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Theorem List for Metamath Proof Explorer - 45301-45400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelsetpreimafvbi 45301* An element of the preimage of a function value is an element of the domain of the function with the same value as another element of the preimage. (Contributed by AV, 9-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    β‡’   ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) β†’ (π‘Œ ∈ 𝑆 ↔ (π‘Œ ∈ 𝐴 ∧ (πΉβ€˜π‘Œ) = (πΉβ€˜π‘‹))))
 
Theoremelsetpreimafveqfv 45302* The elements of the preimage of a function value have the same function values. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    β‡’   ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆)) β†’ (πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ))
 
Theoremeqfvelsetpreimafv 45303* If an element of the domain of the function has the same function value as an element of the preimage of a function value, then it is an element of the same preimage. (Contributed by AV, 9-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    β‡’   ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) β†’ ((π‘Œ ∈ 𝐴 ∧ (πΉβ€˜π‘Œ) = (πΉβ€˜π‘‹)) β†’ π‘Œ ∈ 𝑆))
 
Theoremelsetpreimafvrab 45304* An element of the preimage of a function value expressed as a restricted class abstraction. (Contributed by AV, 9-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    β‡’   ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) β†’ 𝑆 = {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹)})
 
Theoremimaelsetpreimafv 45305* The image of an element of the preimage of a function value is the singleton consisting of the function value at one of its elements. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    β‡’   ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) β†’ (𝐹 β€œ 𝑆) = {(πΉβ€˜π‘‹)})
 
Theoremuniimaelsetpreimafv 45306* The union of the image of an element of the preimage of a function value is an element of the range of the function. (Contributed by AV, 5-Mar-2024.) (Revised by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    β‡’   ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) β†’ βˆͺ (𝐹 β€œ 𝑆) ∈ ran 𝐹)
 
Theoremelsetpreimafveq 45307* If two preimages of function values contain elements with identical function values, then both preimages are equal. (Contributed by AV, 8-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    β‡’   ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑅)) β†’ ((πΉβ€˜π‘‹) = (πΉβ€˜π‘Œ) β†’ 𝑆 = 𝑅))
 
Theoremfundcmpsurinjlem1 45308* Lemma 1 for fundcmpsurinj 45319. (Contributed by AV, 4-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    &   πΊ = (π‘₯ ∈ 𝐴 ↦ (◑𝐹 β€œ {(πΉβ€˜π‘₯)}))    β‡’   ran 𝐺 = 𝑃
 
Theoremfundcmpsurinjlem2 45309* Lemma 2 for fundcmpsurinj 45319. (Contributed by AV, 4-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    &   πΊ = (π‘₯ ∈ 𝐴 ↦ (◑𝐹 β€œ {(πΉβ€˜π‘₯)}))    β‡’   ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) β†’ 𝐺:𝐴–onto→𝑃)
 
Theoremfundcmpsurinjlem3 45310* Lemma 3 for fundcmpsurinj 45319. (Contributed by AV, 3-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    &   π» = (𝑝 ∈ 𝑃 ↦ βˆͺ (𝐹 β€œ 𝑝))    β‡’   ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) β†’ (π»β€˜π‘‹) = βˆͺ (𝐹 β€œ 𝑋))
 
Theoremimasetpreimafvbijlemf 45311* Lemma for imasetpreimafvbij 45316: the mapping 𝐻 is a function into the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    &   π» = (𝑝 ∈ 𝑃 ↦ βˆͺ (𝐹 β€œ 𝑝))    β‡’   (𝐹 Fn 𝐴 β†’ 𝐻:π‘ƒβŸΆ(𝐹 β€œ 𝐴))
 
Theoremimasetpreimafvbijlemfv 45312* Lemma for imasetpreimafvbij 45316: the value of the mapping 𝐻 at a preimage of a value of function 𝐹. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    &   π» = (𝑝 ∈ 𝑃 ↦ βˆͺ (𝐹 β€œ 𝑝))    β‡’   ((𝐹 Fn 𝐴 ∧ π‘Œ ∈ 𝑃 ∧ 𝑋 ∈ π‘Œ) β†’ (π»β€˜π‘Œ) = (πΉβ€˜π‘‹))
 
Theoremimasetpreimafvbijlemfv1 45313* Lemma for imasetpreimafvbij 45316: for a preimage of a value of function 𝐹 there is an element of the preimage so that the value of the mapping 𝐻 at this preimage is the function value at this element. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    &   π» = (𝑝 ∈ 𝑃 ↦ βˆͺ (𝐹 β€œ 𝑝))    β‡’   ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) β†’ βˆƒπ‘¦ ∈ 𝑋 (π»β€˜π‘‹) = (πΉβ€˜π‘¦))
 
Theoremimasetpreimafvbijlemf1 45314* Lemma for imasetpreimafvbij 45316: the mapping 𝐻 is an injective function into the range of function 𝐹. (Contributed by AV, 9-Mar-2024.) (Revised by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    &   π» = (𝑝 ∈ 𝑃 ↦ βˆͺ (𝐹 β€œ 𝑝))    β‡’   (𝐹 Fn 𝐴 β†’ 𝐻:𝑃–1-1β†’(𝐹 β€œ 𝐴))
 
Theoremimasetpreimafvbijlemfo 45315* Lemma for imasetpreimafvbij 45316: the mapping 𝐻 is a function onto the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    &   π» = (𝑝 ∈ 𝑃 ↦ βˆͺ (𝐹 β€œ 𝑝))    β‡’   ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) β†’ 𝐻:𝑃–ontoβ†’(𝐹 β€œ 𝐴))
 
Theoremimasetpreimafvbij 45316* The mapping 𝐻 is a bijective function betwen the set 𝑃 of all preimages of values of function 𝐹 and the range of 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    &   π» = (𝑝 ∈ 𝑃 ↦ βˆͺ (𝐹 β€œ 𝑝))    β‡’   ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) β†’ 𝐻:𝑃–1-1-ontoβ†’(𝐹 β€œ 𝐴))
 
Theoremfundcmpsurbijinjpreimafv 45317* Every function 𝐹:𝐴⟢𝐡 can be decomposed into a surjective function onto 𝑃, a bijective function from 𝑃 and an injective function into the codomain of 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    β‡’   ((𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉) β†’ βˆƒπ‘”βˆƒβ„Žβˆƒπ‘–((𝑔:𝐴–onto→𝑃 ∧ β„Ž:𝑃–1-1-ontoβ†’(𝐹 β€œ 𝐴) ∧ 𝑖:(𝐹 β€œ 𝐴)–1-1→𝐡) ∧ 𝐹 = ((𝑖 ∘ β„Ž) ∘ 𝑔)))
 
Theoremfundcmpsurinjpreimafv 45318* Every function 𝐹:𝐴⟢𝐡 can be decomposed into a surjective function onto 𝑃 and an injective function from 𝑃. (Contributed by AV, 12-Mar-2024.) (Proof shortened by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}    β‡’   ((𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉) β†’ βˆƒπ‘”βˆƒβ„Ž(𝑔:𝐴–onto→𝑃 ∧ β„Ž:𝑃–1-1→𝐡 ∧ 𝐹 = (β„Ž ∘ 𝑔)))
 
Theoremfundcmpsurinj 45319* Every function 𝐹:𝐴⟢𝐡 can be decomposed into a surjective and an injective function. (Contributed by AV, 13-Mar-2024.)
((𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉) β†’ βˆƒπ‘”βˆƒβ„Žβˆƒπ‘(𝑔:𝐴–onto→𝑝 ∧ β„Ž:𝑝–1-1→𝐡 ∧ 𝐹 = (β„Ž ∘ 𝑔)))
 
Theoremfundcmpsurbijinj 45320* Every function 𝐹:𝐴⟢𝐡 can be decomposed into a surjective, a bijective and an injective function. (Contributed by AV, 23-Mar-2024.)
((𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉) β†’ βˆƒπ‘”βˆƒβ„Žβˆƒπ‘–βˆƒπ‘βˆƒπ‘ž((𝑔:𝐴–onto→𝑝 ∧ β„Ž:𝑝–1-1-ontoβ†’π‘ž ∧ 𝑖:π‘žβ€“1-1→𝐡) ∧ 𝐹 = ((𝑖 ∘ β„Ž) ∘ 𝑔)))
 
Theoremfundcmpsurinjimaid 45321* Every function 𝐹:𝐴⟢𝐡 can be decomposed into a surjective function onto the image (𝐹 β€œ 𝐴) of the domain of 𝐹 and an injective function from the image (𝐹 β€œ 𝐴). (Contributed by AV, 17-Mar-2024.)
𝐼 = (𝐹 β€œ 𝐴)    &   πΊ = (π‘₯ ∈ 𝐴 ↦ (πΉβ€˜π‘₯))    &   π» = ( I β†Ύ 𝐼)    β‡’   (𝐹:𝐴⟢𝐡 β†’ (𝐺:𝐴–onto→𝐼 ∧ 𝐻:𝐼–1-1→𝐡 ∧ 𝐹 = (𝐻 ∘ 𝐺)))
 
TheoremfundcmpsurinjALT 45322* Alternate proof of fundcmpsurinj 45319, based on fundcmpsurinjimaid 45321: Every function 𝐹:𝐴⟢𝐡 can be decomposed into a surjective and an injective function. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by AV, 13-Mar-2024.)
((𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉) β†’ βˆƒπ‘”βˆƒβ„Žβˆƒπ‘(𝑔:𝐴–onto→𝑝 ∧ β„Ž:𝑝–1-1→𝐡 ∧ 𝐹 = (β„Ž ∘ 𝑔)))
 
21.43.8  Partitions of real intervals

Based on the theorems of the fourierdlem* series of GS's mathbox.

 
Syntaxciccp 45323 Extend class notation with the partitions of a closed interval of extended reals.
class RePart
 
Definitiondf-iccp 45324* Define partitions of a closed interval of extended reals. Such partitions are finite increasing sequences of extended reals. (Contributed by AV, 8-Jul-2020.)
RePart = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ* ↑m (0...π‘š)) ∣ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1))})
 
Theoremiccpval 45325* Partition consisting of a fixed number 𝑀 of parts. (Contributed by AV, 9-Jul-2020.)
(𝑀 ∈ β„• β†’ (RePartβ€˜π‘€) = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ βˆ€π‘– ∈ (0..^𝑀)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1))})
 
Theoremiccpart 45326* A special partition. Corresponds to fourierdlem2 44072 in GS's mathbox. (Contributed by AV, 9-Jul-2020.)
(𝑀 ∈ β„• β†’ (𝑃 ∈ (RePartβ€˜π‘€) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1)))))
 
Theoremiccpartimp 45327 Implications for a class being a partition. (Contributed by AV, 11-Jul-2020.)
((𝑀 ∈ β„• ∧ 𝑃 ∈ (RePartβ€˜π‘€) ∧ 𝐼 ∈ (0..^𝑀)) β†’ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (π‘ƒβ€˜πΌ) < (π‘ƒβ€˜(𝐼 + 1))))
 
Theoremiccpartres 45328 The restriction of a partition is a partition. (Contributed by AV, 16-Jul-2020.)
((𝑀 ∈ β„• ∧ 𝑃 ∈ (RePartβ€˜(𝑀 + 1))) β†’ (𝑃 β†Ύ (0...𝑀)) ∈ (RePartβ€˜π‘€))
 
Theoremiccpartxr 45329 If there is a partition, then all intermediate points and bounds are extended real numbers. (Contributed by AV, 11-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    &   (πœ‘ β†’ 𝐼 ∈ (0...𝑀))    β‡’   (πœ‘ β†’ (π‘ƒβ€˜πΌ) ∈ ℝ*)
 
Theoremiccpartgtprec 45330 If there is a partition, then all intermediate points and the upper bound are strictly greater than the preceeding intermediate points or lower bound. (Contributed by AV, 11-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    &   (πœ‘ β†’ 𝐼 ∈ (1...𝑀))    β‡’   (πœ‘ β†’ (π‘ƒβ€˜(𝐼 βˆ’ 1)) < (π‘ƒβ€˜πΌ))
 
Theoremiccpartipre 45331 If there is a partition, then all intermediate points are real numbers. (Contributed by AV, 11-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    &   (πœ‘ β†’ 𝐼 ∈ (1..^𝑀))    β‡’   (πœ‘ β†’ (π‘ƒβ€˜πΌ) ∈ ℝ)
 
Theoremiccpartiltu 45332* If there is a partition, then all intermediate points are strictly less than the upper bound. (Contributed by AV, 12-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ βˆ€π‘– ∈ (1..^𝑀)(π‘ƒβ€˜π‘–) < (π‘ƒβ€˜π‘€))
 
Theoremiccpartigtl 45333* If there is a partition, then all intermediate points are strictly greater than the lower bound. (Contributed by AV, 12-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ βˆ€π‘– ∈ (1..^𝑀)(π‘ƒβ€˜0) < (π‘ƒβ€˜π‘–))
 
Theoremiccpartlt 45334 If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 44081 in GS's mathbox. (Contributed by AV, 12-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ (π‘ƒβ€˜0) < (π‘ƒβ€˜π‘€))
 
Theoremiccpartltu 45335* If there is a partition, then all intermediate points and the lower bound are strictly less than the upper bound. (Contributed by AV, 14-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝑀)(π‘ƒβ€˜π‘–) < (π‘ƒβ€˜π‘€))
 
Theoremiccpartgtl 45336* If there is a partition, then all intermediate points and the upper bound are strictly greater than the lower bound. (Contributed by AV, 14-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ βˆ€π‘– ∈ (1...𝑀)(π‘ƒβ€˜0) < (π‘ƒβ€˜π‘–))
 
Theoremiccpartgt 45337* If there is a partition, then all intermediate points and the bounds are strictly ordered. (Contributed by AV, 18-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ βˆ€π‘– ∈ (0...𝑀)βˆ€π‘— ∈ (0...𝑀)(𝑖 < 𝑗 β†’ (π‘ƒβ€˜π‘–) < (π‘ƒβ€˜π‘—)))
 
Theoremiccpartleu 45338* If there is a partition, then all intermediate points and the lower and the upper bound are less than or equal to the upper bound. (Contributed by AV, 14-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ βˆ€π‘– ∈ (0...𝑀)(π‘ƒβ€˜π‘–) ≀ (π‘ƒβ€˜π‘€))
 
Theoremiccpartgel 45339* If there is a partition, then all intermediate points and the upper and the lower bound are greater than or equal to the lower bound. (Contributed by AV, 14-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ βˆ€π‘– ∈ (0...𝑀)(π‘ƒβ€˜0) ≀ (π‘ƒβ€˜π‘–))
 
Theoremiccpartrn 45340 If there is a partition, then all intermediate points and bounds are contained in a closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ ran 𝑃 βŠ† ((π‘ƒβ€˜0)[,](π‘ƒβ€˜π‘€)))
 
Theoremiccpartf 45341 The range of the partition is between its starting point and its ending point. Corresponds to fourierdlem15 44085 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 14-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ 𝑃:(0...𝑀)⟢((π‘ƒβ€˜0)[,](π‘ƒβ€˜π‘€)))
 
Theoremiccpartel 45342 If there is a partition, then all intermediate points and bounds are contained in a closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   ((πœ‘ ∧ 𝐼 ∈ (0...𝑀)) β†’ (π‘ƒβ€˜πΌ) ∈ ((π‘ƒβ€˜0)[,](π‘ƒβ€˜π‘€)))
 
Theoremiccelpart 45343* An element of any partitioned half-open interval of extended reals is an element of a part of this partition. (Contributed by AV, 18-Jul-2020.)
(𝑀 ∈ β„• β†’ βˆ€π‘ ∈ (RePartβ€˜π‘€)(𝑋 ∈ ((π‘β€˜0)[,)(π‘β€˜π‘€)) β†’ βˆƒπ‘– ∈ (0..^𝑀)𝑋 ∈ ((π‘β€˜π‘–)[,)(π‘β€˜(𝑖 + 1)))))
 
Theoremiccpartiun 45344* A half-open interval of extended reals is the union of the parts of its partition. (Contributed by AV, 18-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ ((π‘ƒβ€˜0)[,)(π‘ƒβ€˜π‘€)) = βˆͺ 𝑖 ∈ (0..^𝑀)((π‘ƒβ€˜π‘–)[,)(π‘ƒβ€˜(𝑖 + 1))))
 
Theoremicceuelpartlem 45345 Lemma for icceuelpart 45346. (Contributed by AV, 19-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ ((𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀)) β†’ (𝐼 < 𝐽 β†’ (π‘ƒβ€˜(𝐼 + 1)) ≀ (π‘ƒβ€˜π½))))
 
Theoremicceuelpart 45346* An element of a partitioned half-open interval of extended reals is an element of exactly one part of the partition. (Contributed by AV, 19-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   ((πœ‘ ∧ 𝑋 ∈ ((π‘ƒβ€˜0)[,)(π‘ƒβ€˜π‘€))) β†’ βˆƒ!𝑖 ∈ (0..^𝑀)𝑋 ∈ ((π‘ƒβ€˜π‘–)[,)(π‘ƒβ€˜(𝑖 + 1))))
 
Theoremiccpartdisj 45347* The segments of a partitioned half-open interval of extended reals are a disjoint collection. (Contributed by AV, 19-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    β‡’   (πœ‘ β†’ Disj 𝑖 ∈ (0..^𝑀)((π‘ƒβ€˜π‘–)[,)(π‘ƒβ€˜(𝑖 + 1))))
 
Theoremiccpartnel 45348 A point of a partition is not an element of any open interval determined by the partition. Corresponds to fourierdlem12 44082 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 8-Jul-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))    &   (πœ‘ β†’ 𝑋 ∈ ran 𝑃)    β‡’   ((πœ‘ ∧ 𝐼 ∈ (0..^𝑀)) β†’ Β¬ 𝑋 ∈ ((π‘ƒβ€˜πΌ)(,)(π‘ƒβ€˜(𝐼 + 1))))
 
21.43.9  Shifting functions with an integer range domain
 
Theoremfargshiftfv 45349* If a class is a function, then the values of the "shifted function" correspond to the function values of the class. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (π‘₯ ∈ (0..^(β™―β€˜πΉ)) ↦ (πΉβ€˜(π‘₯ + 1)))    β‡’   ((𝑁 ∈ β„•0 ∧ 𝐹:(1...𝑁)⟢dom 𝐸) β†’ (𝑋 ∈ (0..^𝑁) β†’ (πΊβ€˜π‘‹) = (πΉβ€˜(𝑋 + 1))))
 
Theoremfargshiftf 45350* If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (π‘₯ ∈ (0..^(β™―β€˜πΉ)) ↦ (πΉβ€˜(π‘₯ + 1)))    β‡’   ((𝑁 ∈ β„•0 ∧ 𝐹:(1...𝑁)⟢dom 𝐸) β†’ 𝐺:(0..^(β™―β€˜πΉ))⟢dom 𝐸)
 
Theoremfargshiftf1 45351* If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (π‘₯ ∈ (0..^(β™―β€˜πΉ)) ↦ (πΉβ€˜(π‘₯ + 1)))    β‡’   ((𝑁 ∈ β„•0 ∧ 𝐹:(1...𝑁)–1-1β†’dom 𝐸) β†’ 𝐺:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐸)
 
Theoremfargshiftfo 45352* If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
𝐺 = (π‘₯ ∈ (0..^(β™―β€˜πΉ)) ↦ (πΉβ€˜(π‘₯ + 1)))    β‡’   ((𝑁 ∈ β„•0 ∧ 𝐹:(1...𝑁)–ontoβ†’dom 𝐸) β†’ 𝐺:(0..^(β™―β€˜πΉ))–ontoβ†’dom 𝐸)
 
Theoremfargshiftfva 45353* The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
𝐺 = (π‘₯ ∈ (0..^(β™―β€˜πΉ)) ↦ (πΉβ€˜(π‘₯ + 1)))    β‡’   ((𝑁 ∈ β„•0 ∧ 𝐹:(1...𝑁)⟢dom 𝐸) β†’ (βˆ€π‘˜ ∈ (1...𝑁)(πΈβ€˜(πΉβ€˜π‘˜)) = β¦‹π‘˜ / π‘₯β¦Œπ‘ƒ β†’ βˆ€π‘™ ∈ (0..^𝑁)(πΈβ€˜(πΊβ€˜π‘™)) = ⦋(𝑙 + 1) / π‘₯β¦Œπ‘ƒ))
 
21.43.10  Words over a set (extension)
 
21.43.10.1  Last symbol of a word - extension
 
Theoremlswn0 45354 The last symbol of a not empty word exists. The empty set must be excluded as symbol, because otherwise, it cannot be distinguished between valid cases (βˆ… is the last symbol) and invalid cases (βˆ… means that no last symbol exists. This is because of the special definition of a function in set.mm. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
((π‘Š ∈ Word 𝑉 ∧ βˆ… βˆ‰ 𝑉 ∧ (β™―β€˜π‘Š) β‰  0) β†’ (lastSβ€˜π‘Š) β‰  βˆ…)
 
21.43.11  Unordered pairs
 
21.43.11.1  Interchangeable setvar variables
 
Syntaxwich 45355 Extend wff notation to include the propery of a wff πœ‘ that the setvar variables π‘₯ and 𝑦 are interchangeable. Read this notation as "π‘₯ and 𝑦 are interchangeable in wff πœ‘".
wff [π‘₯⇄𝑦]πœ‘
 
Definitiondf-ich 45356* Define the property of a wff πœ‘ that the setvar variables π‘₯ and 𝑦 are interchangeable. For an alternate definition using implicit substitution and a temporary setvar variable see ichcircshi 45364. Another, equivalent definition using two temporary setvar variables is provided in dfich2 45368. (Contributed by AV, 29-Jul-2023.)
([π‘₯⇄𝑦]πœ‘ ↔ βˆ€π‘₯βˆ€π‘¦([π‘₯ / π‘Ž][𝑦 / π‘₯][π‘Ž / 𝑦]πœ‘ ↔ πœ‘))
 
Theoremnfich1 45357 The first interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.)
β„²π‘₯[π‘₯⇄𝑦]πœ‘
 
Theoremnfich2 45358 The second interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.)
Ⅎ𝑦[π‘₯⇄𝑦]πœ‘
 
Theoremichv 45359* Setvar variables are interchangeable in a wff they do not appear in. (Contributed by SN, 23-Nov-2023.)
[π‘₯⇄𝑦]πœ‘
 
Theoremichf 45360 Setvar variables are interchangeable in a wff they are not free in. (Contributed by SN, 23-Nov-2023.)
β„²π‘₯πœ‘    &   β„²π‘¦πœ‘    β‡’   [π‘₯⇄𝑦]πœ‘
 
Theoremichid 45361 A setvar variable is always interchangeable with itself. (Contributed by AV, 29-Jul-2023.)
[π‘₯⇄π‘₯]πœ‘
 
Theoremicht 45362 A theorem is interchangeable. (Contributed by SN, 4-May-2024.)
πœ‘    β‡’   [π‘₯⇄𝑦]πœ‘
 
Theoremichbidv 45363* Formula building rule for interchangeability (deduction). (Contributed by SN, 4-May-2024.)
(πœ‘ β†’ (πœ“ ↔ πœ’))    β‡’   (πœ‘ β†’ ([π‘₯⇄𝑦]πœ“ ↔ [π‘₯⇄𝑦]πœ’))
 
Theoremichcircshi 45364* The setvar variables are interchangeable if they can be circularily shifted using a third setvar variable, using implicit substitution. (Contributed by AV, 29-Jul-2023.)
(π‘₯ = 𝑧 β†’ (πœ‘ ↔ πœ“))    &   (𝑦 = π‘₯ β†’ (πœ“ ↔ πœ’))    &   (𝑧 = 𝑦 β†’ (πœ’ ↔ πœ‘))    β‡’   [π‘₯⇄𝑦]πœ‘
 
Theoremichan 45365 If two setvar variables are interchangeable in two wffs, then they are interchangeable in the conjunction of these two wffs. Notice that the reverse implication is not necessarily true. Corresponding theorems will hold for other commutative operations, too. (Contributed by AV, 31-Jul-2023.) Use df-ich 45356 instead of dfich2 45368 to reduce axioms. (Revised by SN, 4-May-2024.)
(([π‘Žβ‡„π‘]πœ‘ ∧ [π‘Žβ‡„π‘]πœ“) β†’ [π‘Žβ‡„π‘](πœ‘ ∧ πœ“))
 
Theoremichn 45366 Negation does not affect interchangeability. (Contributed by SN, 30-Aug-2023.)
([π‘Žβ‡„π‘]πœ‘ ↔ [π‘Žβ‡„π‘] Β¬ πœ‘)
 
Theoremichim 45367 Formula building rule for implication in interchangeability. (Contributed by SN, 4-May-2024.)
(([π‘Žβ‡„π‘]πœ‘ ∧ [π‘Žβ‡„π‘]πœ“) β†’ [π‘Žβ‡„π‘](πœ‘ β†’ πœ“))
 
Theoremdfich2 45368* Alternate definition of the propery of a wff πœ‘ that the setvar variables π‘₯ and 𝑦 are interchangeable. (Contributed by AV and WL, 6-Aug-2023.)
([π‘₯⇄𝑦]πœ‘ ↔ βˆ€π‘Žβˆ€π‘([π‘Ž / π‘₯][𝑏 / 𝑦]πœ‘ ↔ [𝑏 / π‘₯][π‘Ž / 𝑦]πœ‘))
 
Theoremichcom 45369* The interchangeability of setvar variables is commutative. (Contributed by AV, 20-Aug-2023.)
([π‘₯⇄𝑦]πœ“ ↔ [𝑦⇄π‘₯]πœ“)
 
Theoremichbi12i 45370* Equivalence for interchangeable setvar variables. (Contributed by AV, 29-Jul-2023.)
((π‘₯ = π‘Ž ∧ 𝑦 = 𝑏) β†’ (πœ“ ↔ πœ’))    β‡’   ([π‘₯⇄𝑦]πœ“ ↔ [π‘Žβ‡„π‘]πœ’)
 
Theoremicheqid 45371 In an equality for the same setvar variable, the setvar variable is interchangeable by itself. Special case of ichid 45361 and icheq 45372 without distinct variables restriction. (Contributed by AV, 29-Jul-2023.)
[π‘₯⇄π‘₯]π‘₯ = π‘₯
 
Theoremicheq 45372* In an equality of setvar variables, the setvar variables are interchangeable. (Contributed by AV, 29-Jul-2023.)
[π‘₯⇄𝑦]π‘₯ = 𝑦
 
Theoremichnfimlem 45373* Lemma for ichnfim 45374: A substitution for a nonfree variable has no effect. (Contributed by Wolf Lammen, 6-Aug-2023.) Avoid ax-13 2372. (Revised by Gino Giotto, 1-May-2024.)
(βˆ€π‘¦β„²π‘₯πœ‘ β†’ ([π‘Ž / π‘₯][𝑏 / 𝑦]πœ‘ ↔ [𝑏 / 𝑦]πœ‘))
 
Theoremichnfim 45374* If in an interchangeability context π‘₯ is not free in πœ‘, the same holds for 𝑦. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)
((βˆ€π‘¦β„²π‘₯πœ‘ ∧ [π‘₯⇄𝑦]πœ‘) β†’ βˆ€π‘₯β„²π‘¦πœ‘)
 
Theoremichnfb 45375* If π‘₯ and 𝑦 are interchangeable in πœ‘, they are both free or both not free in πœ‘. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)
([π‘₯⇄𝑦]πœ‘ β†’ (βˆ€π‘₯β„²π‘¦πœ‘ ↔ βˆ€π‘¦β„²π‘₯πœ‘))
 
Theoremichal 45376* Move a universal quantifier inside interchangeability. (Contributed by SN, 30-Aug-2023.)
(βˆ€π‘₯[π‘Žβ‡„π‘]πœ‘ β†’ [π‘Žβ‡„π‘]βˆ€π‘₯πœ‘)
 
Theoremich2al 45377 Two setvar variables are always interchangeable when there are two universal quantifiers. (Contributed by SN, 23-Nov-2023.)
[π‘₯⇄𝑦]βˆ€π‘₯βˆ€π‘¦πœ‘
 
Theoremich2ex 45378 Two setvar variables are always interchangeable when there are two existential quantifiers. (Contributed by SN, 23-Nov-2023.)
[π‘₯⇄𝑦]βˆƒπ‘₯βˆƒπ‘¦πœ‘
 
Theoremichexmpl1 45379* Example for interchangeable setvar variables in a statement of predicate calculus with equality. (Contributed by AV, 31-Jul-2023.)
[π‘Žβ‡„π‘]βˆƒπ‘Žβˆƒπ‘βˆƒπ‘(π‘Ž = 𝑏 ∧ π‘Ž β‰  𝑐 ∧ 𝑏 β‰  𝑐)
 
Theoremichexmpl2 45380* Example for interchangeable setvar variables in an arithmetic expression. (Contributed by AV, 31-Jul-2023.)
[π‘Žβ‡„π‘]((π‘Ž ∈ β„‚ ∧ 𝑏 ∈ β„‚ ∧ 𝑐 ∈ β„‚) β†’ ((π‘Žβ†‘2) + (𝑏↑2)) = (𝑐↑2))
 
Theoremich2exprop 45381* If the setvar variables are interchangeable in a wff, there is an ordered pair fulfilling the wff iff there is an unordered pair fulfilling the wff. (Contributed by AV, 16-Jul-2023.)
((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ [π‘Žβ‡„π‘]πœ‘) β†’ (βˆƒπ‘Žβˆƒπ‘({𝐴, 𝐡} = {π‘Ž, 𝑏} ∧ πœ‘) ↔ βˆƒπ‘Žβˆƒπ‘(⟨𝐴, 𝐡⟩ = βŸ¨π‘Ž, π‘βŸ© ∧ πœ‘)))
 
Theoremichnreuop 45382* If the setvar variables are interchangeable in a wff, there is never a unique ordered pair with different components fulfilling the wff (because if βŸ¨π‘Ž, π‘βŸ© fulfils the wff, then also βŸ¨π‘, π‘ŽβŸ© fulfils the wff). (Contributed by AV, 27-Aug-2023.)
([π‘Žβ‡„π‘]πœ‘ β†’ Β¬ βˆƒ!𝑝 ∈ (𝑋 Γ— 𝑋)βˆƒπ‘Žβˆƒπ‘(𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ π‘Ž β‰  𝑏 ∧ πœ‘))
 
Theoremichreuopeq 45383* If the setvar variables are interchangeable in a wff, and there is a unique ordered pair fulfilling the wff, then both setvar variables must be equal. (Contributed by AV, 28-Aug-2023.)
([π‘Žβ‡„π‘]πœ‘ β†’ (βˆƒ!𝑝 ∈ (𝑋 Γ— 𝑋)βˆƒπ‘Žβˆƒπ‘(𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ πœ‘) β†’ βˆƒπ‘Žβˆƒπ‘(π‘Ž = 𝑏 ∧ πœ‘)))
 
21.43.11.2  Set of unordered pairs
 
Theoremsprid 45384 Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
{𝑝 ∣ βˆƒπ‘Ž ∈ V βˆƒπ‘ ∈ V 𝑝 = {π‘Ž, 𝑏}} = {𝑝 ∣ βˆƒπ‘Žβˆƒπ‘ 𝑝 = {π‘Ž, 𝑏}}
 
Theoremelsprel 45385* An unordered pair is an element of all unordered pairs. At least one of the two elements of the unordered pair must be a set. Otherwise, the unordered pair would be the empty set, see prprc 4727, which is not an element of all unordered pairs, see spr0nelg 45386. (Contributed by AV, 21-Nov-2021.)
((𝐴 ∈ 𝑉 ∨ 𝐡 ∈ π‘Š) β†’ {𝐴, 𝐡} ∈ {𝑝 ∣ βˆƒπ‘Žβˆƒπ‘ 𝑝 = {π‘Ž, 𝑏}})
 
Theoremspr0nelg 45386* The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
βˆ… βˆ‰ {𝑝 ∣ βˆƒπ‘Žβˆƒπ‘ 𝑝 = {π‘Ž, 𝑏}}
 
Syntaxcspr 45387 Extend class notation with set of pairs.
class Pairs
 
Definitiondf-spr 45388* Define the function which maps a set 𝑣 to the set of pairs consisting of elements of the set 𝑣. (Contributed by AV, 21-Nov-2021.)
Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ βˆƒπ‘Ž ∈ 𝑣 βˆƒπ‘ ∈ 𝑣 𝑝 = {π‘Ž, 𝑏}})
 
Theoremsprval 45389* The set of all unordered pairs over a given set 𝑉. (Contributed by AV, 21-Nov-2021.)
(𝑉 ∈ π‘Š β†’ (Pairsβ€˜π‘‰) = {𝑝 ∣ βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 𝑝 = {π‘Ž, 𝑏}})
 
Theoremsprvalpw 45390* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
(𝑉 ∈ π‘Š β†’ (Pairsβ€˜π‘‰) = {𝑝 ∈ 𝒫 𝑉 ∣ βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 𝑝 = {π‘Ž, 𝑏}})
 
Theoremsprssspr 45391* The set of all unordered pairs over a given set 𝑉 is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
(Pairsβ€˜π‘‰) βŠ† {𝑝 ∣ βˆƒπ‘Žβˆƒπ‘ 𝑝 = {π‘Ž, 𝑏}}
 
Theoremspr0el 45392 The empty set is not an unordered pair over any set 𝑉. (Contributed by AV, 21-Nov-2021.)
βˆ… βˆ‰ (Pairsβ€˜π‘‰)
 
Theoremsprvalpwn0 45393* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
(𝑉 ∈ π‘Š β†’ (Pairsβ€˜π‘‰) = {𝑝 ∈ (𝒫 𝑉 βˆ– {βˆ…}) ∣ βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 𝑝 = {π‘Ž, 𝑏}})
 
Theoremsprel 45394* An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
(𝑋 ∈ (Pairsβ€˜π‘‰) β†’ βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 𝑋 = {π‘Ž, 𝑏})
 
Theoremprssspr 45395* An element of a subset of the set of all unordered pairs over a given set 𝑉, is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
((𝑃 βŠ† (Pairsβ€˜π‘‰) ∧ 𝑋 ∈ 𝑃) β†’ βˆƒπ‘Ž ∈ 𝑉 βˆƒπ‘ ∈ 𝑉 𝑋 = {π‘Ž, 𝑏})
 
Theoremprelspr 45396 An unordered pair of elements of a fixed set 𝑉 belongs to the set of all unordered pairs over the set 𝑉. (Contributed by AV, 21-Nov-2021.)
((𝑉 ∈ π‘Š ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ {𝑋, π‘Œ} ∈ (Pairsβ€˜π‘‰))
 
Theoremprsprel 45397 The elements of a pair from the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
(({𝑋, π‘Œ} ∈ (Pairsβ€˜π‘‰) ∧ (𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘Š)) β†’ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉))
 
Theoremprsssprel 45398 The elements of a pair from a subset of the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 21-Nov-2021.)
((𝑃 βŠ† (Pairsβ€˜π‘‰) ∧ {𝑋, π‘Œ} ∈ 𝑃 ∧ (𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘Š)) β†’ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉))
 
Theoremsprvalpwle2 45399* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 24-Nov-2021.)
(𝑉 ∈ π‘Š β†’ (Pairsβ€˜π‘‰) = {𝑝 ∈ (𝒫 𝑉 βˆ– {βˆ…}) ∣ (β™―β€˜π‘) ≀ 2})
 
Theoremsprsymrelfvlem 45400* Lemma for sprsymrelf 45405 and sprsymrelfv 45404. (Contributed by AV, 19-Nov-2021.)
(𝑃 βŠ† (Pairsβ€˜π‘‰) β†’ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘ ∈ 𝑃 𝑐 = {π‘₯, 𝑦}} ∈ 𝒫 (𝑉 Γ— 𝑉))
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