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Theorem List for Metamath Proof Explorer - 35301-35400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-trans 35301 Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
 
Definitiondf-bigcup 35302 Define the Bigcup function, which, per fvbigcup 35346, carries a set to its union. (Contributed by Scott Fenton, 11-Apr-2012.)
Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
 
Definitiondf-fix 35303 Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
Fix 𝐴 = dom (𝐴 ∩ I )
 
Definitiondf-limits 35304 Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
 
Definitiondf-funs 35305 Define the class of all functions. See elfuns 35359 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)
Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )))
 
Definitiondf-singleton 35306 Define the singleton function. See brsingle 35361 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
 
Definitiondf-singles 35307 Define the class of all singletons. See elsingles 35362 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
Singletons = ran Singleton
 
Definitiondf-image 35308 Define the image functor. This function takes a set 𝐴 to a function 𝑥 ↦ (𝐴𝑥), providing that the latter exists. See imageval 35374 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V)))
 
Definitiondf-cart 35309 Define the cartesian product function. See brcart 35376 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
 
Definitiondf-img 35310 Define the image function. See brimg 35381 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
 
Definitiondf-domain 35311 Define the domain function. See brdomain 35377 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Domain = Image(1st ↾ (V × V))
 
Definitiondf-range 35312 Define the range function. See brrange 35378 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Range = Image(2nd ↾ (V × V))
 
Definitiondf-cup 35313 Define the little cup function. See brcup 35383 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∪ (2nd ∘ E )) ⊗ V)))
 
Definitiondf-cap 35314 Define the little cap function. See brcap 35384 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∩ (2nd ∘ E )) ⊗ V)))
 
Definitiondf-restrict 35315 Define the restriction function. See brrestrict 35393 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
 
Definitiondf-succf 35316 Define the successor function. See brsuccf 35385 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Succ = (Cup ∘ ( I ⊗ Singleton))
 
Definitiondf-apply 35317 Define the application function. See brapply 35382 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 35318 Define the functional part of a class 𝐹. This is the maximal part of 𝐹 that is a function. See funpartfun 35387 and funpartfv 35389 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
 
Definitiondf-fullfun 35319 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 35320 Define the upper bound relationship functor. See brub 35398 for value. (Contributed by Scott Fenton, 3-May-2018.)
UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ E ))
 
Definitiondf-lb 35321 Define the lower bound relationship functor. See brlb 35399 for value. (Contributed by Scott Fenton, 3-May-2018.)
LB𝑅 = UB𝑅
 
Theoremtxpss3v 35322 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 35323 A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel (𝐴𝐵)
 
Theorembrtxp 35324 Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 35322, 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 35325* 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 35326 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 35327 A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
 
Theorempprodss4v 35328 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 35329 Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 35328, 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 35330* 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 35331* 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 35332 The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel SSet
 
Theorembrsset 35333 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 35334 I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
I = ( SSet SSet )
 
Theoremeltrans 35335 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
𝐴 ∈ V       (𝐴 Trans ↔ Tr 𝐴)
 
Theoremdfon3 35336 A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.)
On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
 
Theoremdfon4 35337 Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))
 
Theorembrtxpsd 35338* 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 35339* 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 35340* 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 35341 The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
Rel Bigcup
 
Theorembrbigcup 35342 Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.)
𝐵 ∈ V       (𝐴 Bigcup 𝐵 𝐴 = 𝐵)
 
Theoremdfbigcup2 35343 Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
Bigcup = (𝑥 ∈ V ↦ 𝑥)
 
Theoremfobigcup 35344 Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
Bigcup :V–onto→V
 
Theoremfnbigcup 35345 Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)
Bigcup Fn V
 
Theoremfvbigcup 35346 For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       ( Bigcup 𝐴) = 𝐴
 
Theoremelfix 35347 Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       (𝐴 Fix 𝑅𝐴𝑅𝐴)
 
Theoremelfix2 35348 Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
Rel 𝑅       (𝐴 Fix 𝑅𝐴𝑅𝐴)
 
Theoremdffix2 35349 The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 = ran (𝐴 ∩ I )
 
Theoremfixssdm 35350 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 ⊆ dom 𝐴
 
Theoremfixssrn 35351 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 ⊆ ran 𝐴
 
Theoremfixcnv 35352 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 35353 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix (𝐴𝐵) = ( Fix 𝐴 Fix 𝐵)
 
Theoremellimits 35354 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       (𝐴 Limits ↔ Lim 𝐴)
 
Theoremlimitssson 35355 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 35356 A quantifier-free definition of ω that does not depend on ax-inf 9639. (Note: label was changed from dfom5 9651 to dfom5b 35356 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.)
ω = (On ∩ Limits )
 
Theoremsscoid 35357 A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
(𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))
 
Theoremdffun10 35358 Another potential definition of functionality. 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 35359 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐹 ∈ V       (𝐹 Funs ↔ Fun 𝐹)
 
Theoremelfunsg 35360 Closed form of elfuns 35359. (Contributed by Scott Fenton, 2-May-2014.)
(𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))
 
Theorembrsingle 35361 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 35362* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥})
 
Theoremfnsingle 35363 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 35364 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 35365* 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 35366 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
𝐴 ∈ V       {𝐴} ∈ Singletons
 
Theoremdfiota3 35367 A definition of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
(℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )
 
Theoremdffv5 35368 Another quantifier-free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐹𝐴) = ({(𝐹 “ {𝐴})} ∩ Singletons )
 
Theoremunisnif 35369 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 35370 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 35371 Closed form of brimage 35370. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴)))
 
Theoremfunimage 35372 Image𝐴 is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Image𝐴
 
Theoremfnimage 35373* 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 35374* 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 35375 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 35376 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 35377 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 35378 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 35379 Closed form of brdomain 35377. (Contributed by Scott Fenton, 2-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Domain𝐵𝐵 = dom 𝐴))
 
Theorembrrangeg 35380 Closed form of brrange 35378. (Contributed by Scott Fenton, 3-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))
 
Theorembrimg 35381 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 35382 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 35383 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 35384 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 35385 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)
 
Theoremfunpartlem 35386* Lemma for funpartfun 35387. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
 
Theoremfunpartfun 35387 The functional part of 𝐹 is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Fun Funpart𝐹
 
Theoremfunpartss 35388 The functional part of 𝐹 is a subset of 𝐹. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Funpart𝐹𝐹
 
Theoremfunpartfv 35389 The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(Funpart𝐹𝐴) = (𝐹𝐴)
 
Theoremfullfunfnv 35390 The full functional part of 𝐹 is a function over V. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
FullFun𝐹 Fn V
 
Theoremfullfunfv 35391 The function value of the full function of 𝐹 agrees with 𝐹. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(FullFun𝐹𝐴) = (𝐹𝐴)
 
Theorembrfullfun 35392 A binary relation form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴FullFun𝐹𝐵𝐵 = (𝐹𝐴))
 
Theorembrrestrict 35393 Binary relation form of the Restrict function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Restrict𝐶𝐶 = (𝐴𝐵))
 
Theoremdfrecs2 35394 A quantifier-free definition of recs. (Contributed by Scott Fenton, 17-Jul-2020.)
recs(𝐹) = (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))))
 
Theoremdfrdg4 35395 A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
rec(𝐹, 𝐴) = (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))))
 
Theoremdfint3 35396 Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
𝐴 = (V ∖ ((V ∖ E ) “ 𝐴))
 
Theoremimagesset 35397 The Image functor applied to the converse of the subset relationship yields a subset of the subset relationship. (Contributed by Scott Fenton, 14-Apr-2018.)
Image SSet SSet
 
Theorembrub 35398* Binary relation form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)
𝑆 ∈ V    &   𝐴 ∈ V       (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)
 
Theorembrlb 35399* Binary relation form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)
𝑆 ∈ V    &   𝐴 ∈ V       (𝑆LB𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
 
21.10.17  Alternate ordered pairs
 
Syntaxcaltop 35400 Declare the syntax for an alternate ordered pair.
class 𝐴, 𝐵
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