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Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsupubti 6801* A supremum is an upper bound. See also supclti 6800 and suplubti 6802.

This proof demonstrates how to expand an iota-based definition (df-iota 5024) using riotacl2 5675.

(Contributed by Jim Kingdon, 24-Nov-2021.)

((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))

Theoremsuplubti 6802* A supremum is the least upper bound. See also supclti 6800 and supubti 6801. (Contributed by Jim Kingdon, 24-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))

Theoremsuplub2ti 6803* Bidirectional form of suplubti 6802. (Contributed by Jim Kingdon, 17-Jan-2022.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))    &   (𝜑𝑅 Or 𝐴)    &   (𝜑𝐵𝐴)       ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧𝐵 𝐶𝑅𝑧))

Theoremsupelti 6804* Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐶 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))    &   (𝜑𝐶𝐴)       (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐶)

Theoremsup00 6805 The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
sup(𝐵, ∅, 𝑅) = ∅

Theoremsupmaxti 6806* The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐶𝐴)    &   (𝜑𝐶𝐵)    &   ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)

Theoremsupsnti 6807* The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐵𝐴)       (𝜑 → sup({𝐵}, 𝐴, 𝑅) = 𝐵)

Theoremisotilem 6808* Lemma for isoti 6809. (Contributed by Jim Kingdon, 26-Nov-2021.)
(𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))

Theoremisoti 6809* An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.)
(𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))

Theoremsupisolem 6810* Lemma for supisoti 6812. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)       ((𝜑𝐷𝐴) → ((∀𝑦𝐶 ¬ 𝐷𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐷 → ∃𝑧𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝐷)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝐷) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))

Theoremsupisoex 6811* Lemma for supisoti 6812. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))       (𝜑 → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))

Theoremsupisoti 6812* Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))    &   ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))

Theoreminfeq1 6813 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Theoreminfeq1d 6814 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Theoreminfeq1i 6815 Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
𝐵 = 𝐶       inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)

Theoreminfeq2 6816 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))

Theoreminfeq3 6817 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))

Theoreminfeq123d 6818 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝐴 = 𝐷)    &   (𝜑𝐵 = 𝐸)    &   (𝜑𝐶 = 𝐹)       (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))

Theoremnfinf 6819 Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑅       𝑥inf(𝐴, 𝐵, 𝑅)

Theoremcnvinfex 6820* Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.)
(𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))

Theoremcnvti 6821* If a relation satisfies a condition corresponding to tightness of an apartness generated by an order, so does its converse. (Contributed by Jim Kingdon, 17-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))

Theoremeqinfti 6822* Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))

Theoremeqinftid 6823* Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐶𝐴)    &   ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)    &   ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)

Theoreminfvalti 6824* Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → inf(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))))

Theoreminfclti 6825* An infimum belongs to its base class (closure law). See also inflbti 6826 and infglbti 6827. (Contributed by Jim Kingdon, 17-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)

Theoreminflbti 6826* An infimum is a lower bound. See also infclti 6825 and infglbti 6827. (Contributed by Jim Kingdon, 18-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → (𝐶𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))

Theoreminfglbti 6827* An infimum is the greatest lower bound. See also infclti 6825 and inflbti 6826. (Contributed by Jim Kingdon, 18-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ((𝐶𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧𝐵 𝑧𝑅𝐶))

Theoreminfnlbti 6828* A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ((𝐶𝐴 ∧ ∀𝑧𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))

Theoreminfminti 6829* The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by Jim Kingdon, 18-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐶𝐴)    &   (𝜑𝐶𝐵)    &   ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)

Theoreminfmoti 6830* Any class 𝐵 has at most one infimum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by Jim Kingdon, 18-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → ∃*𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))

Theoreminfeuti 6831* An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))

Theoreminfsnti 6832* The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐵𝐴)       (𝜑 → inf({𝐵}, 𝐴, 𝑅) = 𝐵)

Theoreminf00 6833 The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
inf(𝐵, ∅, 𝑅) = ∅

Theoreminfisoti 6834* Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))    &   ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → inf((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))

2.6.33  Ordinal isomorphism

Theoremordiso2 6835 Generalize ordiso 6836 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)

Theoremordiso 6836* Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 ↔ ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵)))

2.6.34  Disjoint union

2.6.34.1  Disjoint union

Syntaxcdju 6837 Extend class notation to include disjoint union of two classes.
class (𝐴𝐵)

Definitiondf-dju 6838 Disjoint union of two classes. This is a way of creating a class which contains elements corresponding to each element of 𝐴 or 𝐵, tagging each one with whether it came from 𝐴 or 𝐵. (Contributed by Jim Kingdon, 20-Jun-2022.)
(𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))

Theoremdjueq12 6839 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremdjueq1 6840 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremdjueq2 6841 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremnfdju 6842 Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremdjuex 6843 The disjoint union of sets is a set. See also the more precise djuss 6870. (Contributed by AV, 28-Jun-2022.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Theoremdjuexb 6844 The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)

2.6.34.2  Left and right injections of a disjoint union

In this section, we define the left and right injections of a disjoint union and prove their main properties. These injections are restrictions of the "template" functions inl and inr, which appear in most applications in the form (inl ↾ 𝐴) and (inr ↾ 𝐵).

Syntaxcinl 6845 Extend class notation to include left injection of a disjoint union.
class inl

Syntaxcinr 6846 Extend class notation to include right injection of a disjoint union.
class inr

Definitiondf-inl 6847 Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)

Definitiondf-inr 6848 Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)

Theoremdjulclr 6849 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
(𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴𝐵))

Theoremdjurclr 6850 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
(𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴𝐵))

Theoremdjulcl 6851 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
(𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))

Theoremdjurcl 6852 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
(𝐶𝐵 → (inr‘𝐶) ∈ (𝐴𝐵))

Theoremdjuf1olem 6853* Lemma for djulf1o 6858 and djurf1o 6859. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
𝑋 ∈ V    &   𝐹 = (𝑥𝐴 ↦ ⟨𝑋, 𝑥⟩)       𝐹:𝐴1-1-onto→({𝑋} × 𝐴)

Theoremdjuf1olemr 6854* Lemma for djulf1or 6856 and djurf1or 6857. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 6853. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
𝑋 ∈ V    &   𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)       (𝐹𝐴):𝐴1-1-onto→({𝑋} × 𝐴)

Theoremdjulclb 6855 Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
(𝐶𝑉 → (𝐶𝐴 ↔ (inl‘𝐶) ∈ (𝐴𝐵)))

Theoremdjulf1or 6856 The left injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
(inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)

Theoremdjurf1or 6857 The right injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
(inr ↾ 𝐴):𝐴1-1-onto→({1o} × 𝐴)

Theoremdjulf1o 6858 The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
inl:V–1-1-onto→({∅} × V)

Theoremdjurf1o 6859 The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
inr:V–1-1-onto→({1o} × V)

Theoreminresflem 6860* Lemma for inlresf1 6861 and inrresf1 6862. (Contributed by BJ, 4-Jul-2022.)
𝐹:𝐴1-1-onto→({𝑋} × 𝐴)    &   (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵)       𝐹:𝐴1-1𝐵

Theoreminlresf1 6861 The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
(inl ↾ 𝐴):𝐴1-1→(𝐴𝐵)

Theoreminrresf1 6862 The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
(inr ↾ 𝐵):𝐵1-1→(𝐴𝐵)

Theoremdjuinr 6863 The ranges of any left and right injections are disjoint. Remark: the extra generality offered by the two restrictions makes the theorem more readily usable (e.g., by djudom 6893 and djufun 6904) while the simpler statement (ran inl ∩ ran inr) = ∅ is easily recovered from it by substituting V for both 𝐴 and 𝐵 as done in casefun 6885). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
(ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅

Theoremdjuin 6864 The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅

Theoreminl11 6865 Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)
((𝐴𝑉𝐵𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵))

Theoremdjuunr 6866 The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 6-Jul-2022.)
(ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴𝐵)

Theoremdjuun 6867 The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴𝐵)

Theoremeldju 6868* Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)
(𝐶 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))

Theoremdjur 6869* A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.)
(𝐶 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥𝐵 𝐶 = (inr‘𝑥)))

2.6.34.3  Universal property of the disjoint union

Theoremdjuss 6870 A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
(𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))

Theoremeldju1st 6871 The first component of an element of a disjoint union is either or 1o. (Contributed by AV, 26-Jun-2022.)
(𝑋 ∈ (𝐴𝐵) → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))

Theoremeldju2ndl 6872 The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.)
((𝑋 ∈ (𝐴𝐵) ∧ (1st𝑋) = ∅) → (2nd𝑋) ∈ 𝐴)

Theoremeldju2ndr 6873 The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.)
((𝑋 ∈ (𝐴𝐵) ∧ (1st𝑋) ≠ ∅) → (2nd𝑋) ∈ 𝐵)

Theorem1stinl 6874 The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.)
(𝑋𝑉 → (1st ‘(inl‘𝑋)) = ∅)

Theorem2ndinl 6875 The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)
(𝑋𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)

Theorem1stinr 6876 The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.)
(𝑋𝑉 → (1st ‘(inr‘𝑋)) = 1o)

Theorem2ndinr 6877 The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)
(𝑋𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)

Theoremdjune 6878 Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.)
((𝐴𝑉𝐵𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))

Theoremupdjudhf 6879* The mapping of an element of the disjoint union to the value of the corresponding function is a function. (Contributed by AV, 26-Jun-2022.)
(𝜑𝐹:𝐴𝐶)    &   (𝜑𝐺:𝐵𝐶)    &   𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))       (𝜑𝐻:(𝐴𝐵)⟶𝐶)

Theoremupdjudhcoinlf 6880* The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-Jun-2022.)
(𝜑𝐹:𝐴𝐶)    &   (𝜑𝐺:𝐵𝐶)    &   𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))       (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)

Theoremupdjudhcoinrg 6881* The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.)
(𝜑𝐹:𝐴𝐶)    &   (𝜑𝐺:𝐵𝐶)    &   𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))       (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)

Theoremupdjud 6882* Universal property of the disjoint union. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.)
(𝜑𝐹:𝐴𝐶)    &   (𝜑𝐺:𝐵𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → ∃!(:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺))

Syntaxcdjucase 6883 Syntax for the "case" construction.
class case(𝑅, 𝑆)

Definitiondf-case 6884 The "case" construction: if 𝐹:𝐴𝑋 and 𝐺:𝐵𝑋 are functions, then case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋 is the natural function obtained by a definition by cases, hence the name. It is the unique function whose existence is asserted by the universal property of disjoint unions updjud 6882. The definition is adapted to make sense also for binary relations (where the universal property also holds). (Contributed by MC and BJ, 10-Jul-2022.)
case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))

Theoremcasefun 6885 The "case" construction of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
(𝜑 → Fun 𝐹)    &   (𝜑 → Fun 𝐺)       (𝜑 → Fun case(𝐹, 𝐺))

Theoremcasedm 6886 The domain of the "case" construction is the disjoint union of the domains. TODO (although less important): ran case(𝐹, 𝐺) = (ran 𝐹 ∪ ran 𝐺). (Contributed by BJ, 10-Jul-2022.)
dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)

Theoremcaserel 6887 The "case" construction of two relations is a relation, with bounds on its domain and codomain. Typically, the "case" construction is used when both relations have a common codomain. (Contributed by BJ, 10-Jul-2022.)
case(𝑅, 𝑆) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))

Theoremcasef 6888 The "case" construction of two functions is a function on the disjoint union of their domains. (Contributed by BJ, 10-Jul-2022.)
(𝜑𝐹:𝐴𝑋)    &   (𝜑𝐺:𝐵𝑋)       (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋)

Theoremcaseinj 6889 The "case" construction of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)
(𝜑 → Fun 𝑅)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → (ran 𝑅 ∩ ran 𝑆) = ∅)       (𝜑 → Fun case(𝑅, 𝑆))

Theoremcasef1 6890 The "case" construction of two injective functions with disjoint ranges is an injective function. (Contributed by BJ, 10-Jul-2022.)
(𝜑𝐹:𝐴1-1𝑋)    &   (𝜑𝐺:𝐵1-1𝑋)    &   (𝜑 → (ran 𝐹 ∩ ran 𝐺) = ∅)       (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)–1-1𝑋)

Theoremcaseinl 6891 Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.)
(𝜑𝐹 Fn 𝐵)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝐴𝐵)       (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹𝐴))

Theoremcaseinr 6892 Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.)
(𝜑 → Fun 𝐹)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝐵)       (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = (𝐺𝐴))

2.6.34.4  Dominance and equinumerosity properties of disjoint union

Theoremdjudom 6893 Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ≼ (𝐵𝐷))

Theoremomp1eomlem 6894* Lemma for omp1eom 6895. (Contributed by Jim Kingdon, 11-Jul-2023.)
𝐹 = (𝑥 ∈ ω ↦ if(𝑥 = ∅, (inr‘𝑥), (inl‘ 𝑥)))    &   𝑆 = (𝑥 ∈ ω ↦ suc 𝑥)    &   𝐺 = case(𝑆, ( I ↾ 1o))       𝐹:ω–1-1-onto→(ω ⊔ 1o)

Theoremomp1eom 6895 Adding one to ω. (Contributed by Jim Kingdon, 10-Jul-2023.)
(ω ⊔ 1o) ≈ ω

Theoremendjusym 6896 Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ≈ (𝐵𝐴))

Theoremeninl 6897 Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
(𝐴𝑉 → (inl “ 𝐴) ≈ 𝐴)

Theoremeninr 6898 Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
(𝐴𝑉 → (inr “ 𝐴) ≈ 𝐴)

Theoremdifinfsnlem 6899* Lemma for difinfsn 6900. The case where we need to swap 𝐵 and (inr‘∅) in building the mapping 𝐺. (Contributed by Jim Kingdon, 9-Aug-2023.)
(𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)    &   (𝜑𝐵𝐴)    &   (𝜑𝐹:(ω ⊔ 1o)–1-1𝐴)    &   (𝜑 → (𝐹‘(inr‘∅)) ≠ 𝐵)    &   𝐺 = (𝑛 ∈ ω ↦ if((𝐹‘(inl‘𝑛)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑛))))       (𝜑𝐺:ω–1-1→(𝐴 ∖ {𝐵}))

Theoremdifinfsn 6900* An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ (𝐴 ∖ {𝐵}))

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