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Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiunin1f 30301 Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4973 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.)
𝑥𝐶        𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)

Theoremssiun3 30302* Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)
(∀𝑦𝐶𝑥𝐴 𝑦𝐵𝐶 𝑥𝐴 𝐵)

Theoremssiun2sf 30303 Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
𝑥𝐴    &   𝑥𝐶    &   𝑥𝐷    &   (𝑥 = 𝐶𝐵 = 𝐷)       (𝐶𝐴𝐷 𝑥𝐴 𝐵)

Theoremiuninc 30304* The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.)
(𝜑𝐹 Fn ℕ)    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))       ((𝜑𝑖 ∈ ℕ) → 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))

Theoremiundifdifd 30305* The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.)
(𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥))))

Theoremiundifdif 30306* The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 30305. (Contributed by Thierry Arnoux, 4-Sep-2016.)
𝑂 ∈ V    &   𝐴 ⊆ 𝒫 𝑂       (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))

Theoremiunrdx 30307* Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)
(𝜑𝐹:𝐴onto𝐶)    &   ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)       (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)

Theoremiunpreima 30308* Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)
(Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))

Theoremiunrnmptss 30309* A subset relation for an indexed union over the range of function expressed as a mapping. (Contributed by Thierry Arnoux, 27-Mar-2018.)
(𝑦 = 𝐵𝐶 = 𝐷)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 𝑦 ∈ ran (𝑥𝐴𝐵)𝐶 𝑥𝐴 𝐷)

Theoremiunxunsn 30310* Appending a set to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑥 = 𝑋𝐵 = 𝐶)       (𝑋𝑉 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = ( 𝑥𝐴 𝐵𝐶))

Theoremiunxunpr 30311* Appending two sets to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)       ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))

Theoremdisjnf 30312* In case 𝑥 is not free in 𝐵, disjointness is not so interesting since it reduces to cases where 𝐴 is a singleton. (Google Groups discussion with Peter Mazsa.) (Contributed by Thierry Arnoux, 26-Jul-2018.)
(Disj 𝑥𝐴 𝐵 ↔ (𝐵 = ∅ ∨ ∃*𝑥 𝑥𝐴))

Theoremcbvdisjf 30313* Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑥𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)

Theoremdisjss1f 30314 A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))

Theoremdisjeq1f 30315 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))

Theoremdisjxun0 30316* Simplify a disjoint union. (Contributed by Thierry Arnoux, 27-Nov-2023.)
((𝜑𝑥𝐵) → 𝐶 = ∅)       (𝜑 → (Disj 𝑥 ∈ (𝐴𝐵)𝐶Disj 𝑥𝐴 𝐶))

Theoremdisjdifprg 30317* A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝐴𝑉𝐵𝑊) → Disj 𝑥 ∈ {(𝐵𝐴), 𝐴}𝑥)

Theoremdisjdifprg2 30318* A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝐴𝑉Disj 𝑥 ∈ {(𝐴𝐵), (𝐴𝐵)}𝑥)

Theoremdisji2f 30319* Property of a disjoint collection: if 𝐵(𝑥) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑥𝑌, then 𝐵 and 𝐶 are disjoint. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑥𝐶    &   (𝑥 = 𝑌𝐵 = 𝐶)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ 𝑥𝑌) → (𝐵𝐶) = ∅)

Theoremdisjif 30320* Property of a disjoint collection: if 𝐵(𝑥) and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑥 = 𝑌. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑥𝐶    &   (𝑥 = 𝑌𝐵 = 𝐶)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)

Theoremdisjorf 30321* Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑖𝐴    &   𝑗𝐴    &   (𝑖 = 𝑗𝐵 = 𝐶)       (Disj 𝑖𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅))

Theoremdisjorsf 30322* Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝐴       (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))

Theoremdisjif2 30323* Property of a disjoint collection: if 𝐵(𝑥) and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑥 = 𝑌. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝑌𝐵 = 𝐶)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)

Theoremdisjabrex 30324* Rewriting a disjoint collection into a partition of its image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
(Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)

Theoremdisjabrexf 30325* Rewriting a disjoint collection into a partition of its image set. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Revised by Thierry Arnoux, 9-Mar-2017.)
𝑥𝐴       (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)

Theoremdisjpreima 30326* A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
((Fun 𝐹Disj 𝑥𝐴 𝐵) → Disj 𝑥𝐴 (𝐹𝐵))

Theoremdisjrnmpt 30327* Rewriting a disjoint collection using the range of a mapping. (Contributed by Thierry Arnoux, 27-May-2020.)
(Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ ran (𝑥𝐴𝐵)𝑦)

Theoremdisjin 30328 If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐶𝐴))

Theoremdisjin2 30329 If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐴𝐶))

Theoremdisjxpin 30330* Derive a disjunction over a Cartesian product from the disjunctions over its first and second elements. (Contributed by Thierry Arnoux, 9-Mar-2018.)
(𝑥 = (1st𝑝) → 𝐶 = 𝐸)    &   (𝑦 = (2nd𝑝) → 𝐷 = 𝐹)    &   (𝜑Disj 𝑥𝐴 𝐶)    &   (𝜑Disj 𝑦𝐵 𝐷)       (𝜑Disj 𝑝 ∈ (𝐴 × 𝐵)(𝐸𝐹))

Theoremiundisjf 30331* Rewrite a countable union as a disjoint union. Cf. iundisj 24141. (Contributed by Thierry Arnoux, 31-Dec-2016.)
𝑘𝐴    &   𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)        𝑛 ∈ ℕ 𝐴 = 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)

Theoremiundisj2f 30332* A disjoint union is disjoint. Cf. iundisj2 24142. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑘𝐴    &   𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)       Disj 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)

Theoremdisjrdx 30333* Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.)
(𝜑𝐹:𝐴1-1-onto𝐶)    &   ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)       (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑦𝐶 𝐷))

Theoremdisjex 30334* Two ways to say that two classes are disjoint (or equal). (Contributed by Thierry Arnoux, 4-Oct-2016.)
((∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵) ↔ (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))

Theoremdisjexc 30335* A variant of disjex 30334, applicable for more generic families. (Contributed by Thierry Arnoux, 4-Oct-2016.)
(𝑥 = 𝑦𝐴 = 𝐵)       ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝑥 = 𝑦) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))

Theoremdisjunsn 30336* Append an element to a disjoint collection. Similar to ralunsn 4816, gsumunsn 19072, etc. (Contributed by Thierry Arnoux, 28-Mar-2018.)
(𝑥 = 𝑀𝐵 = 𝐶)       ((𝑀𝑉 ∧ ¬ 𝑀𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {𝑀})𝐵 ↔ (Disj 𝑥𝐴 𝐵 ∧ ( 𝑥𝐴 𝐵𝐶) = ∅)))

Theoremdisjun0 30337* Adding the empty element preserves disjointness. (Contributed by Thierry Arnoux, 30-May-2020.)
(Disj 𝑥𝐴 𝑥Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)

Theoremdisjiunel 30338* A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.)
(𝜑Disj 𝑥𝐴 𝐵)    &   (𝑥 = 𝑌𝐵 = 𝐷)    &   (𝜑𝐸𝐴)    &   (𝜑𝑌 ∈ (𝐴𝐸))       (𝜑 → ( 𝑥𝐸 𝐵𝐷) = ∅)

Theoremdisjuniel 30339* A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.)
(𝜑Disj 𝑥𝐴 𝑥)    &   (𝜑𝐵𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))       (𝜑 → ( 𝐵𝐶) = ∅)

20.3.4  Relations and Functions

Theoremxpdisjres 30340 Restriction of a constant function (or other Cartesian product) outside of its domain. (Contributed by Thierry Arnoux, 25-Jan-2017.)
((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) ↾ 𝐶) = ∅)

Theoremopeldifid 30341 Ordered pair elementhood outside of the diagonal. (Contributed by Thierry Arnoux, 1-Jan-2020.)
(Rel 𝐴 → (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴𝑋𝑌)))

Theoremdifres 30342 Case when class difference in unaffected by restriction. (Contributed by Thierry Arnoux, 1-Jan-2020.)
(𝐴 ⊆ (𝐵 × V) → (𝐴 ∖ (𝐶𝐵)) = (𝐴𝐶))

Theoremimadifxp 30343 Image of the difference with a Cartesian product. (Contributed by Thierry Arnoux, 13-Dec-2017.)
(𝐶𝐴 → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))

Theoremrelfi 30344 A relation (set) is finite if and only if both its domain and range are finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
(Rel 𝐴 → (𝐴 ∈ Fin ↔ (dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin)))

Theoremreldisjun 30345 Split a relation into two disjoint parts based on its domain. (Contributed by Thierry Arnoux, 9-Oct-2023.)
((Rel 𝑅 ∧ dom 𝑅 = (𝐴𝐵) ∧ (𝐴𝐵) = ∅) → 𝑅 = ((𝑅𝐴) ∪ (𝑅𝐵)))

Theorem0res 30346 Restriction of the empty function. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(∅ ↾ 𝐴) = ∅

Theoremfunresdm1 30347 Restriction of a disjoint union to the domain of the first term. (Contributed by Thierry Arnoux, 9-Dec-2021.)
((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴𝐵) ↾ dom 𝐴) = 𝐴)

Theoremfnunres1 30348 Restriction of a disjoint union to the domain of the first function. (Contributed by Thierry Arnoux, 9-Dec-2021.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)

Theoremfcoinver 30349 Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 30350. (Contributed by Thierry Arnoux, 3-Jan-2020.)
(𝐹 Fn 𝑋 → (𝐹𝐹) Er 𝑋)

Theoremfcoinvbr 30350 Binary relation for the equivalence relation from fcoinver 30349. (Contributed by Thierry Arnoux, 3-Jan-2020.)
= (𝐹𝐹)       ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))

Theorembrabgaf 30351* The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) (Revised by Thierry Arnoux, 17-May-2020.)
𝑥𝜓    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝜓))

Theorembrelg 30352 Two things in a binary relation belong to the relation's domain. (Contributed by Thierry Arnoux, 29-Aug-2017.)
((𝑅 ⊆ (𝐶 × 𝐷) ∧ 𝐴𝑅𝐵) → (𝐴𝐶𝐵𝐷))

Theorembr8d 30353* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by Thierry Arnoux, 21-Mar-2019.)
(𝑎 = 𝐴 → (𝜓𝜒))    &   (𝑏 = 𝐵 → (𝜒𝜃))    &   (𝑐 = 𝐶 → (𝜃𝜏))    &   (𝑑 = 𝐷 → (𝜏𝜂))    &   (𝑒 = 𝐸 → (𝜂𝜁))    &   (𝑓 = 𝐹 → (𝜁𝜎))    &   (𝑔 = 𝐺 → (𝜎𝜌))    &   ( = 𝐻 → (𝜌𝜇))    &   (𝜑𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃𝑒𝑃𝑓𝑃𝑔𝑃𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ⟩⟩ ∧ 𝜓)})    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐺𝑃)    &   (𝜑𝐻𝑃)       (𝜑 → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜇))

Theoremopabdm 30354* Domain of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑})

Theoremopabrn 30355* Range of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑})

Theoremopabssi 30356* Sufficient condition for a collection of ordered pairs to be a subclass of a relation. (Contributed by Peter Mazsa, 21-Oct-2019.) (Revised by Thierry Arnoux, 18-Feb-2022.)
(𝜑 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)       {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ 𝐴

Theoremopabid2ss 30357* One direction of opabid2 5693 which holds without a Rel 𝐴 requirement. (Contributed by Thierry Arnoux, 18-Feb-2022.)
{⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ⊆ 𝐴

Theoremssrelf 30358* A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Thierry Arnoux, 6-Nov-2017.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝐴    &   𝑦𝐴    &   𝑥𝐵    &   𝑦𝐵       (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Theoremeqrelrd2 30359* A version of eqrelrdv2 5661 with explicit non-free declarations. (Contributed by Thierry Arnoux, 28-Aug-2017.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝐴    &   𝑦𝐴    &   𝑥𝐵    &   𝑦𝐵    &   (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Theoremerbr3b 30360 Biconditional for equivalent elements. (Contributed by Thierry Arnoux, 6-Jan-2020.)
((𝑅 Er 𝑋𝐴𝑅𝐵) → (𝐴𝑅𝐶𝐵𝑅𝐶))

Theoremiunsnima 30361 Image of a singleton by an indexed union involving that singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)       ((𝜑𝑥𝐴) → ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵)

Theoremac6sf2 30362* Alternate version of ac6 9894 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.) (Revised by Thierry Arnoux, 17-May-2020.)
𝑦𝐵    &   𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremfnresin 30363 Restriction of a function with a subclass of its domain. (Contributed by Thierry Arnoux, 10-Oct-2017.)
(𝐹 Fn 𝐴 → (𝐹𝐵) Fn (𝐴𝐵))

Theoremf1o3d 30364* Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)
(𝜑𝐹 = (𝑥𝐴𝐶))    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑦𝐵) → 𝐷𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))       (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))

Theoremeldmne0 30365 A function of nonempty domain is not empty. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑋 ∈ dom 𝐹𝐹 ≠ ∅)

Theoremf1rnen 30366 Equinumerosity of the range of an injective function. (Contributed by Thierry Arnoux, 7-Jul-2023.)
((𝐹:𝐴1-1𝐵𝐴𝑉) → ran 𝐹𝐴)

Theoremrinvf1o 30367 Sufficient conditions for the restriction of an involution to be a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
Fun 𝐹    &   𝐹 = 𝐹    &   (𝐹𝐴) ⊆ 𝐵    &   (𝐹𝐵) ⊆ 𝐴    &   𝐴 ⊆ dom 𝐹    &   𝐵 ⊆ dom 𝐹       (𝐹𝐴):𝐴1-1-onto𝐵

Theoremfresf1o 30368 Conditions for a restriction to be a one-to-one onto function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1-onto𝐶)

Theoremnfpconfp 30369 The set of fixed points of 𝐹 is the complement of the set of points moved by 𝐹. (Contributed by Thierry Arnoux, 17-Nov-2023.)
(𝐹 Fn 𝐴 → (𝐴 ∖ dom (𝐹 ∖ I )) = dom (𝐹 ∩ I ))

Theoremfmptco1f1o 30370* The action of composing (to the right) with a bijection is itself a bijection of functions. (Contributed by Thierry Arnoux, 3-Jan-2021.)
𝐴 = (𝑅m 𝐸)    &   𝐵 = (𝑅m 𝐷)    &   𝐹 = (𝑓𝐴 ↦ (𝑓𝑇))    &   (𝜑𝐷𝑉)    &   (𝜑𝐸𝑊)    &   (𝜑𝑅𝑋)    &   (𝜑𝑇:𝐷1-1-onto𝐸)       (𝜑𝐹:𝐴1-1-onto𝐵)

Theoremcofmpt2 30371* Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 15-Jul-2023.)
((𝜑𝑦 = (𝐹𝑥)) → 𝐶 = 𝐷)    &   ((𝜑𝑦𝐵) → 𝐶𝐸)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐷𝑉)       (𝜑 → ((𝑦𝐵𝐶) ∘ 𝐹) = (𝑥𝐴𝐷))

Theoremf1mptrn 30372* Express injection for a mapping operation. (Contributed by Thierry Arnoux, 3-May-2020.)
((𝜑𝑥𝐴) → 𝐵𝐶)    &   ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = 𝐵)       (𝜑 → Fun (𝑥𝐴𝐵))

Theoremdfimafnf 30373* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)
𝑥𝐴    &   𝑥𝐹       ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})

Theoremfunimass4f 30374 Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐹       ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))

Theoremelimampt 30375* Membership in the image of a mapping. (Contributed by Thierry Arnoux, 3-Jan-2022.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐶𝑊)    &   (𝜑𝐷𝐴)       (𝜑 → (𝐶 ∈ (𝐹𝐷) ↔ ∃𝑥𝐷 𝐶 = 𝐵))

Theoremsuppss2f 30376* Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.) (Revised by AV, 1-Sep-2020.)
𝑘𝜑    &   𝑘𝐴    &   𝑘𝑊    &   ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)    &   (𝜑𝐴𝑉)       (𝜑 → ((𝑘𝐴𝐵) supp 𝑍) ⊆ 𝑊)

Theoremfovcld 30377 Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)
(𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)       ((𝜑𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)

Theoremofrn 30378 The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑+ :(𝐵 × 𝐵)⟶𝐶)    &   (𝜑𝐴𝑉)       (𝜑 → ran (𝐹f + 𝐺) ⊆ 𝐶)

Theoremofrn2 30379 The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑+ :(𝐵 × 𝐵)⟶𝐶)    &   (𝜑𝐴𝑉)       (𝜑 → ran (𝐹f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))

Theoremoff2 30380* The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)
((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐵𝑇)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑 → (𝐴𝐵) = 𝐶)       (𝜑 → (𝐹f 𝑅𝐺):𝐶𝑈)

Theoremofresid 30381 Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑𝐴𝑉)       (𝜑 → (𝐹f 𝑅𝐺) = (𝐹f (𝑅 ↾ (𝐵 × 𝐵))𝐺))

Theoremfimarab 30382* Expressing the image of a set as a restricted abstract builder. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝐹:𝐴𝐵𝑋𝐴) → (𝐹𝑋) = {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦})

Theoremunipreima 30383* Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)
(Fun 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))

Theoremsspreima 30384 The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)
((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))

Theoremopfv 30385 Value of a function producing ordered pairs. (Contributed by Thierry Arnoux, 3-Jan-2017.)
(((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩)

Theoremxppreima 30386 The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.)
((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (𝐹 “ (𝑌 × 𝑍)) = (((1st𝐹) “ 𝑌) ∩ ((2nd𝐹) “ 𝑍)))

Theoremxppreima2 30387* The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)       (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))

Theoremelunirn2 30388 Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.)
((Fun 𝐹𝐵 ∈ (𝐹𝐴)) → 𝐵 ran 𝐹)

Theoremabfmpunirn 30389* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})    &   {𝑦𝜑} ∈ V    &   (𝑦 = 𝐵 → (𝜑𝜓))       (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 𝜓))

Theoremrabfmpunirn 30390* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑊𝜑})    &   𝑊 ∈ V    &   (𝑦 = 𝐵 → (𝜑𝜓))       (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 (𝐵𝑊𝜓))

Theoremabfmpeld 30391* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝜓})    &   (𝜑 → {𝑦𝜓} ∈ V)    &   (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))       (𝜑 → ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒)))

Theoremabfmpel 30392* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})    &   {𝑦𝜑} ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜓))

TheoremfmptdF 30393 Domain and codomain of the mapping operation; deduction form. This version of fmptd 6871 uses bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐶    &   ((𝜑𝑥𝐴) → 𝐵𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑𝐹:𝐴𝐶)

Theoremfmptcof2 30394* Composition of two functions expressed as ordered-pair class abstractions. (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝑆    &   𝑦𝑇    &   𝑥𝐴    &   𝑥𝐵    &   𝑥𝜑    &   (𝜑 → ∀𝑥𝐴 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))

Theoremfcomptf 30395* Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 6888. (Contributed by Thierry Arnoux, 30-Jun-2017.)
𝑥𝐵       ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))

Theoremacunirnmpt 30396* Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 6-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝐶 = ran (𝑗𝐴𝐵)       (𝜑 → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵))

Theoremacunirnmpt2 30397* Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝐶 = ran (𝑗𝐴𝐵)    &   (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)       (𝜑 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))

Theoremacunirnmpt2f 30398* Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝑗𝐴    &   𝑗𝐶    &   𝑗𝐷    &   𝐶 = 𝑗𝐴 𝐵    &   (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)    &   ((𝜑𝑗𝐴) → 𝐵𝑊)       (𝜑 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))

Theoremaciunf1lem 30399* Choice in an index union. (Contributed by Thierry Arnoux, 8-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝑗𝐴    &   ((𝜑𝑗𝐴) → 𝐵𝑊)       (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 𝑗𝐴 𝐵(2nd ‘(𝑓𝑥)) = 𝑥))

Theoremaciunf1 30400* Choice in an index union. (Contributed by Thierry Arnoux, 4-May-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵𝑊)       (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 𝑘))

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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 44801-44886
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