobijfs 28889* | Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 8196. (Contributed by Thierry Arnoux, 25-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑇 ∈ 𝑊) & ⊢ (𝜑 → 𝑂 ∈ 𝑆) & ⊢ 𝑄 = (𝐺‘𝑂) & ⊢ 𝑋 = {𝑔 ∈ (𝑆 ↑𝑚 𝑅) ∣ 𝑔 finSupp 𝑂} & ⊢ 𝑌 = {ℎ ∈ (𝑇 ↑𝑚 𝑅) ∣ ℎ finSupp 𝑄} ⇒ ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌) | ||
Theorem | suppss3 28890* | Deduce a function's support's inclusion in another function's support. (Contributed by Thierry Arnoux, 7-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑍) → 𝐵 = 𝑍) ⇒ ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍)) | ||
Theorem | ffs2 28891 | Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 7194. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
⊢ 𝐶 = (𝐵 ∖ {𝑍}) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ 𝐶)) | ||
Theorem | ffsrn 28892 | The range of a finitely supported function is finite. (Contributed by Thierry Arnoux, 27-Aug-2017.) |
⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) ⇒ ⊢ (𝜑 → ran 𝐹 ∈ Fin) | ||
Theorem | resf1o 28893* | Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.) |
⊢ 𝑋 = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶} & ⊢ 𝐹 = (𝑓 ∈ 𝑋 ↦ (𝑓 ↾ 𝐶)) ⇒ ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝐹:𝑋–1-1-onto→(𝐵 ↑𝑚 𝐶)) | ||
Theorem | maprnin 28894* | Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐵 ∩ 𝐶) ↑𝑚 𝐴) = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ran 𝑓 ⊆ 𝐶} | ||
Theorem | fpwrelmapffslem 28895* | Lemma for fpwrelmapffs 28897. For this theorem, the sets 𝐴 and 𝐵 could be infinite, but the relation 𝑅 itself is finite. (Contributed by Thierry Arnoux, 1-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝜑 → 𝐹:𝐴⟶𝒫 𝐵) & ⊢ (𝜑 → 𝑅 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}) ⇒ ⊢ (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin))) | ||
Theorem | fpwrelmap 28896* | Define a canonical mapping between functions from 𝐴 into subsets of 𝐵 and the relations with domain 𝐴 and range within 𝐵. Note that the same relation is used in axdc2lem 9153 and marypha2lem1 8224. (Contributed by Thierry Arnoux, 28-Aug-2017.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝑀 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ⇒ ⊢ 𝑀:(𝒫 𝐵 ↑𝑚 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵) | ||
Theorem | fpwrelmapffs 28897* | Define a canonical mapping between finite relations (finite subsets of a cartesian product) and functions with finite support into finite subsets. (Contributed by Thierry Arnoux, 28-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝑀 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) & ⊢ 𝑆 = {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑𝑚 𝐴) ∣ (𝑓 supp ∅) ∈ Fin} ⇒ ⊢ (𝑀 ↾ 𝑆):𝑆–1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin) | ||
Theorem | addeq0 28898 | Two complex which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 2-May-2017.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) = 0 ↔ 𝐴 = -𝐵)) | ||
Theorem | subeqxfrd 28899 | Transfer two terms of a subtraction in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐵 − 𝐷)) | ||
Theorem | znsqcld 28900 | Squaring of nonzero relative numbers. (Contributed by Thierry Arnoux, 2-Feb-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ≠ 0) ⇒ ⊢ (𝜑 → (𝑁↑2) ∈ ℕ) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |