HomeHome Metamath Proof Explorer
Theorem List (p. 322 of 429)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27903)
  Hilbert Space Explorer  Hilbert Space Explorer
(27904-29428)
  Users' Mathboxes  Users' Mathboxes
(29429-42879)
 

Theorem List for Metamath Proof Explorer - 32101-32200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-cup 32101 Define the little cup function. See brcup 32171 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∪ (2nd ∘ E )) ⊗ V)))
 
Definitiondf-cap 32102 Define the little cap function. See brcap 32172 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∩ (2nd ∘ E )) ⊗ V)))
 
Definitiondf-restrict 32103 Define the restriction function. See brrestrict 32181 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
 
Definitiondf-succf 32104 Define the successor function. See brsuccf 32173 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Succ = (Cup ∘ ( I ⊗ Singleton))
 
Definitiondf-apply 32105 Define the application function. See brapply 32170 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 32106 Define the functional part of a class 𝐹. This is the maximal part of 𝐹 that is a function. See funpartfun 32175 and funpartfv 32177 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
 
Definitiondf-fullfun 32107 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 32108 Define the upper bound relationship functor. See brub 32186 for value. (Contributed by Scott Fenton, 3-May-2018.)
UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ E ))
 
Definitiondf-lb 32109 Define the lower bound relationship functor. See brlb 32187 for value. (Contributed by Scott Fenton, 3-May-2018.)
LB𝑅 = UB𝑅
 
Theoremtxpss3v 32110 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 32111 A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel (𝐴𝐵)
 
Theorembrtxp 32112 Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 32110, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.)
𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V       (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌𝑋𝐵𝑍))
 
Theorembrtxp2 32113* 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 32114 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 32115 A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
 
Theorempprodss4v 32116 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.)
pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))
 
Theorembrpprod 32117 Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 32116, 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 32118* 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 32119* 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 32120 The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel SSet
 
Theorembrsset 32121 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 32122 I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
I = ( SSet SSet )
 
Theoremeltrans 32123 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
𝐴 ∈ V       (𝐴 Trans ↔ Tr 𝐴)
 
Theoremdfon3 32124 A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.)
On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
 
Theoremdfon4 32125 Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))
 
Theorembrtxpsd 32126* 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 32127* 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 32128* 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 32129 The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
Rel Bigcup
 
Theorembrbigcup 32130 Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.)
𝐵 ∈ V       (𝐴 Bigcup 𝐵 𝐴 = 𝐵)
 
Theoremdfbigcup2 32131 Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
Bigcup = (𝑥 ∈ V ↦ 𝑥)
 
Theoremfobigcup 32132 Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
Bigcup :V–onto→V
 
Theoremfnbigcup 32133 Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)
Bigcup Fn V
 
Theoremfvbigcup 32134 For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       ( Bigcup 𝐴) = 𝐴
 
Theoremelfix 32135 Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       (𝐴 Fix 𝑅𝐴𝑅𝐴)
 
Theoremelfix2 32136 Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
Rel 𝑅       (𝐴 Fix 𝑅𝐴𝑅𝐴)
 
Theoremdffix2 32137 The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 = ran (𝐴 ∩ I )
 
Theoremfixssdm 32138 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 ⊆ dom 𝐴
 
Theoremfixssrn 32139 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 ⊆ ran 𝐴
 
Theoremfixcnv 32140 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 32141 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix (𝐴𝐵) = ( Fix 𝐴 Fix 𝐵)
 
Theoremellimits 32142 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       (𝐴 Limits ↔ Lim 𝐴)
 
Theoremlimitssson 32143 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 32144 A quantifier-free definition of ω that does not depend on ax-inf 8573. (Note: label was changed from dfom5 8585 to dfom5b 32144 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.)
ω = (On ∩ Limits )
 
Theoremsscoid 32145 A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
(𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))
 
Theoremdffun10 32146 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 32147 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐹 ∈ V       (𝐹 Funs ↔ Fun 𝐹)
 
Theoremelfunsg 32148 Closed form of elfuns 32147. (Contributed by Scott Fenton, 2-May-2014.)
(𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))
 
Theorembrsingle 32149 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 32150* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥})
 
Theoremfnsingle 32151 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 32152 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 32153* 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 32154 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
𝐴 ∈ V       {𝐴} ∈ Singletons
 
Theoremdfiota3 32155 A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
(℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )
 
Theoremdffv5 32156 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐹𝐴) = ({(𝐹 “ {𝐴})} ∩ Singletons )
 
Theoremunisnif 32157 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 32158 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 32159 Closed form of brimage 32158. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴)))
 
Theoremfunimage 32160 Image𝐴 is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Image𝐴
 
Theoremfnimage 32161* 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 32162* 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 32163 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 32164 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 32165 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 32166 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 32167 Closed form of brdomain 32165. (Contributed by Scott Fenton, 2-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Domain𝐵𝐵 = dom 𝐴))
 
Theorembrrangeg 32168 Closed form of brrange 32166. (Contributed by Scott Fenton, 3-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))
 
Theorembrimg 32169 Binary relation form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Img𝐶𝐶 = (𝐴𝐵))
 
Theorembrapply 32170 Binary relation form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Apply𝐶𝐶 = (𝐴𝐵))
 
Theorembrcup 32171 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 32172 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 32173 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 32174* Lemma for funpartfun 32175. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
 
Theoremfunpartfun 32175 The functional part of 𝐹 is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Funpart𝐹
 
Theoremfunpartss 32176 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 32177 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 32178 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 32179 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 32180 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 32181 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 32182 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 32183 A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
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 32184 Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
𝐴 = (V ∖ ((V ∖ E ) “ 𝐴))
 
Theoremimagesset 32185 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 32186* Binary relation form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)
𝑆 ∈ V    &   𝐴 ∈ V       (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)
 
Theorembrlb 32187* Binary relation form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)
𝑆 ∈ V    &   𝐴 ∈ V       (𝑆LB𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
 
20.8.31  Alternate ordered pairs
 
Syntaxcaltop 32188 Declare the syntax for an alternate ordered pair.
class 𝐴, 𝐵
 
Syntaxcaltxp 32189 Declare the syntax for an alternate Cartesian product.
class (𝐴 ×× 𝐵)
 
Definitiondf-altop 32190 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 32201), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)
𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
 
Definitiondf-altxp 32191* Define Cartesian products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.)
(𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
 
Theoremaltopex 32192 Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)
𝐴, 𝐵⟫ ∈ V
 
Theoremaltopthsn 32193 Two alternate ordered pairs are equal iff the singletons of their respective elements are equal. Note that this holds regardless of sethood of any of the elements. (Contributed by Scott Fenton, 16-Apr-2012.)
(⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
 
Theoremaltopeq12 32194 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
((𝐴 = 𝐵𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)
 
Theoremaltopeq1 32195 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)
 
Theoremaltopeq2 32196 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫)
 
Theoremaltopth1 32197 Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐴𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐴 = 𝐶))
 
Theoremaltopth2 32198 Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐵𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐵 = 𝐷))
 
Theoremaltopthg 32199 Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)
((𝐴𝑉𝐵𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
Theoremaltopthbg 32200 Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)
((𝐴𝑉𝐷𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
  Copyright terms: Public domain < Previous  Next >