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Theorem List for Metamath Proof Explorer - 30201-30300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdmdcompli 30201 A condition equivalent to the dual modular pair property. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝑀* 𝐵 ↔ (𝐴 ∩ (⊥‘(𝐴𝐵))) 𝑀* (𝐵 ∩ (⊥‘(𝐴𝐵))))

Theoremmddmdin0i 30202* If dual modular implies modular whenever meet is zero, then dual modular implies modular for arbitrary lattice elements. This theorem is needed for the remark after Lemma 7 of [Holland] p. 1524 to hold. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝑥C𝑦C ((𝑥 𝑀* 𝑦 ∧ (𝑥𝑦) = 0) → 𝑥 𝑀 𝑦)       (𝐴 𝑀* 𝐵𝐴 𝑀 𝐵)

Theoremcdjreui 30203* A member of the sum of disjoint subspaces has a unique decomposition. Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 20-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S       ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → ∃!𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))

Theoremcdj1i 30204* Two ways to express "𝐴 and 𝐵 are completely disjoint subspaces." (1) => (2) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 21-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S       (∃𝑤 ∈ ℝ (0 < 𝑤 ∧ ∀𝑦𝐴𝑣𝐵 ((norm𝑦) + (norm𝑣)) ≤ (𝑤 · (norm‘(𝑦 + 𝑣)))) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦𝐴𝑧𝐵 ((norm𝑦) = 1 → 𝑥 ≤ (norm‘(𝑦 𝑧)))))

Theoremcdj3lem1 30205* A property of "𝐴 and 𝐵 are completely disjoint subspaces." Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S       (∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦𝐴𝑧𝐵 ((norm𝑦) + (norm𝑧)) ≤ (𝑥 · (norm‘(𝑦 + 𝑧)))) → (𝐴𝐵) = 0)

Theoremcdj3lem2 30206* Lemma for cdj3i 30212. Value of the first-component function 𝑆. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))       ((𝐶𝐴𝐷𝐵 ∧ (𝐴𝐵) = 0) → (𝑆‘(𝐶 + 𝐷)) = 𝐶)

Theoremcdj3lem2a 30207* Lemma for cdj3i 30212. Closure of the first-component function 𝑆. (Contributed by NM, 25-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))       ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → (𝑆𝐶) ∈ 𝐴)

Theoremcdj3lem2b 30208* Lemma for cdj3i 30212. The first-component function 𝑆 is bounded if the subspaces are completely disjoint. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))       (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))

Theoremcdj3lem3 30209* Lemma for cdj3i 30212. Value of the second-component function 𝑇. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑇 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))       ((𝐶𝐴𝐷𝐵 ∧ (𝐴𝐵) = 0) → (𝑇‘(𝐶 + 𝐷)) = 𝐷)

Theoremcdj3lem3a 30210* Lemma for cdj3i 30212. Closure of the second-component function 𝑇. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑇 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))       ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → (𝑇𝐶) ∈ 𝐵)

Theoremcdj3lem3b 30211* Lemma for cdj3i 30212. The second-component function 𝑇 is bounded if the subspaces are completely disjoint. (Contributed by NM, 31-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑇 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))       (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))

Theoremcdj3i 30212* Two ways to express "𝐴 and 𝐵 are completely disjoint subspaces." (1) <=> (3) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 1-Jun-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))    &   𝑇 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))    &   (𝜑 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))    &   (𝜓 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))       (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) ↔ ((𝐴𝐵) = 0𝜑𝜓))

PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)

20.1  Mathboxes for user contributions

20.1.1  Mathbox guidelines

Theoremmathbox 30213 (This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of set.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of set.mm.

Mathboxes are provided to help keep your work synchronized with changes in set.mm while allowing you to work independently without affecting other contributors. Even though in a sense your mathbox belongs to you, it is still part of the shared body of knowledge contained in set.mm, and occasionally other people may make maintenance edits to your mathbox for things like keeping it synchronized with the rest of set.mm, reducing proof lengths, moving your theorems to the main part of set.mm when needed, and fixing typos or other errors. If you want to preserve it the way you left it, you can keep a local copy or keep track of the GitHub commit number.

Guidelines:

1. See conventions 28173 for our general style guidelines. For contributing via GitHub, see https://github.com/metamath/set.mm/blob/develop/CONTRIBUTING.md 28173. The Metamath program command "verify markup *" will check that you have followed many of of the conventions we use.

2. If at all possible, please use only nullary class constants for new definitions, for example as in df-div 11292.

3. Each \$p and \$a statement must be immediately preceded with the comment that will be shown on its web page description. The Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command will take care of indentation conventions and line wrapping.

4. All mathbox content will be on public display and should hopefully reflect the overall quality of the website.

5. Mathboxes must be independent from one another (checked by "verify markup *"). If you need a theorem from another mathbox, typically it is moved to the main part of set.mm. New users should consult with more experienced users before doing this.

6. If a contributor is no longer active, we will continue the usual maintenance edits. As time goes on, often theorems will be moved to main or removed in favor of similar replacements. But we are also willing to maintain mathboxes in place, as work by others from years ago may form the foundation of future work; you could even argue that all of mathematics is like that.

7. For theorems of importance (for example, a Metamath 100 theorem or a dependency of one), we prefer to eventually move them out of mathboxes (although a mathbox is perfectly appropriate as proofs are being developed and refined). (Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

𝜑       𝜑

20.2  Mathbox for Stefan Allan

Theoremsa-abvi 30214 A theorem about the universal class. Inference associated with bj-abv 34218 (which is proved from fewer axioms). (Contributed by Stefan Allan, 9-Dec-2008.)
𝜑       V = {𝑥𝜑}

Theoremxfree 30215 A partial converse to 19.9t 2199. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)
(∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑𝜑))

Theoremxfree2 30216 A partial converse to 19.9t 2199. (Contributed by Stefan Allan, 21-Dec-2008.)
(∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

TheoremaddltmulALT 30217 A proof readability experiment for addltmul 11867. (Contributed by Stefan Allan, 30-Oct-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (2 < 𝐴 ∧ 2 < 𝐵)) → (𝐴 + 𝐵) < (𝐴 · 𝐵))

20.3  Mathbox for Thierry Arnoux

20.3.1  Propositional Calculus - misc additions

Theorembian1d 30218 Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       (𝜑 → ((𝜒𝜓) ↔ (𝜒𝜃)))

Theoremor3di 30219 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
((𝜑 ∨ (𝜓𝜒𝜏)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜏)))

Theoremor3dir 30220 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
(((𝜑𝜓𝜒) ∨ 𝜏) ↔ ((𝜑𝜏) ∧ (𝜓𝜏) ∧ (𝜒𝜏)))

Theorem3o1cs 30221 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theorem3o2cs 30222 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜓𝜃)

Theorem3o3cs 30223 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜒𝜃)

20.3.2  Predicate Calculus

20.3.2.1  Predicate Calculus - misc additions

Theoremsbc2iedf 30224* Conversion of implicit substitution to explicit class substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝜒    &   𝑦𝜒    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))

Theoremrspc2daf 30225* Double restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝜒    &   𝑦𝜒    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝑉𝑦𝑊 𝜓)       (𝜑𝜒)

TheoremnelbOLD 30226* Obsolete version of nelb 3268 as of 23-Jan-2024. (Contributed by Thierry Arnoux, 20-Nov-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)

20.3.2.2  Restricted quantification - misc additions

Theoremralcom4f 30227* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
𝑦𝐴       (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)

Theoremrexcom4f 30228* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
𝑦𝐴       (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)

Theorem19.9d2rf 30229 A deduction version of one direction of 19.9 2200 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜓)

Theorem19.9d2r 30230* A deduction version of one direction of 19.9 2200 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜓)

Theoremr19.29ffa 30231* A commonly used pattern based on r19.29 3254, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)       ((𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → 𝜒)

20.3.2.3  Equality

Theoremeqtrb 30232 A transposition of equality. (Contributed by Thierry Arnoux, 20-Aug-2023.)
((𝐴 = 𝐵𝐴 = 𝐶) ↔ (𝐴 = 𝐵𝐵 = 𝐶))

20.3.2.4  Double restricted existential uniqueness quantification

Theoremopsbc2ie 30233* Conversion of implicit substitution to explicit class substitution for ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))       (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜒))

Theoremopreu2reuALT 30234* Correspondence between uniqueness of ordered pairs and double restricted existential uniqueness quantification. Alternate proof of one direction only, use opreu2reurex 6140 instead. (Contributed by Thierry Arnoux, 4-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))       ((∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒) → ∃!𝑝 ∈ (𝐴 × 𝐵)𝜑)

20.3.2.5  Double restricted existential uniqueness quantification syntax

Syntaxw2reu 30235 Syntax for double restricted existential uniqueness quantification.
wff ∃!𝑥𝐴 , 𝑦𝐵𝜑

Definitiondf-2reu 30236 Define the double restricted existential uniqueness quantifier. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))

Theorem2reucom 30237 Double restricted existential uniqueness commutes. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ ∃!𝑦𝐵 , 𝑥𝐴𝜑)

Theorem2reu2rex1 30238 Double restricted existential uniqueness implies double restricted existence. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)

Theorem2reureurex 30239 Double restricted existential uniqueness implies restricted existential uniqueness with restricted existence. (Contributed by AV, 5-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 → ∃!𝑥𝐴𝑦𝐵 𝜑)

Theorem2reu2reu2 30240* Double restricted existential uniqueness implies two nested restricted existential uniqueness. (Contributed by AV, 5-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)

Theoremopreu2reu1 30241* Equivalent definition of the double restricted existential uniqueness quantifier, using uniqueness of ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(𝑝 = ⟨𝑥, 𝑦⟩ → (𝜒𝜑))       (∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒)

Theoremsq2reunnltb 30242* There exists a unique decomposition of a prime as a sum of squares of two different positive integers iff the prime is of the form 4𝑘 + 1. Double restricted existential uniqueness variant of 2sqreunnltb 26031. (Contributed by AV, 5-Jul-2023.)
(𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑎 ∈ ℕ , 𝑏 ∈ ℕ(𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))

Theoremaddsqnot2reu 30243* For each complex number 𝐶, there does not uniquely exist two complex numbers 𝑎 and 𝑏, with 𝑏 squared and added to 𝑎 resulting in the given complex number 𝐶. Double restricted existential uniqueness variant of addsqn2reurex2 26015. (Contributed by AV, 5-Jul-2023.)
(𝐶 ∈ ℂ → ¬ ∃!𝑎 ∈ ℂ , 𝑏 ∈ ℂ(𝑎 + (𝑏↑2)) = 𝐶)

20.3.2.6  Substitution (without distinct variables) - misc additions

Theoremsbceqbidf 30244 Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))

Theoremsbcies 30245* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝐴 = (𝐸𝑊)    &   (𝑎 = 𝐴 → (𝜑𝜓))       (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎]𝜓𝜑))

20.3.2.7  Existential "at most one" - misc additions

Theoremmoel 30246* "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.)
(∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦)

Theoremmo5f 30247* Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)
𝑖𝜑    &   𝑗𝜑       (∃*𝑥𝜑 ↔ ∀𝑖𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))

Theoremnmo 30248* Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)
𝑦𝜑       (¬ ∃*𝑥𝜑 ↔ ∀𝑦𝑥(𝜑𝑥𝑦))

20.3.2.8  Existential uniqueness - misc additions

Theoremreuxfrdf 30249* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfrd 3739 (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) (Revised by Thierry Arnoux, 30-Mar-2018.)
𝑦𝐵    &   ((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃*𝑦𝐶 𝑥 = 𝐴)       (𝜑 → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐶 𝜓))

Theoremrexunirn 30250* Restricted existential quantification over the union of the range of a function. Cf. rexrn 6848 and eluni2 4836. (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑥𝐴𝐵𝑉)       (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦 ran 𝐹𝜑)

20.3.2.9  Restricted "at most one" - misc additions

Theoremrmoxfrd 30251* Transfer "at most one" restricted quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐵 𝜓 ↔ ∃*𝑦𝐶 𝜒))

Theoremrmoun 30252 "At most one" restricted existential quantifier for a union implies the same quantifier on both sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
(∃*𝑥 ∈ (𝐴𝐵)𝜑 → (∃*𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑))

Theoremrmounid 30253* Case where an "at most one" restricted existential quantifier for a union is equivalent to such a quantifier for one of the sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
((𝜑𝑥𝐵) → ¬ 𝜓)       (𝜑 → (∃*𝑥 ∈ (𝐴𝐵)𝜓 ↔ ∃*𝑥𝐴 𝜓))

20.3.3  General Set Theory

20.3.3.1  Class abstractions (a.k.a. class builders)

Theoremdmrab 30254* Domain of a restricted class abstraction over a cartesian product. (Contributed by Thierry Arnoux, 3-Jul-2023.)
(𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥𝐴 ∣ ∃𝑦𝐵 𝜓}

Theoremdifrab2 30255 Difference of two restricted class abstractions. Compare with difrab 4277. (Contributed by Thierry Arnoux, 3-Jan-2022.)
({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}

TheoremrabexgfGS 30256 Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.)
𝑥𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theoremrabsnel 30257* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.)
𝐵 ∈ V       ({𝑥𝐴𝜑} = {𝐵} → 𝐵𝐴)

Theoremrabeqsnd 30258* Conditions for a restricted class abstraction to be a singleton, in deduction form. (Contributed by Thierry Arnoux, 2-Dec-2021.)
(𝑥 = 𝐵 → (𝜓𝜒))    &   (𝜑𝐵𝐴)    &   (𝜑𝜒)    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝑥 = 𝐵)       (𝜑 → {𝑥𝐴𝜓} = {𝐵})

Theoremforesf1o 30259* From a surjective function, *choose* a subset of the domain, such that the restricted function is bijective. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝐴𝑉𝐹:𝐴onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)

Theoremrabfodom 30260* Domination relation for restricted abstract class builders, based on a surjective function. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝜑𝑥𝐴𝑦 = (𝐹𝑥)) → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴onto𝐵)       (𝜑 → {𝑦𝐵𝜒} ≼ {𝑥𝐴𝜓})

20.3.3.2  Image Sets

Theoremabrexdomjm 30261* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑦𝐴 → ∃*𝑥𝜑)       (𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝜑} ≼ 𝐴)

Theoremabrexdom2jm 30262* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ≼ 𝐴)

Theoremabrexexd 30263* Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)
𝑥𝐴    &   (𝜑𝐴 ∈ V)       (𝜑 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)

Theoremelabreximd 30264* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑥𝜑    &   𝑥𝜒    &   (𝐴 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐶) → 𝜓)       ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)

Theoremelabreximdv 30265* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
(𝐴 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐶) → 𝜓)       ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)

Theoremabrexss 30266* A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)
𝑥𝐶       (∀𝑥𝐴 𝐵𝐶 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶)

20.3.3.3  Set relations and operations - misc additions

Theoremelunsn 30267 Elementhood to a union with a singleton. (Contributed by Thierry Arnoux, 14-Dec-2023.)
(𝐴𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴𝐵𝐴 = 𝐶)))

Theoremnelun 30268 Negated membership for a union. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝐴 = (𝐵𝐶) → (¬ 𝑋𝐴 ↔ (¬ 𝑋𝐵 ∧ ¬ 𝑋𝐶)))

Theoremdisjdifr 30269 A class and its relative complement are disjoint. (Contributed by Thierry Arnoux, 29-Nov-2023.)
((𝐵𝐴) ∩ 𝐴) = ∅

Theoremrabss3d 30270* Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})

Theoreminin 30271 Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Theoreminindif 30272 See inundif 4427. (Contributed by Thierry Arnoux, 13-Sep-2017.)
((𝐴𝐶) ∩ (𝐴𝐶)) = ∅

Theoremdifininv 30273 Condition for the intersections of two sets with a given set to be equal. (Contributed by Thierry Arnoux, 28-Dec-2021.)
((((𝐴𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶𝐴) ∩ 𝐵) = ∅) → (𝐴𝐵) = (𝐶𝐵))

Theoremdifeq 30274 Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
((𝐴𝐵) = 𝐶 ↔ ((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)))

Theoremeqdif 30275 If both set differences of two sets are empty, those sets are equal. (Contributed by Thierry Arnoux, 16-Nov-2023.)
(((𝐴𝐵) = ∅ ∧ (𝐵𝐴) = ∅) → 𝐴 = 𝐵)

Theoremdifxp1ss 30276 Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
((𝐴𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)

Theoremdifxp2ss 30277 Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
(𝐴 × (𝐵𝐶)) ⊆ (𝐴 × 𝐵)

Theoremundifr 30278 Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023.)
(𝐴𝐵 ↔ ((𝐵𝐴) ∪ 𝐴) = 𝐵)

Theoremindifundif 30279 A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)
(((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵)) = (𝐴 ∖ (𝐵𝐶))

Theoremelpwincl1 30280 Closure of intersection with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐴 ∈ 𝒫 𝐶)       (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)

Theoremelpwdifcl 30281 Closure of class difference with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐴 ∈ 𝒫 𝐶)       (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)

Theoremelpwiuncl 30282* Closure of indexed union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 27-May-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ 𝒫 𝐶)       (𝜑 𝑘𝐴 𝐵 ∈ 𝒫 𝐶)

20.3.3.4  Unordered pairs

Theoremeqsnd 30283* Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.)
((𝜑𝑥𝐴) → 𝑥 = 𝐵)    &   (𝜑𝐵𝐴)       (𝜑𝐴 = {𝐵})

Theoremelpreq 30284 Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)
(𝜑𝑋 ∈ {𝐴, 𝐵})    &   (𝜑𝑌 ∈ {𝐴, 𝐵})    &   (𝜑 → (𝑋 = 𝐴𝑌 = 𝐴))       (𝜑𝑋 = 𝑌)

Theoremnelpr 30285 A set 𝐴 not in a pair is neither element of the pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝐴𝑉 → (¬ 𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴𝐵𝐴𝐶)))

Theoreminpr0 30286 Rewrite an empty intersection with a pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ (¬ 𝐵𝐴 ∧ ¬ 𝐶𝐴))

Theoremneldifpr1 30287 The first element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})

Theoremneldifpr2 30288 The second element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})

Theoremunidifsnel 30289 The other element of a pair is an element of the pair. (Contributed by Thierry Arnoux, 26-Aug-2017.)
((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ 𝑃)

Theoremunidifsnne 30290 The other element of a pair is not the known element. (Contributed by Thierry Arnoux, 26-Aug-2017.)
((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≠ 𝑋)

20.3.3.5  Conditional operator - misc additions

Theoremifeqeqx 30291* An equality theorem tailored for ballotlemsf1o 31766. (Contributed by Thierry Arnoux, 14-Apr-2017.)
(𝑥 = 𝑋𝐴 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝑎)    &   (𝑥 = 𝑋 → (𝜒𝜃))    &   (𝑥 = 𝑌 → (𝜒𝜓))    &   (𝜑𝑎 = 𝐶)    &   ((𝜑𝜓) → 𝜃)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑊)       ((𝜑𝑥 = if(𝜓, 𝑋, 𝑌)) → 𝑎 = if(𝜒, 𝐴, 𝐵))

Theoremelimifd 30292 Elimination of a conditional operator contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒𝜃)))    &   (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒𝜏)))       (𝜑 → (𝜒 ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))

Theoremelim2if 30293 Elimination of two conditional operators contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒𝜃))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒𝜏))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒𝜂))       (𝜒 ↔ ((𝜑𝜃) ∨ (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))))

Theoremelim2ifim 30294 Elimination of two conditional operators for an implication. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒𝜃))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒𝜏))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒𝜂))    &   (𝜑𝜃)    &   ((¬ 𝜑𝜓) → 𝜏)    &   ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜂)       𝜒

Theoremifeq3da 30295 Given an expression 𝐶 containing if(𝜓, 𝐸, 𝐹), substitute (hypotheses .1 and .2) and evaluate (hypotheses .3 and .4) it for both cases at the same time. (Contributed by Thierry Arnoux, 13-Dec-2021.)
(if(𝜓, 𝐸, 𝐹) = 𝐸𝐶 = 𝐺)    &   (if(𝜓, 𝐸, 𝐹) = 𝐹𝐶 = 𝐻)    &   (𝜑𝐺 = 𝐴)    &   (𝜑𝐻 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)

20.3.3.6  Set union

Theoremuniinn0 30296* Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.)
(( 𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)

Theoremuniin1 30297* Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
𝑥𝐴 (𝑥𝐵) = ( 𝐴𝐵)

Theoremuniin2 30298* Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
𝑥𝐵 (𝐴𝑥) = (𝐴 𝐵)

Theoremdifuncomp 30299 Express a class difference using unions and class complements. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝐴𝐶 → (𝐴𝐵) = (𝐶 ∖ ((𝐶𝐴) ∪ 𝐵)))

Theoremelpwunicl 30300 Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020.) (Proof shortened by BJ, 6-Apr-2024.)
(𝜑𝐴 ∈ 𝒫 𝒫 𝐵)       (𝜑 𝐴 ∈ 𝒫 𝐵)

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44899
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