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Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsupeq2 7001 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐡 = 𝐢 β†’ sup(𝐴, 𝐡, 𝑅) = sup(𝐴, 𝐢, 𝑅))
 
Theoremsupeq3 7002 Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 = 𝑆 β†’ sup(𝐴, 𝐡, 𝑅) = sup(𝐴, 𝐡, 𝑆))
 
Theoremsupeq123d 7003 Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
(πœ‘ β†’ 𝐴 = 𝐷)    &   (πœ‘ β†’ 𝐡 = 𝐸)    &   (πœ‘ β†’ 𝐢 = 𝐹)    β‡’   (πœ‘ β†’ sup(𝐴, 𝐡, 𝐢) = sup(𝐷, 𝐸, 𝐹))
 
Theoremnfsup 7004 Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    &   β„²π‘₯𝑅    β‡’   β„²π‘₯sup(𝐴, 𝐡, 𝑅)
 
Theoremsupmoti 7005* Any class 𝐡 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). The hypothesis is satisfied by real numbers (see lttri3 8050) or other orders which correspond to tight apartnesses. (Contributed by Jim Kingdon, 23-Nov-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    β‡’   (πœ‘ β†’ βˆƒ*π‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐡 𝑦𝑅𝑧)))
 
Theoremsupeuti 7006* 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.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐡 𝑦𝑅𝑧)))    β‡’   (πœ‘ β†’ βˆƒ!π‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐡 𝑦𝑅𝑧)))
 
Theoremsupval2ti 7007* Alternate expression for the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐡 𝑦𝑅𝑧)))    β‡’   (πœ‘ β†’ sup(𝐡, 𝐴, 𝑅) = (β„©π‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐡 𝑦𝑅𝑧))))
 
Theoremeqsupti 7008* Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    β‡’   (πœ‘ β†’ ((𝐢 ∈ 𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 Β¬ 𝐢𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅𝐢 β†’ βˆƒπ‘§ ∈ 𝐡 𝑦𝑅𝑧)) β†’ sup(𝐡, 𝐴, 𝑅) = 𝐢))
 
Theoremeqsuptid 7009* Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 24-Nov-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   ((πœ‘ ∧ 𝑦 ∈ 𝐡) β†’ Β¬ 𝐢𝑅𝑦)    &   ((πœ‘ ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐢)) β†’ βˆƒπ‘§ ∈ 𝐡 𝑦𝑅𝑧)    β‡’   (πœ‘ β†’ sup(𝐡, 𝐴, 𝑅) = 𝐢)
 
Theoremsupclti 7010* A supremum belongs to its base class (closure law). See also supubti 7011 and suplubti 7012. (Contributed by Jim Kingdon, 24-Nov-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐡 𝑦𝑅𝑧)))    β‡’   (πœ‘ β†’ sup(𝐡, 𝐴, 𝑅) ∈ 𝐴)
 
Theoremsupubti 7011* A supremum is an upper bound. See also supclti 7010 and suplubti 7012.

This proof demonstrates how to expand an iota-based definition (df-iota 5190) using riotacl2 5857.

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

((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐡 𝑦𝑅𝑧)))    β‡’   (πœ‘ β†’ (𝐢 ∈ 𝐡 β†’ Β¬ sup(𝐡, 𝐴, 𝑅)𝑅𝐢))
 
Theoremsuplubti 7012* A supremum is the least upper bound. See also supclti 7010 and supubti 7011. (Contributed by Jim Kingdon, 24-Nov-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐡 𝑦𝑅𝑧)))    β‡’   (πœ‘ β†’ ((𝐢 ∈ 𝐴 ∧ 𝐢𝑅sup(𝐡, 𝐴, 𝑅)) β†’ βˆƒπ‘§ ∈ 𝐡 𝐢𝑅𝑧))
 
Theoremsuplub2ti 7013* Bidirectional form of suplubti 7012. (Contributed by Jim Kingdon, 17-Jan-2022.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐡 𝑦𝑅𝑧)))    &   (πœ‘ β†’ 𝑅 Or 𝐴)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐴)    β‡’   ((πœ‘ ∧ 𝐢 ∈ 𝐴) β†’ (𝐢𝑅sup(𝐡, 𝐴, 𝑅) ↔ βˆƒπ‘§ ∈ 𝐡 𝐢𝑅𝑧))
 
Theoremsupelti 7014* Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐢 (βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐡 𝑦𝑅𝑧)))    &   (πœ‘ β†’ 𝐢 βŠ† 𝐴)    β‡’   (πœ‘ β†’ sup(𝐡, 𝐴, 𝑅) ∈ 𝐢)
 
Theoremsup00 7015 The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
sup(𝐡, βˆ…, 𝑅) = βˆ…
 
Theoremsupmaxti 7016* 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.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   (πœ‘ β†’ 𝐢 ∈ 𝐡)    &   ((πœ‘ ∧ 𝑦 ∈ 𝐡) β†’ Β¬ 𝐢𝑅𝑦)    β‡’   (πœ‘ β†’ sup(𝐡, 𝐴, 𝑅) = 𝐢)
 
Theoremsupsnti 7017* The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    β‡’   (πœ‘ β†’ sup({𝐡}, 𝐴, 𝑅) = 𝐡)
 
Theoremisotilem 7018* Lemma for isoti 7019. (Contributed by Jim Kingdon, 26-Nov-2021.)
(𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ = 𝑦 ↔ (Β¬ π‘₯𝑆𝑦 ∧ Β¬ 𝑦𝑆π‘₯)) β†’ βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐴 (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒))))
 
Theoremisoti 7019* An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.)
(𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐡) β†’ (βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐴 (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)) ↔ βˆ€π‘’ ∈ 𝐡 βˆ€π‘£ ∈ 𝐡 (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑆𝑣 ∧ Β¬ 𝑣𝑆𝑒))))
 
Theoremsupisolem 7020* Lemma for supisoti 7022. (Contributed by Mario Carneiro, 24-Dec-2016.)
(πœ‘ β†’ 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐡))    &   (πœ‘ β†’ 𝐢 βŠ† 𝐴)    β‡’   ((πœ‘ ∧ 𝐷 ∈ 𝐴) β†’ ((βˆ€π‘¦ ∈ 𝐢 Β¬ 𝐷𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅𝐷 β†’ βˆƒπ‘§ ∈ 𝐢 𝑦𝑅𝑧)) ↔ (βˆ€π‘€ ∈ (𝐹 β€œ 𝐢) Β¬ (πΉβ€˜π·)𝑆𝑀 ∧ βˆ€π‘€ ∈ 𝐡 (𝑀𝑆(πΉβ€˜π·) β†’ βˆƒπ‘£ ∈ (𝐹 β€œ 𝐢)𝑀𝑆𝑣))))
 
Theoremsupisoex 7021* Lemma for supisoti 7022. (Contributed by Mario Carneiro, 24-Dec-2016.)
(πœ‘ β†’ 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐡))    &   (πœ‘ β†’ 𝐢 βŠ† 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐢 Β¬ π‘₯𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐢 𝑦𝑅𝑧)))    β‡’   (πœ‘ β†’ βˆƒπ‘’ ∈ 𝐡 (βˆ€π‘€ ∈ (𝐹 β€œ 𝐢) Β¬ 𝑒𝑆𝑀 ∧ βˆ€π‘€ ∈ 𝐡 (𝑀𝑆𝑒 β†’ βˆƒπ‘£ ∈ (𝐹 β€œ 𝐢)𝑀𝑆𝑣)))
 
Theoremsupisoti 7022* Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
(πœ‘ β†’ 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐡))    &   (πœ‘ β†’ 𝐢 βŠ† 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐢 Β¬ π‘₯𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐢 𝑦𝑅𝑧)))    &   ((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    β‡’   (πœ‘ β†’ sup((𝐹 β€œ 𝐢), 𝐡, 𝑆) = (πΉβ€˜sup(𝐢, 𝐴, 𝑅)))
 
Theoreminfeq1 7023 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐡 = 𝐢 β†’ inf(𝐡, 𝐴, 𝑅) = inf(𝐢, 𝐴, 𝑅))
 
Theoreminfeq1d 7024 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
(πœ‘ β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ inf(𝐡, 𝐴, 𝑅) = inf(𝐢, 𝐴, 𝑅))
 
Theoreminfeq1i 7025 Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
𝐡 = 𝐢    β‡’   inf(𝐡, 𝐴, 𝑅) = inf(𝐢, 𝐴, 𝑅)
 
Theoreminfeq2 7026 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐡 = 𝐢 β†’ inf(𝐴, 𝐡, 𝑅) = inf(𝐴, 𝐢, 𝑅))
 
Theoreminfeq3 7027 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝑅 = 𝑆 β†’ inf(𝐴, 𝐡, 𝑅) = inf(𝐴, 𝐡, 𝑆))
 
Theoreminfeq123d 7028 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
(πœ‘ β†’ 𝐴 = 𝐷)    &   (πœ‘ β†’ 𝐡 = 𝐸)    &   (πœ‘ β†’ 𝐢 = 𝐹)    β‡’   (πœ‘ β†’ inf(𝐴, 𝐡, 𝐢) = inf(𝐷, 𝐸, 𝐹))
 
Theoremnfinf 7029 Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    &   β„²π‘₯𝑅    β‡’   β„²π‘₯inf(𝐴, 𝐡, 𝑅)
 
Theoremcnvinfex 7030* Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.)
(πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦𝑅π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝑦)))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯◑𝑅𝑦 ∧ βˆ€π‘¦ ∈ 𝐴 (𝑦◑𝑅π‘₯ β†’ βˆƒπ‘§ ∈ 𝐡 𝑦◑𝑅𝑧)))
 
Theoremcnvti 7031* 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.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    β‡’   ((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒◑𝑅𝑣 ∧ Β¬ 𝑣◑𝑅𝑒)))
 
Theoremeqinfti 7032* Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    β‡’   (πœ‘ β†’ ((𝐢 ∈ 𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦𝑅𝐢 ∧ βˆ€π‘¦ ∈ 𝐴 (𝐢𝑅𝑦 β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝑦)) β†’ inf(𝐡, 𝐴, 𝑅) = 𝐢))
 
Theoremeqinftid 7033* Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   ((πœ‘ ∧ 𝑦 ∈ 𝐡) β†’ Β¬ 𝑦𝑅𝐢)    &   ((πœ‘ ∧ (𝑦 ∈ 𝐴 ∧ 𝐢𝑅𝑦)) β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝑦)    β‡’   (πœ‘ β†’ inf(𝐡, 𝐴, 𝑅) = 𝐢)
 
Theoreminfvalti 7034* Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦𝑅π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝑦)))    β‡’   (πœ‘ β†’ inf(𝐡, 𝐴, 𝑅) = (β„©π‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦𝑅π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝑦))))
 
Theoreminfclti 7035* An infimum belongs to its base class (closure law). See also inflbti 7036 and infglbti 7037. (Contributed by Jim Kingdon, 17-Dec-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦𝑅π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝑦)))    β‡’   (πœ‘ β†’ inf(𝐡, 𝐴, 𝑅) ∈ 𝐴)
 
Theoreminflbti 7036* An infimum is a lower bound. See also infclti 7035 and infglbti 7037. (Contributed by Jim Kingdon, 18-Dec-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦𝑅π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝑦)))    β‡’   (πœ‘ β†’ (𝐢 ∈ 𝐡 β†’ Β¬ 𝐢𝑅inf(𝐡, 𝐴, 𝑅)))
 
Theoreminfglbti 7037* An infimum is the greatest lower bound. See also infclti 7035 and inflbti 7036. (Contributed by Jim Kingdon, 18-Dec-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦𝑅π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝑦)))    β‡’   (πœ‘ β†’ ((𝐢 ∈ 𝐴 ∧ inf(𝐡, 𝐴, 𝑅)𝑅𝐢) β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝐢))
 
Theoreminfnlbti 7038* A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦𝑅π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝑦)))    β‡’   (πœ‘ β†’ ((𝐢 ∈ 𝐴 ∧ βˆ€π‘§ ∈ 𝐡 Β¬ 𝑧𝑅𝐢) β†’ Β¬ inf(𝐡, 𝐴, 𝑅)𝑅𝐢))
 
Theoreminfminti 7039* 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.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   (πœ‘ β†’ 𝐢 ∈ 𝐡)    &   ((πœ‘ ∧ 𝑦 ∈ 𝐡) β†’ Β¬ 𝑦𝑅𝐢)    β‡’   (πœ‘ β†’ inf(𝐡, 𝐴, 𝑅) = 𝐢)
 
Theoreminfmoti 7040* Any class 𝐡 has at most one infimum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by Jim Kingdon, 18-Dec-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    β‡’   (πœ‘ β†’ βˆƒ*π‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦𝑅π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝑦)))
 
Theoreminfeuti 7041* An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦𝑅π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝑦)))    β‡’   (πœ‘ β†’ βˆƒ!π‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦𝑅π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ βˆƒπ‘§ ∈ 𝐡 𝑧𝑅𝑦)))
 
Theoreminfsnti 7042* The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)
((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    β‡’   (πœ‘ β†’ inf({𝐡}, 𝐴, 𝑅) = 𝐡)
 
Theoreminf00 7043 The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
inf(𝐡, βˆ…, 𝑅) = βˆ…
 
Theoreminfisoti 7044* Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)
(πœ‘ β†’ 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐡))    &   (πœ‘ β†’ 𝐢 βŠ† 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐢 Β¬ 𝑦𝑅π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ βˆƒπ‘§ ∈ 𝐢 𝑧𝑅𝑦)))    &   ((πœ‘ ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 = 𝑣 ↔ (Β¬ 𝑒𝑅𝑣 ∧ Β¬ 𝑣𝑅𝑒)))    β‡’   (πœ‘ β†’ inf((𝐹 β€œ 𝐢), 𝐡, 𝑆) = (πΉβ€˜inf(𝐢, 𝐴, 𝑅)))
 
Theoremsupex2g 7045 Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 ∈ 𝐢 β†’ sup(𝐡, 𝐴, 𝑅) ∈ V)
 
Theoreminfex2g 7046 Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
(𝐴 ∈ 𝐢 β†’ inf(𝐡, 𝐴, 𝑅) ∈ V)
 
2.6.35  Ordinal isomorphism
 
Theoremordiso2 7047 Generalize ordiso 7048 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹 Isom E , E (𝐴, 𝐡) ∧ Ord 𝐴 ∧ Ord 𝐡) β†’ 𝐴 = 𝐡)
 
Theoremordiso 7048* Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 = 𝐡 ↔ βˆƒπ‘“ 𝑓 Isom E , E (𝐴, 𝐡)))
 
2.6.36  Disjoint union
 
2.6.36.1  Disjoint union
 
Syntaxcdju 7049 Extend class notation to include disjoint union of two classes.
class (𝐴 βŠ” 𝐡)
 
Definitiondf-dju 7050 Disjoint union of two classes. This is a way of creating a class which contains elements corresponding to each element of 𝐴 or 𝐡, tagging each one with whether it came from 𝐴 or 𝐡. (Contributed by Jim Kingdon, 20-Jun-2022.)
(𝐴 βŠ” 𝐡) = (({βˆ…} Γ— 𝐴) βˆͺ ({1o} Γ— 𝐡))
 
Theoremdjueq12 7051 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
((𝐴 = 𝐡 ∧ 𝐢 = 𝐷) β†’ (𝐴 βŠ” 𝐢) = (𝐡 βŠ” 𝐷))
 
Theoremdjueq1 7052 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
(𝐴 = 𝐡 β†’ (𝐴 βŠ” 𝐢) = (𝐡 βŠ” 𝐢))
 
Theoremdjueq2 7053 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
(𝐴 = 𝐡 β†’ (𝐢 βŠ” 𝐴) = (𝐢 βŠ” 𝐡))
 
Theoremnfdju 7054 Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    β‡’   β„²π‘₯(𝐴 βŠ” 𝐡)
 
Theoremdjuex 7055 The disjoint union of sets is a set. See also the more precise djuss 7082. (Contributed by AV, 28-Jun-2022.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐴 βŠ” 𝐡) ∈ V)
 
Theoremdjuexb 7056 The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)
((𝐴 ∈ V ∧ 𝐡 ∈ V) ↔ (𝐴 βŠ” 𝐡) ∈ 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 β†Ύ 𝐴) and (inr β†Ύ 𝐡).

 
Syntaxcinl 7057 Extend class notation to include left injection of a disjoint union.
class inl
 
Syntaxcinr 7058 Extend class notation to include right injection of a disjoint union.
class inr
 
Definitiondf-inl 7059 Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
inl = (π‘₯ ∈ V ↦ βŸ¨βˆ…, π‘₯⟩)
 
Definitiondf-inr 7060 Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
inr = (π‘₯ ∈ V ↦ ⟨1o, π‘₯⟩)
 
Theoremdjulclr 7061 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
(𝐢 ∈ 𝐴 β†’ ((inl β†Ύ 𝐴)β€˜πΆ) ∈ (𝐴 βŠ” 𝐡))
 
Theoremdjurclr 7062 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
(𝐢 ∈ 𝐡 β†’ ((inr β†Ύ 𝐡)β€˜πΆ) ∈ (𝐴 βŠ” 𝐡))
 
Theoremdjulcl 7063 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
(𝐢 ∈ 𝐴 β†’ (inlβ€˜πΆ) ∈ (𝐴 βŠ” 𝐡))
 
Theoremdjurcl 7064 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
(𝐢 ∈ 𝐡 β†’ (inrβ€˜πΆ) ∈ (𝐴 βŠ” 𝐡))
 
Theoremdjuf1olem 7065* Lemma for djulf1o 7070 and djurf1o 7071. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
𝑋 ∈ V    &   πΉ = (π‘₯ ∈ 𝐴 ↦ βŸ¨π‘‹, π‘₯⟩)    β‡’   πΉ:𝐴–1-1-ontoβ†’({𝑋} Γ— 𝐴)
 
Theoremdjuf1olemr 7066* Lemma for djulf1or 7068 and djurf1or 7069. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 7065. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
𝑋 ∈ V    &   πΉ = (π‘₯ ∈ V ↦ βŸ¨π‘‹, π‘₯⟩)    β‡’   (𝐹 β†Ύ 𝐴):𝐴–1-1-ontoβ†’({𝑋} Γ— 𝐴)
 
Theoremdjulclb 7067 Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
(𝐢 ∈ 𝑉 β†’ (𝐢 ∈ 𝐴 ↔ (inlβ€˜πΆ) ∈ (𝐴 βŠ” 𝐡)))
 
Theoremdjulf1or 7068 The left injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
(inl β†Ύ 𝐴):𝐴–1-1-ontoβ†’({βˆ…} Γ— 𝐴)
 
Theoremdjurf1or 7069 The right injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
(inr β†Ύ 𝐴):𝐴–1-1-ontoβ†’({1o} Γ— 𝐴)
 
Theoremdjulf1o 7070 The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
inl:V–1-1-ontoβ†’({βˆ…} Γ— V)
 
Theoremdjurf1o 7071 The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
inr:V–1-1-ontoβ†’({1o} Γ— V)
 
Theoreminresflem 7072* Lemma for inlresf1 7073 and inrresf1 7074. (Contributed by BJ, 4-Jul-2022.)
𝐹:𝐴–1-1-ontoβ†’({𝑋} Γ— 𝐴)    &   (π‘₯ ∈ 𝐴 β†’ (πΉβ€˜π‘₯) ∈ 𝐡)    β‡’   πΉ:𝐴–1-1→𝐡
 
Theoreminlresf1 7073 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 β†Ύ 𝐴):𝐴–1-1β†’(𝐴 βŠ” 𝐡)
 
Theoreminrresf1 7074 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 β†Ύ 𝐡):𝐡–1-1β†’(𝐴 βŠ” 𝐡)
 
Theoremdjuinr 7075 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 7105 and djufun 7116) while the simpler statement (ran inl ∩ ran inr) = βˆ… is easily recovered from it by substituting V for both 𝐴 and 𝐡 as done in casefun 7097). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
(ran (inl β†Ύ 𝐴) ∩ ran (inr β†Ύ 𝐡)) = βˆ…
 
Theoremdjuin 7076 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 β€œ 𝐴) ∩ (inr β€œ 𝐡)) = βˆ…
 
Theoreminl11 7077 Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ((inlβ€˜π΄) = (inlβ€˜π΅) ↔ 𝐴 = 𝐡))
 
Theoremdjuunr 7078 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 β†Ύ 𝐴) βˆͺ ran (inr β†Ύ 𝐡)) = (𝐴 βŠ” 𝐡)
 
Theoremdjuun 7079 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 β€œ 𝐴) βˆͺ (inr β€œ 𝐡)) = (𝐴 βŠ” 𝐡)
 
Theoremeldju 7080* Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)
(𝐢 ∈ (𝐴 βŠ” 𝐡) ↔ (βˆƒπ‘₯ ∈ 𝐴 𝐢 = ((inl β†Ύ 𝐴)β€˜π‘₯) ∨ βˆƒπ‘₯ ∈ 𝐡 𝐢 = ((inr β†Ύ 𝐡)β€˜π‘₯)))
 
Theoremdjur 7081* 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.)
(𝐢 ∈ (𝐴 βŠ” 𝐡) ↔ (βˆƒπ‘₯ ∈ 𝐴 𝐢 = (inlβ€˜π‘₯) ∨ βˆƒπ‘₯ ∈ 𝐡 𝐢 = (inrβ€˜π‘₯)))
 
2.6.36.3  Universal property of the disjoint union
 
Theoremdjuss 7082 A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
(𝐴 βŠ” 𝐡) βŠ† ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡))
 
Theoremeldju1st 7083 The first component of an element of a disjoint union is either βˆ… or 1o. (Contributed by AV, 26-Jun-2022.)
(𝑋 ∈ (𝐴 βŠ” 𝐡) β†’ ((1st β€˜π‘‹) = βˆ… ∨ (1st β€˜π‘‹) = 1o))
 
Theoremeldju2ndl 7084 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 βŠ” 𝐡) ∧ (1st β€˜π‘‹) = βˆ…) β†’ (2nd β€˜π‘‹) ∈ 𝐴)
 
Theoremeldju2ndr 7085 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 βŠ” 𝐡) ∧ (1st β€˜π‘‹) β‰  βˆ…) β†’ (2nd β€˜π‘‹) ∈ 𝐡)
 
Theorem1stinl 7086 The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.)
(𝑋 ∈ 𝑉 β†’ (1st β€˜(inlβ€˜π‘‹)) = βˆ…)
 
Theorem2ndinl 7087 The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)
(𝑋 ∈ 𝑉 β†’ (2nd β€˜(inlβ€˜π‘‹)) = 𝑋)
 
Theorem1stinr 7088 The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.)
(𝑋 ∈ 𝑉 β†’ (1st β€˜(inrβ€˜π‘‹)) = 1o)
 
Theorem2ndinr 7089 The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)
(𝑋 ∈ 𝑉 β†’ (2nd β€˜(inrβ€˜π‘‹)) = 𝑋)
 
Theoremdjune 7090 Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (inlβ€˜π΄) β‰  (inrβ€˜π΅))
 
Theoremupdjudhf 7091* 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.)
(πœ‘ β†’ 𝐹:𝐴⟢𝐢)    &   (πœ‘ β†’ 𝐺:𝐡⟢𝐢)    &   π» = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))    β‡’   (πœ‘ β†’ 𝐻:(𝐴 βŠ” 𝐡)⟢𝐢)
 
Theoremupdjudhcoinlf 7092* 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.)
(πœ‘ β†’ 𝐹:𝐴⟢𝐢)    &   (πœ‘ β†’ 𝐺:𝐡⟢𝐢)    &   π» = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))    β‡’   (πœ‘ β†’ (𝐻 ∘ (inl β†Ύ 𝐴)) = 𝐹)
 
Theoremupdjudhcoinrg 7093* 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.)
(πœ‘ β†’ 𝐹:𝐴⟢𝐢)    &   (πœ‘ β†’ 𝐺:𝐡⟢𝐢)    &   π» = (π‘₯ ∈ (𝐴 βŠ” 𝐡) ↦ if((1st β€˜π‘₯) = βˆ…, (πΉβ€˜(2nd β€˜π‘₯)), (πΊβ€˜(2nd β€˜π‘₯))))    β‡’   (πœ‘ β†’ (𝐻 ∘ (inr β†Ύ 𝐡)) = 𝐺)
 
Theoremupdjud 7094* Universal property of the disjoint union. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.)
(πœ‘ β†’ 𝐹:𝐴⟢𝐢)    &   (πœ‘ β†’ 𝐺:𝐡⟢𝐢)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ π‘Š)    β‡’   (πœ‘ β†’ βˆƒ!β„Ž(β„Ž:(𝐴 βŠ” 𝐡)⟢𝐢 ∧ (β„Ž ∘ (inl β†Ύ 𝐴)) = 𝐹 ∧ (β„Ž ∘ (inr β†Ύ 𝐡)) = 𝐺))
 
Syntaxcdjucase 7095 Syntax for the "case" construction.
class case(𝑅, 𝑆)
 
Definitiondf-case 7096 The "case" construction: if 𝐹:π΄βŸΆπ‘‹ and 𝐺:π΅βŸΆπ‘‹ are functions, then case(𝐹, 𝐺):(𝐴 βŠ” 𝐡)βŸΆπ‘‹ 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 7094. 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) = ((𝑅 ∘ β—‘inl) βˆͺ (𝑆 ∘ β—‘inr))
 
Theoremcasefun 7097 The "case" construction of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
(πœ‘ β†’ Fun 𝐹)    &   (πœ‘ β†’ Fun 𝐺)    β‡’   (πœ‘ β†’ Fun case(𝐹, 𝐺))
 
Theoremcasedm 7098 The domain of the "case" construction is the disjoint union of the domains. TODO (although less important): ran case(𝐹, 𝐺) = (ran 𝐹 βˆͺ ran 𝐺). (Contributed by BJ, 10-Jul-2022.)
dom case(𝐹, 𝐺) = (dom 𝐹 βŠ” dom 𝐺)
 
Theoremcaserel 7099 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) βŠ† ((dom 𝑅 βŠ” dom 𝑆) Γ— (ran 𝑅 βˆͺ ran 𝑆))
 
Theoremcasef 7100 The "case" construction of two functions is a function on the disjoint union of their domains. (Contributed by BJ, 10-Jul-2022.)
(πœ‘ β†’ 𝐹:π΄βŸΆπ‘‹)    &   (πœ‘ β†’ 𝐺:π΅βŸΆπ‘‹)    β‡’   (πœ‘ β†’ case(𝐹, 𝐺):(𝐴 βŠ” 𝐡)βŸΆπ‘‹)
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