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Type | Label | Description |
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Statement | ||
Theorem | elsetpreimafvbi 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.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} β β’ ((πΉ Fn π΄ β§ π β π β§ π β π) β (π β π β (π β π΄ β§ (πΉβπ) = (πΉβπ)))) | ||
Theorem | elsetpreimafveqfv 45302* | The elements of the preimage of a function value have the same function values. (Contributed by AV, 5-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} β β’ ((πΉ Fn π΄ β§ (π β π β§ π β π β§ π β π)) β (πΉβπ) = (πΉβπ)) | ||
Theorem | eqfvelsetpreimafv 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.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} β β’ ((πΉ Fn π΄ β§ π β π β§ π β π) β ((π β π΄ β§ (πΉβπ) = (πΉβπ)) β π β π)) | ||
Theorem | elsetpreimafvrab 45304* | An element of the preimage of a function value expressed as a restricted class abstraction. (Contributed by AV, 9-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} β β’ ((πΉ Fn π΄ β§ π β π β§ π β π) β π = {π₯ β π΄ β£ (πΉβπ₯) = (πΉβπ)}) | ||
Theorem | imaelsetpreimafv 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.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} β β’ ((πΉ Fn π΄ β§ π β π β§ π β π) β (πΉ β π) = {(πΉβπ)}) | ||
Theorem | uniimaelsetpreimafv 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.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} β β’ ((πΉ Fn π΄ β§ π β π) β βͺ (πΉ β π) β ran πΉ) | ||
Theorem | elsetpreimafveq 45307* | If two preimages of function values contain elements with identical function values, then both preimages are equal. (Contributed by AV, 8-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} β β’ ((πΉ Fn π΄ β§ (π β π β§ π β π) β§ (π β π β§ π β π )) β ((πΉβπ) = (πΉβπ) β π = π )) | ||
Theorem | fundcmpsurinjlem1 45308* | Lemma 1 for fundcmpsurinj 45319. (Contributed by AV, 4-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} & β’ πΊ = (π₯ β π΄ β¦ (β‘πΉ β {(πΉβπ₯)})) β β’ ran πΊ = π | ||
Theorem | fundcmpsurinjlem2 45309* | Lemma 2 for fundcmpsurinj 45319. (Contributed by AV, 4-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} & β’ πΊ = (π₯ β π΄ β¦ (β‘πΉ β {(πΉβπ₯)})) β β’ ((πΉ Fn π΄ β§ π΄ β π) β πΊ:π΄βontoβπ) | ||
Theorem | fundcmpsurinjlem3 45310* | Lemma 3 for fundcmpsurinj 45319. (Contributed by AV, 3-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} & β’ π» = (π β π β¦ βͺ (πΉ β π)) β β’ ((Fun πΉ β§ π β π) β (π»βπ) = βͺ (πΉ β π)) | ||
Theorem | imasetpreimafvbijlemf 45311* | Lemma for imasetpreimafvbij 45316: the mapping π» is a function into the range of function πΉ. (Contributed by AV, 22-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} & β’ π» = (π β π β¦ βͺ (πΉ β π)) β β’ (πΉ Fn π΄ β π»:πβΆ(πΉ β π΄)) | ||
Theorem | imasetpreimafvbijlemfv 45312* | Lemma for imasetpreimafvbij 45316: the value of the mapping π» at a preimage of a value of function πΉ. (Contributed by AV, 5-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} & β’ π» = (π β π β¦ βͺ (πΉ β π)) β β’ ((πΉ Fn π΄ β§ π β π β§ π β π) β (π»βπ) = (πΉβπ)) | ||
Theorem | imasetpreimafvbijlemfv1 45313* | Lemma for imasetpreimafvbij 45316: for a preimage of a value of function πΉ there is an element of the preimage so that the value of the mapping π» at this preimage is the function value at this element. (Contributed by AV, 5-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} & β’ π» = (π β π β¦ βͺ (πΉ β π)) β β’ ((πΉ Fn π΄ β§ π β π) β βπ¦ β π (π»βπ) = (πΉβπ¦)) | ||
Theorem | imasetpreimafvbijlemf1 45314* | Lemma for imasetpreimafvbij 45316: the mapping π» is an injective function into the range of function πΉ. (Contributed by AV, 9-Mar-2024.) (Revised by AV, 22-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} & β’ π» = (π β π β¦ βͺ (πΉ β π)) β β’ (πΉ Fn π΄ β π»:πβ1-1β(πΉ β π΄)) | ||
Theorem | imasetpreimafvbijlemfo 45315* | Lemma for imasetpreimafvbij 45316: the mapping π» is a function onto the range of function πΉ. (Contributed by AV, 22-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} & β’ π» = (π β π β¦ βͺ (πΉ β π)) β β’ ((πΉ Fn π΄ β§ π΄ β π) β π»:πβontoβ(πΉ β π΄)) | ||
Theorem | imasetpreimafvbij 45316* | The mapping π» is a bijective function betwen the set π of all preimages of values of function πΉ and the range of πΉ. (Contributed by AV, 22-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} & β’ π» = (π β π β¦ βͺ (πΉ β π)) β β’ ((πΉ Fn π΄ β§ π΄ β π) β π»:πβ1-1-ontoβ(πΉ β π΄)) | ||
Theorem | fundcmpsurbijinjpreimafv 45317* | Every function πΉ:π΄βΆπ΅ can be decomposed into a surjective function onto π, a bijective function from π and an injective function into the codomain of πΉ. (Contributed by AV, 22-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} β β’ ((πΉ:π΄βΆπ΅ β§ π΄ β π) β βπβββπ((π:π΄βontoβπ β§ β:πβ1-1-ontoβ(πΉ β π΄) β§ π:(πΉ β π΄)β1-1βπ΅) β§ πΉ = ((π β β) β π))) | ||
Theorem | fundcmpsurinjpreimafv 45318* | Every function πΉ:π΄βΆπ΅ can be decomposed into a surjective function onto π and an injective function from π. (Contributed by AV, 12-Mar-2024.) (Proof shortened by AV, 22-Mar-2024.) |
β’ π = {π§ β£ βπ₯ β π΄ π§ = (β‘πΉ β {(πΉβπ₯)})} β β’ ((πΉ:π΄βΆπ΅ β§ π΄ β π) β βπββ(π:π΄βontoβπ β§ β:πβ1-1βπ΅ β§ πΉ = (β β π))) | ||
Theorem | fundcmpsurinj 45319* | Every function πΉ:π΄βΆπ΅ can be decomposed into a surjective and an injective function. (Contributed by AV, 13-Mar-2024.) |
β’ ((πΉ:π΄βΆπ΅ β§ π΄ β π) β βπβββπ(π:π΄βontoβπ β§ β:πβ1-1βπ΅ β§ πΉ = (β β π))) | ||
Theorem | fundcmpsurbijinj 45320* | Every function πΉ:π΄βΆπ΅ can be decomposed into a surjective, a bijective and an injective function. (Contributed by AV, 23-Mar-2024.) |
β’ ((πΉ:π΄βΆπ΅ β§ π΄ β π) β βπβββπβπβπ((π:π΄βontoβπ β§ β:πβ1-1-ontoβπ β§ π:πβ1-1βπ΅) β§ πΉ = ((π β β) β π))) | ||
Theorem | fundcmpsurinjimaid 45321* | Every function πΉ:π΄βΆπ΅ can be decomposed into a surjective function onto the image (πΉ β π΄) of the domain of πΉ and an injective function from the image (πΉ β π΄). (Contributed by AV, 17-Mar-2024.) |
β’ πΌ = (πΉ β π΄) & β’ πΊ = (π₯ β π΄ β¦ (πΉβπ₯)) & β’ π» = ( I βΎ πΌ) β β’ (πΉ:π΄βΆπ΅ β (πΊ:π΄βontoβπΌ β§ π»:πΌβ1-1βπ΅ β§ πΉ = (π» β πΊ))) | ||
Theorem | fundcmpsurinjALT 45322* | Alternate proof of fundcmpsurinj 45319, based on fundcmpsurinjimaid 45321: Every function πΉ:π΄βΆπ΅ can be decomposed into a surjective and an injective function. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by AV, 13-Mar-2024.) |
β’ ((πΉ:π΄βΆπ΅ β§ π΄ β π) β βπβββπ(π:π΄βontoβπ β§ β:πβ1-1βπ΅ β§ πΉ = (β β π))) | ||
Based on the theorems of the fourierdlem* series of GS's mathbox. | ||
Syntax | ciccp 45323 | Extend class notation with the partitions of a closed interval of extended reals. |
class RePart | ||
Definition | df-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.) |
β’ RePart = (π β β β¦ {π β (β* βm (0...π)) β£ βπ β (0..^π)(πβπ) < (πβ(π + 1))}) | ||
Theorem | iccpval 45325* | Partition consisting of a fixed number π of parts. (Contributed by AV, 9-Jul-2020.) |
β’ (π β β β (RePartβπ) = {π β (β* βm (0...π)) β£ βπ β (0..^π)(πβπ) < (πβ(π + 1))}) | ||
Theorem | iccpart 45326* | A special partition. Corresponds to fourierdlem2 44072 in GS's mathbox. (Contributed by AV, 9-Jul-2020.) |
β’ (π β β β (π β (RePartβπ) β (π β (β* βm (0...π)) β§ βπ β (0..^π)(πβπ) < (πβ(π + 1))))) | ||
Theorem | iccpartimp 45327 | Implications for a class being a partition. (Contributed by AV, 11-Jul-2020.) |
β’ ((π β β β§ π β (RePartβπ) β§ πΌ β (0..^π)) β (π β (β* βm (0...π)) β§ (πβπΌ) < (πβ(πΌ + 1)))) | ||
Theorem | iccpartres 45328 | The restriction of a partition is a partition. (Contributed by AV, 16-Jul-2020.) |
β’ ((π β β β§ π β (RePartβ(π + 1))) β (π βΎ (0...π)) β (RePartβπ)) | ||
Theorem | iccpartxr 45329 | If there is a partition, then all intermediate points and bounds are extended real numbers. (Contributed by AV, 11-Jul-2020.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) & β’ (π β πΌ β (0...π)) β β’ (π β (πβπΌ) β β*) | ||
Theorem | iccpartgtprec 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.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) & β’ (π β πΌ β (1...π)) β β’ (π β (πβ(πΌ β 1)) < (πβπΌ)) | ||
Theorem | iccpartipre 45331 | If there is a partition, then all intermediate points are real numbers. (Contributed by AV, 11-Jul-2020.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) & β’ (π β πΌ β (1..^π)) β β’ (π β (πβπΌ) β β) | ||
Theorem | iccpartiltu 45332* | If there is a partition, then all intermediate points are strictly less than the upper bound. (Contributed by AV, 12-Jul-2020.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β βπ β (1..^π)(πβπ) < (πβπ)) | ||
Theorem | iccpartigtl 45333* | If there is a partition, then all intermediate points are strictly greater than the lower bound. (Contributed by AV, 12-Jul-2020.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β βπ β (1..^π)(πβ0) < (πβπ)) | ||
Theorem | iccpartlt 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.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β (πβ0) < (πβπ)) | ||
Theorem | iccpartltu 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.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β βπ β (0..^π)(πβπ) < (πβπ)) | ||
Theorem | iccpartgtl 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.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β βπ β (1...π)(πβ0) < (πβπ)) | ||
Theorem | iccpartgt 45337* | If there is a partition, then all intermediate points and the bounds are strictly ordered. (Contributed by AV, 18-Jul-2020.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β βπ β (0...π)βπ β (0...π)(π < π β (πβπ) < (πβπ))) | ||
Theorem | iccpartleu 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.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β βπ β (0...π)(πβπ) β€ (πβπ)) | ||
Theorem | iccpartgel 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.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β βπ β (0...π)(πβ0) β€ (πβπ)) | ||
Theorem | iccpartrn 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.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β ran π β ((πβ0)[,](πβπ))) | ||
Theorem | iccpartf 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.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β π:(0...π)βΆ((πβ0)[,](πβπ))) | ||
Theorem | iccpartel 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.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ ((π β§ πΌ β (0...π)) β (πβπΌ) β ((πβ0)[,](πβπ))) | ||
Theorem | iccelpart 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.) |
β’ (π β β β βπ β (RePartβπ)(π β ((πβ0)[,)(πβπ)) β βπ β (0..^π)π β ((πβπ)[,)(πβ(π + 1))))) | ||
Theorem | iccpartiun 45344* | A half-open interval of extended reals is the union of the parts of its partition. (Contributed by AV, 18-Jul-2020.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β ((πβ0)[,)(πβπ)) = βͺ π β (0..^π)((πβπ)[,)(πβ(π + 1)))) | ||
Theorem | icceuelpartlem 45345 | Lemma for icceuelpart 45346. (Contributed by AV, 19-Jul-2020.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β ((πΌ β (0..^π) β§ π½ β (0..^π)) β (πΌ < π½ β (πβ(πΌ + 1)) β€ (πβπ½)))) | ||
Theorem | icceuelpart 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.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ ((π β§ π β ((πβ0)[,)(πβπ))) β β!π β (0..^π)π β ((πβπ)[,)(πβ(π + 1)))) | ||
Theorem | iccpartdisj 45347* | The segments of a partitioned half-open interval of extended reals are a disjoint collection. (Contributed by AV, 19-Jul-2020.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) β β’ (π β Disj π β (0..^π)((πβπ)[,)(πβ(π + 1)))) | ||
Theorem | iccpartnel 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.) |
β’ (π β π β β) & β’ (π β π β (RePartβπ)) & β’ (π β π β ran π) β β’ ((π β§ πΌ β (0..^π)) β Β¬ π β ((πβπΌ)(,)(πβ(πΌ + 1)))) | ||
Theorem | fargshiftfv 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.) |
β’ πΊ = (π₯ β (0..^(β―βπΉ)) β¦ (πΉβ(π₯ + 1))) β β’ ((π β β0 β§ πΉ:(1...π)βΆdom πΈ) β (π β (0..^π) β (πΊβπ) = (πΉβ(π + 1)))) | ||
Theorem | fargshiftf 45350* | If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.) |
β’ πΊ = (π₯ β (0..^(β―βπΉ)) β¦ (πΉβ(π₯ + 1))) β β’ ((π β β0 β§ πΉ:(1...π)βΆdom πΈ) β πΊ:(0..^(β―βπΉ))βΆdom πΈ) | ||
Theorem | fargshiftf1 45351* | If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.) |
β’ πΊ = (π₯ β (0..^(β―βπΉ)) β¦ (πΉβ(π₯ + 1))) β β’ ((π β β0 β§ πΉ:(1...π)β1-1βdom πΈ) β πΊ:(0..^(β―βπΉ))β1-1βdom πΈ) | ||
Theorem | fargshiftfo 45352* | If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.) |
β’ πΊ = (π₯ β (0..^(β―βπΉ)) β¦ (πΉβ(π₯ + 1))) β β’ ((π β β0 β§ πΉ:(1...π)βontoβdom πΈ) β πΊ:(0..^(β―βπΉ))βontoβdom πΈ) | ||
Theorem | fargshiftfva 45353* | The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.) |
β’ πΊ = (π₯ β (0..^(β―βπΉ)) β¦ (πΉβ(π₯ + 1))) β β’ ((π β β0 β§ πΉ:(1...π)βΆdom πΈ) β (βπ β (1...π)(πΈβ(πΉβπ)) = β¦π / π₯β¦π β βπ β (0..^π)(πΈβ(πΊβπ)) = β¦(π + 1) / π₯β¦π)) | ||
Theorem | lswn0 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 (β is the last symbol) and invalid cases (β 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.) |
β’ ((π β Word π β§ β β π β§ (β―βπ) β 0) β (lastSβπ) β β ) | ||
Syntax | wich 45355 | Extend wff notation to include the propery of a wff π that the setvar variables π₯ and π¦ are interchangeable. Read this notation as "π₯ and π¦ are interchangeable in wff π". |
wff [π₯βπ¦]π | ||
Definition | df-ich 45356* | Define the property of a wff π that the setvar variables π₯ and π¦ 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.) |
β’ ([π₯βπ¦]π β βπ₯βπ¦([π₯ / π][π¦ / π₯][π / π¦]π β π)) | ||
Theorem | nfich1 45357 | The first interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.) |
β’ β²π₯[π₯βπ¦]π | ||
Theorem | nfich2 45358 | The second interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.) |
β’ β²π¦[π₯βπ¦]π | ||
Theorem | ichv 45359* | Setvar variables are interchangeable in a wff they do not appear in. (Contributed by SN, 23-Nov-2023.) |
β’ [π₯βπ¦]π | ||
Theorem | ichf 45360 | Setvar variables are interchangeable in a wff they are not free in. (Contributed by SN, 23-Nov-2023.) |
β’ β²π₯π & β’ β²π¦π β β’ [π₯βπ¦]π | ||
Theorem | ichid 45361 | A setvar variable is always interchangeable with itself. (Contributed by AV, 29-Jul-2023.) |
β’ [π₯βπ₯]π | ||
Theorem | icht 45362 | A theorem is interchangeable. (Contributed by SN, 4-May-2024.) |
β’ π β β’ [π₯βπ¦]π | ||
Theorem | ichbidv 45363* | Formula building rule for interchangeability (deduction). (Contributed by SN, 4-May-2024.) |
β’ (π β (π β π)) β β’ (π β ([π₯βπ¦]π β [π₯βπ¦]π)) | ||
Theorem | ichcircshi 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.) |
β’ (π₯ = π§ β (π β π)) & β’ (π¦ = π₯ β (π β π)) & β’ (π§ = π¦ β (π β π)) β β’ [π₯βπ¦]π | ||
Theorem | ichan 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.) |
β’ (([πβπ]π β§ [πβπ]π) β [πβπ](π β§ π)) | ||
Theorem | ichn 45366 | Negation does not affect interchangeability. (Contributed by SN, 30-Aug-2023.) |
β’ ([πβπ]π β [πβπ] Β¬ π) | ||
Theorem | ichim 45367 | Formula building rule for implication in interchangeability. (Contributed by SN, 4-May-2024.) |
β’ (([πβπ]π β§ [πβπ]π) β [πβπ](π β π)) | ||
Theorem | dfich2 45368* | Alternate definition of the propery of a wff π that the setvar variables π₯ and π¦ are interchangeable. (Contributed by AV and WL, 6-Aug-2023.) |
β’ ([π₯βπ¦]π β βπβπ([π / π₯][π / π¦]π β [π / π₯][π / π¦]π)) | ||
Theorem | ichcom 45369* | The interchangeability of setvar variables is commutative. (Contributed by AV, 20-Aug-2023.) |
β’ ([π₯βπ¦]π β [π¦βπ₯]π) | ||
Theorem | ichbi12i 45370* | Equivalence for interchangeable setvar variables. (Contributed by AV, 29-Jul-2023.) |
β’ ((π₯ = π β§ π¦ = π) β (π β π)) β β’ ([π₯βπ¦]π β [πβπ]π) | ||
Theorem | icheqid 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.) |
β’ [π₯βπ₯]π₯ = π₯ | ||
Theorem | icheq 45372* | In an equality of setvar variables, the setvar variables are interchangeable. (Contributed by AV, 29-Jul-2023.) |
β’ [π₯βπ¦]π₯ = π¦ | ||
Theorem | ichnfimlem 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.) |
β’ (βπ¦β²π₯π β ([π / π₯][π / π¦]π β [π / π¦]π)) | ||
Theorem | ichnfim 45374* | If in an interchangeability context π₯ is not free in π, the same holds for π¦. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.) |
β’ ((βπ¦β²π₯π β§ [π₯βπ¦]π) β βπ₯β²π¦π) | ||
Theorem | ichnfb 45375* | If π₯ and π¦ are interchangeable in π, they are both free or both not free in π. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.) |
β’ ([π₯βπ¦]π β (βπ₯β²π¦π β βπ¦β²π₯π)) | ||
Theorem | ichal 45376* | Move a universal quantifier inside interchangeability. (Contributed by SN, 30-Aug-2023.) |
β’ (βπ₯[πβπ]π β [πβπ]βπ₯π) | ||
Theorem | ich2al 45377 | Two setvar variables are always interchangeable when there are two universal quantifiers. (Contributed by SN, 23-Nov-2023.) |
β’ [π₯βπ¦]βπ₯βπ¦π | ||
Theorem | ich2ex 45378 | Two setvar variables are always interchangeable when there are two existential quantifiers. (Contributed by SN, 23-Nov-2023.) |
β’ [π₯βπ¦]βπ₯βπ¦π | ||
Theorem | ichexmpl1 45379* | Example for interchangeable setvar variables in a statement of predicate calculus with equality. (Contributed by AV, 31-Jul-2023.) |
β’ [πβπ]βπβπβπ(π = π β§ π β π β§ π β π) | ||
Theorem | ichexmpl2 45380* | Example for interchangeable setvar variables in an arithmetic expression. (Contributed by AV, 31-Jul-2023.) |
β’ [πβπ]((π β β β§ π β β β§ π β β) β ((πβ2) + (πβ2)) = (πβ2)) | ||
Theorem | ich2exprop 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.) |
β’ ((π΄ β π β§ π΅ β π β§ [πβπ]π) β (βπβπ({π΄, π΅} = {π, π} β§ π) β βπβπ(β¨π΄, π΅β© = β¨π, πβ© β§ π))) | ||
Theorem | ichnreuop 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 β¨π, πβ© fulfils the wff, then also β¨π, πβ© fulfils the wff). (Contributed by AV, 27-Aug-2023.) |
β’ ([πβπ]π β Β¬ β!π β (π Γ π)βπβπ(π = β¨π, πβ© β§ π β π β§ π)) | ||
Theorem | ichreuopeq 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.) |
β’ ([πβπ]π β (β!π β (π Γ π)βπβπ(π = β¨π, πβ© β§ π) β βπβπ(π = π β§ π))) | ||
Theorem | sprid 45384 | Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.) |
β’ {π β£ βπ β V βπ β V π = {π, π}} = {π β£ βπβπ π = {π, π}} | ||
Theorem | elsprel 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.) |
β’ ((π΄ β π β¨ π΅ β π) β {π΄, π΅} β {π β£ βπβπ π = {π, π}}) | ||
Theorem | spr0nelg 45386* | The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.) |
β’ β β {π β£ βπβπ π = {π, π}} | ||
Syntax | cspr 45387 | Extend class notation with set of pairs. |
class Pairs | ||
Definition | df-spr 45388* | Define the function which maps a set π£ to the set of pairs consisting of elements of the set π£. (Contributed by AV, 21-Nov-2021.) |
β’ Pairs = (π£ β V β¦ {π β£ βπ β π£ βπ β π£ π = {π, π}}) | ||
Theorem | sprval 45389* | The set of all unordered pairs over a given set π. (Contributed by AV, 21-Nov-2021.) |
β’ (π β π β (Pairsβπ) = {π β£ βπ β π βπ β π π = {π, π}}) | ||
Theorem | sprvalpw 45390* | The set of all unordered pairs over a given set π, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.) |
β’ (π β π β (Pairsβπ) = {π β π« π β£ βπ β π βπ β π π = {π, π}}) | ||
Theorem | sprssspr 45391* | The set of all unordered pairs over a given set π is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021.) |
β’ (Pairsβπ) β {π β£ βπβπ π = {π, π}} | ||
Theorem | spr0el 45392 | The empty set is not an unordered pair over any set π. (Contributed by AV, 21-Nov-2021.) |
β’ β β (Pairsβπ) | ||
Theorem | sprvalpwn0 45393* | The set of all unordered pairs over a given set π, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.) |
β’ (π β π β (Pairsβπ) = {π β (π« π β {β }) β£ βπ β π βπ β π π = {π, π}}) | ||
Theorem | sprel 45394* | An element of the set of all unordered pairs over a given set π is a pair of elements of the set π. (Contributed by AV, 22-Nov-2021.) |
β’ (π β (Pairsβπ) β βπ β π βπ β π π = {π, π}) | ||
Theorem | prssspr 45395* | An element of a subset of the set of all unordered pairs over a given set π, is a pair of elements of the set π. (Contributed by AV, 22-Nov-2021.) |
β’ ((π β (Pairsβπ) β§ π β π) β βπ β π βπ β π π = {π, π}) | ||
Theorem | prelspr 45396 | An unordered pair of elements of a fixed set π belongs to the set of all unordered pairs over the set π. (Contributed by AV, 21-Nov-2021.) |
β’ ((π β π β§ (π β π β§ π β π)) β {π, π} β (Pairsβπ)) | ||
Theorem | prsprel 45397 | The elements of a pair from the set of all unordered pairs over a given set π are elements of the set π. (Contributed by AV, 22-Nov-2021.) |
β’ (({π, π} β (Pairsβπ) β§ (π β π β§ π β π)) β (π β π β§ π β π)) | ||
Theorem | prsssprel 45398 | The elements of a pair from a subset of the set of all unordered pairs over a given set π are elements of the set π. (Contributed by AV, 21-Nov-2021.) |
β’ ((π β (Pairsβπ) β§ {π, π} β π β§ (π β π β§ π β π)) β (π β π β§ π β π)) | ||
Theorem | sprvalpwle2 45399* | The set of all unordered pairs over a given set π, expressed by a restricted class abstraction. (Contributed by AV, 24-Nov-2021.) |
β’ (π β π β (Pairsβπ) = {π β (π« π β {β }) β£ (β―βπ) β€ 2}) | ||
Theorem | sprsymrelfvlem 45400* | Lemma for sprsymrelf 45405 and sprsymrelfv 45404. (Contributed by AV, 19-Nov-2021.) |
β’ (π β (Pairsβπ) β {β¨π₯, π¦β© β£ βπ β π π = {π₯, π¦}} β π« (π Γ π)) |
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