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Theorem List for Metamath Proof Explorer - 33201-33300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcimg 33201 Declare the syntax for the image function.
class Img
 
Syntaxcdomain 33202 Declare the syntax for the domain function.
class Domain
 
Syntaxcrange 33203 Declare the syntax for the range function.
class Range
 
Syntaxcapply 33204 Declare the syntax for the application function.
class Apply
 
Syntaxccup 33205 Declare the syntax for the cup function.
class Cup
 
Syntaxccap 33206 Declare the syntax for the cap function.
class Cap
 
Syntaxcsuccf 33207 Declare the syntax for the successor function.
class Succ
 
Syntaxcfunpart 33208 Declare the syntax for the functional part functor.
class Funpart𝐹
 
Syntaxcfullfn 33209 Declare the syntax for the full function functor.
class FullFun𝐹
 
Syntaxcrestrict 33210 Declare the syntax for the restriction function.
class Restrict
 
Syntaxcub 33211 Declare the syntax for the upper bound relationship functor.
class UB𝑅
 
Syntaxclb 33212 Declare the syntax for the lower bound relationship functor.
class LB𝑅
 
Definitiondf-txp 33213 Define the tail cross of two classes. Membership in this class is defined by txpss3v 33237 and brtxp 33239. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
 
Definitiondf-pprod 33214 Define the parallel product of two classes. Membership in this class is defined by pprodss4v 33243 and brpprod 33244. (Contributed by Scott Fenton, 11-Apr-2014.)
pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
 
Definitiondf-sset 33215 Define the subset class. For the value, see brsset 33248. (Contributed by Scott Fenton, 31-Mar-2012.)
SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
 
Definitiondf-trans 33216 Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
 
Definitiondf-bigcup 33217 Define the Bigcup function, which, per fvbigcup 33261, carries a set to its union. (Contributed by Scott Fenton, 11-Apr-2012.)
Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
 
Definitiondf-fix 33218 Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
Fix 𝐴 = dom (𝐴 ∩ I )
 
Definitiondf-limits 33219 Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
 
Definitiondf-funs 33220 Define the class of all functions. See elfuns 33274 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)
Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )))
 
Definitiondf-singleton 33221 Define the singleton function. See brsingle 33276 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
 
Definitiondf-singles 33222 Define the class of all singletons. See elsingles 33277 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
Singletons = ran Singleton
 
Definitiondf-image 33223 Define the image functor. This function takes a set 𝐴 to a function 𝑥 ↦ (𝐴𝑥), providing that the latter exists. See imageval 33289 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V)))
 
Definitiondf-cart 33224 Define the cartesian product function. See brcart 33291 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
 
Definitiondf-img 33225 Define the image function. See brimg 33296 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
 
Definitiondf-domain 33226 Define the domain function. See brdomain 33292 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Domain = Image(1st ↾ (V × V))
 
Definitiondf-range 33227 Define the range function. See brrange 33293 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Range = Image(2nd ↾ (V × V))
 
Definitiondf-cup 33228 Define the little cup function. See brcup 33298 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∪ (2nd ∘ E )) ⊗ V)))
 
Definitiondf-cap 33229 Define the little cap function. See brcap 33299 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∩ (2nd ∘ E )) ⊗ V)))
 
Definitiondf-restrict 33230 Define the restriction function. See brrestrict 33308 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
 
Definitiondf-succf 33231 Define the successor function. See brsuccf 33300 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Succ = (Cup ∘ ( I ⊗ Singleton))
 
Definitiondf-apply 33232 Define the application function. See brapply 33297 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
Apply = (( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
 
Definitiondf-funpart 33233 Define the functional part of a class 𝐹. This is the maximal part of 𝐹 that is a function. See funpartfun 33302 and funpartfv 33304 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
 
Definitiondf-fullfun 33234 Define the full function over 𝐹. This is a function with domain V that always agrees with 𝐹 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
 
Definitiondf-ub 33235 Define the upper bound relationship functor. See brub 33313 for value. (Contributed by Scott Fenton, 3-May-2018.)
UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ E ))
 
Definitiondf-lb 33236 Define the lower bound relationship functor. See brlb 33314 for value. (Contributed by Scott Fenton, 3-May-2018.)
LB𝑅 = UB𝑅
 
Theoremtxpss3v 33237 A tail Cartesian product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) ⊆ (V × (V × V))
 
Theoremtxprel 33238 A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel (𝐴𝐵)
 
Theorembrtxp 33239 Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 33237, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V       (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌𝑋𝐵𝑍))
 
Theorembrtxp2 33240* The binary relation over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.)
𝐴 ∈ V       (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦))
 
Theoremdfpprod2 33241 Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod(𝐴, 𝐵) = (((1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))
 
Theorempprodcnveq 33242 A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
 
Theorempprodss4v 33243 The parallel product is a subclass of ((V × V) × (V × V)). (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))
 
Theorembrpprod 33244 Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 33243, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V    &   𝑊 ∈ V       (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍𝑌𝐵𝑊))
 
Theorembrpprod3a 33245* Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V       (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤))
 
Theorembrpprod3b 33246* Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.)
𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V       (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))
 
Theoremrelsset 33247 The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel SSet
 
Theorembrsset 33248 For sets, the SSet binary relation is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
𝐵 ∈ V       (𝐴 SSet 𝐵𝐴𝐵)
 
Theoremidsset 33249 I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
I = ( SSet SSet )
 
Theoremeltrans 33250 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
𝐴 ∈ V       (𝐴 Trans ↔ Tr 𝐴)
 
Theoremdfon3 33251 A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.)
On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
 
Theoremdfon4 33252 Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))
 
Theorembrtxpsd 33253* Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑅𝐴))
 
Theorembrtxpsd2 33254* Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))    &   𝐴𝐶𝐵       (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
 
Theorembrtxpsd3 33255* A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))    &   𝐴𝐶𝐵    &   (𝑥𝑋𝑥𝑆𝐴)       (𝐴𝑅𝐵𝐵 = 𝑋)
 
Theoremrelbigcup 33256 The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
Rel Bigcup
 
Theorembrbigcup 33257 Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.)
𝐵 ∈ V       (𝐴 Bigcup 𝐵 𝐴 = 𝐵)
 
Theoremdfbigcup2 33258 Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
Bigcup = (𝑥 ∈ V ↦ 𝑥)
 
Theoremfobigcup 33259 Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
Bigcup :V–onto→V
 
Theoremfnbigcup 33260 Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)
Bigcup Fn V
 
Theoremfvbigcup 33261 For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       ( Bigcup 𝐴) = 𝐴
 
Theoremelfix 33262 Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       (𝐴 Fix 𝑅𝐴𝑅𝐴)
 
Theoremelfix2 33263 Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
Rel 𝑅       (𝐴 Fix 𝑅𝐴𝑅𝐴)
 
Theoremdffix2 33264 The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 = ran (𝐴 ∩ I )
 
Theoremfixssdm 33265 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 ⊆ dom 𝐴
 
Theoremfixssrn 33266 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 ⊆ ran 𝐴
 
Theoremfixcnv 33267 The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 = Fix 𝐴
 
Theoremfixun 33268 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix (𝐴𝐵) = ( Fix 𝐴 Fix 𝐵)
 
Theoremellimits 33269 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       (𝐴 Limits ↔ Lim 𝐴)
 
Theoremlimitssson 33270 The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
Limits ⊆ On
 
Theoremdfom5b 33271 A quantifier-free definition of ω that does not depend on ax-inf 9090. (Note: label was changed from dfom5 9102 to dfom5b 33271 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.)
ω = (On ∩ Limits )
 
Theoremsscoid 33272 A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
(𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))
 
Theoremdffun10 33273 Another potential definition of functionhood. Based on statements in http://people.math.gatech.edu/~belinfan/research/autoreas/otter/sum/fs/. (Contributed by Scott Fenton, 30-Aug-2017.)
(Fun 𝐹𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))))
 
Theoremelfuns 33274 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐹 ∈ V       (𝐹 Funs ↔ Fun 𝐹)
 
Theoremelfunsg 33275 Closed form of elfuns 33274. (Contributed by Scott Fenton, 2-May-2014.)
(𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))
 
Theorembrsingle 33276 The binary relation form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Singleton𝐵𝐵 = {𝐴})
 
Theoremelsingles 33277* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥})
 
Theoremfnsingle 33278 The singleton relationship is a function over the universe. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singleton Fn V
 
Theoremfvsingle 33279 The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.)
(Singleton‘𝐴) = {𝐴}
 
Theoremdfsingles2 33280* Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singletons = {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}
 
Theoremsnelsingles 33281 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
𝐴 ∈ V       {𝐴} ∈ Singletons
 
Theoremdfiota3 33282 A definition of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
(℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )
 
Theoremdffv5 33283 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐹𝐴) = ({(𝐹 “ {𝐴})} ∩ Singletons )
 
Theoremunisnif 33284 Express union of singleton in terms of if. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
{𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
 
Theorembrimage 33285 Binary relation form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴))
 
Theorembrimageg 33286 Closed form of brimage 33285. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴)))
 
Theoremfunimage 33287 Image𝐴 is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Image𝐴
 
Theoremfnimage 33288* Image𝑅 is a function over the set-like portion of 𝑅. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
 
Theoremimageval 33289* The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image𝑅 = (𝑥 ∈ V ↦ (𝑅𝑥))
 
Theoremfvimage 33290 Value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴𝑉 ∧ (𝑅𝐴) ∈ 𝑊) → (Image𝑅𝐴) = (𝑅𝐴))
 
Theorembrcart 33291 Binary relation form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Cart𝐶𝐶 = (𝐴 × 𝐵))
 
Theorembrdomain 33292 Binary relation form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Domain𝐵𝐵 = dom 𝐴)
 
Theorembrrange 33293 Binary relation form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Range𝐵𝐵 = ran 𝐴)
 
Theorembrdomaing 33294 Closed form of brdomain 33292. (Contributed by Scott Fenton, 2-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Domain𝐵𝐵 = dom 𝐴))
 
Theorembrrangeg 33295 Closed form of brrange 33293. (Contributed by Scott Fenton, 3-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))
 
Theorembrimg 33296 Binary relation form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Img𝐶𝐶 = (𝐴𝐵))
 
Theorembrapply 33297 Binary relation form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Apply𝐶𝐶 = (𝐴𝐵))
 
Theorembrcup 33298 Binary relation form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Cup𝐶𝐶 = (𝐴𝐵))
 
Theorembrcap 33299 Binary relation form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Cap𝐶𝐶 = (𝐴𝐵))
 
Theorembrsuccf 33300 Binary relation form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Succ𝐵𝐵 = suc 𝐴)
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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 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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 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