P. 71 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)

Type	Label	Description
Statement
2.6.34  Supremum and infimum
Syntax	csup 7001	Extend class notation to include supremum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class sup(𝐴, 𝐵, 𝑅)
Syntax	cinf 7002	Extend class notation to include infimum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class inf(𝐴, 𝐵, 𝑅)
Definition	df-sup 7003*	Define the supremum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the supremum exists. (Contributed by NM, 22-May-1999.)
⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
Definition	df-inf 7004	Define the infimum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the infimum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)
Theorem	supeq1 7005	Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
Theorem	supeq1d 7006	Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
Theorem	supeq1i 7007	Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
Theorem	supeq2 7008	Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
Theorem	supeq3 7009	Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))
Theorem	supeq123d 7010	Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ (𝜑 → 𝐴 = 𝐷)    &   ⊢ (𝜑 → 𝐵 = 𝐸)    &   ⊢ (𝜑 → 𝐶 = 𝐹)    ⇒   ⊢ (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
Theorem	nfsup 7011	Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝑅    ⇒   ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅)
Theorem	supmoti 7012*	Any class 𝐵 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). The hypothesis is satisfied by real numbers (see lttri3 8057) or other orders which correspond to tight apartnesses. (Contributed by Jim Kingdon, 23-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    ⇒   ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
Theorem	supeuti 7013*	A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by Jim Kingdon, 23-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
Theorem	supval2ti 7014*	Alternate expression for the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
Theorem	eqsupti 7015*	Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶))
Theorem	eqsuptid 7016*	Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 24-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	supclti 7017*	A supremum belongs to its base class (closure law). See also supubti 7018 and suplubti 7019. (Contributed by Jim Kingdon, 24-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
Theorem	supubti 7018*	A supremum is an upper bound. See also supclti 7017 and suplubti 7019. This proof demonstrates how to expand an iota-based definition (df-iota 5193) using riotacl2 5861. (Contributed by Jim Kingdon, 24-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
Theorem	suplubti 7019*	A supremum is the least upper bound. See also supclti 7017 and supubti 7018. (Contributed by Jim Kingdon, 24-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
Theorem	suplub2ti 7020*	Bidirectional form of suplubti 7019. (Contributed by Jim Kingdon, 17-Jan-2022.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    &   ⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
Theorem	supelti 7021*	Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐶 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐶)
Theorem	sup00 7022	The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
⊢ sup(𝐵, ∅, 𝑅) = ∅
Theorem	supmaxti 7023*	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 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	supsnti 7024*	The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → sup({𝐵}, 𝐴, 𝑅) = 𝐵)
Theorem	isotilem 7025*	Lemma for isoti 7026. (Contributed by Jim Kingdon, 26-Nov-2021.)
⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))
Theorem	isoti 7026*	An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.)
⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))
Theorem	supisolem 7027*	Lemma for supisoti 7029. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐴) → ((∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐷 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
Theorem	supisoex 7028*	Lemma for supisoti 7029. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
Theorem	supisoti 7029*	Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))    &   ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    ⇒   ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
Theorem	infeq1 7030	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
Theorem	infeq1d 7031	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
Theorem	infeq1i 7032	Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)
Theorem	infeq2 7033	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))
Theorem	infeq3 7034	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))
Theorem	infeq123d 7035	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝐴 = 𝐷)    &   ⊢ (𝜑 → 𝐵 = 𝐸)    &   ⊢ (𝜑 → 𝐶 = 𝐹)    ⇒   ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))
Theorem	nfinf 7036	Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝑅    ⇒   ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅)
Theorem	cnvinfex 7037*	Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
Theorem	cnvti 7038*	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 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    ⇒   ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢)))
Theorem	eqinfti 7039*	Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))
Theorem	eqinftid 7040*	Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	infvalti 7041*	Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))
Theorem	infclti 7042*	An infimum belongs to its base class (closure law). See also inflbti 7043 and infglbti 7044. (Contributed by Jim Kingdon, 17-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)
Theorem	inflbti 7043*	An infimum is a lower bound. See also infclti 7042 and infglbti 7044. (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))
Theorem	infglbti 7044*	An infimum is the greatest lower bound. See also infclti 7042 and inflbti 7043. (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
Theorem	infnlbti 7045*	A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))
Theorem	infminti 7046*	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 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	infmoti 7047*	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 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    ⇒   ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
Theorem	infeuti 7048*	An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
Theorem	infsnti 7049*	The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → inf({𝐵}, 𝐴, 𝑅) = 𝐵)
Theorem	inf00 7050	The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
⊢ inf(𝐵, ∅, 𝑅) = ∅
Theorem	infisoti 7051*	Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦)))    &   ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    ⇒   ⊢ (𝜑 → inf((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))
Theorem	supex2g 7052	Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ 𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V)
Theorem	infex2g 7053	Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
⊢ (𝐴 ∈ 𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V)
2.6.35  Ordinal isomorphism
Theorem	ordiso2 7054	Generalize ordiso 7055 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)
Theorem	ordiso 7055*	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)))
2.6.36  Disjoint union
2.6.36.1  Disjoint union
Syntax	cdju 7056	Extend class notation to include disjoint union of two classes.
class (𝐴 ⊔ 𝐵)
Definition	df-dju 7057	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.)
⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
Theorem	djueq12 7058	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷))
Theorem	djueq1 7059	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶))
Theorem	djueq2 7060	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵))
Theorem	nfdju 7061	Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵)
Theorem	djuex 7062	The disjoint union of sets is a set. See also the more precise djuss 7089. (Contributed by AV, 28-Jun-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V)
Theorem	djuexb 7063	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) ↔ (𝐴 ⊔ 𝐵) ∈ V)
2.6.36.2  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).
Syntax	cinl 7064	Extend class notation to include left injection of a disjoint union.
class inl
Syntax	cinr 7065	Extend class notation to include right injection of a disjoint union.
class inr
Definition	df-inl 7066	Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
Definition	df-inr 7067	Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
Theorem	djulclr 7068	Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djurclr 7069	Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djulcl 7070	Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djurcl 7071	Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djuf1olem 7072*	Lemma for djulf1o 7077 and djurf1o 7078. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
⊢ 𝑋 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ⟨𝑋, 𝑥⟩)    ⇒   ⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)
Theorem	djuf1olemr 7073*	Lemma for djulf1or 7075 and djurf1or 7076. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 7072. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
⊢ 𝑋 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)    ⇒   ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→({𝑋} × 𝐴)
Theorem	djulclb 7074	Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ 𝐴 ↔ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)))
Theorem	djulf1or 7075	The left injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
Theorem	djurf1or 7076	The right injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
⊢ (inr ↾ 𝐴):𝐴–1-1-onto→({1o} × 𝐴)
Theorem	djulf1o 7077	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)
Theorem	djurf1o 7078	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)
Theorem	inresflem 7079*	Lemma for inlresf1 7080 and inrresf1 7081. (Contributed by BJ, 4-Jul-2022.)
⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)    &   ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝐵)    ⇒   ⊢ 𝐹:𝐴–1-1→𝐵
Theorem	inlresf1 7080	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):𝐴–1-1→(𝐴 ⊔ 𝐵)
Theorem	inrresf1 7081	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):𝐵–1-1→(𝐴 ⊔ 𝐵)
Theorem	djuinr 7082	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 7112 and djufun 7123) 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 7104). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅
Theorem	djuin 7083	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)) = ∅
Theorem	inl11 7084	Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵))
Theorem	djuunr 7085	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)) = (𝐴 ⊔ 𝐵)
Theorem	djuun 7086	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)) = (𝐴 ⊔ 𝐵)
Theorem	eldju 7087*	Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)
⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
Theorem	djur 7088*	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 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))
2.6.36.3  Universal property of the disjoint union
Theorem	djuss 7089	A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
Theorem	eldju1st 7090	The first component of an element of a disjoint union is either ∅ or 1o. (Contributed by AV, 26-Jun-2022.)
⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))
Theorem	eldju2ndl 7091	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 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) = ∅) → (2nd ‘𝑋) ∈ 𝐴)
Theorem	eldju2ndr 7092	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 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) ≠ ∅) → (2nd ‘𝑋) ∈ 𝐵)
Theorem	1stinl 7093	The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅)
Theorem	2ndinl 7094	The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)
Theorem	1stinr 7095	The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o)
Theorem	2ndinr 7096	The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)
Theorem	djune 7097	Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))
Theorem	updjudhf 7098*	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.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))    ⇒   ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
Theorem	updjudhcoinlf 7099*	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.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))    ⇒   ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)
Theorem	updjudhcoinrg 7100*	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.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))    ⇒   ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)