![]() |
Metamath
Proof Explorer Theorem List (p. 354 of 481) | < 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: | ![]() (1-30606) |
![]() (30607-32129) |
![]() (32130-48017) |
Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-trans 35301 | Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E )) | ||
Definition | df-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))) | ||
Definition | df-fix 35303 | Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ Fix 𝐴 = dom (𝐴 ∩ I ) | ||
Definition | df-limits 35304 | Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅}) | ||
Definition | df-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 ))) | ||
Definition | df-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))) | ||
Definition | df-singles 35307 | Define the class of all singletons. See elsingles 35362 for membership. (Contributed by Scott Fenton, 19-Feb-2013.) |
⊢ Singletons = ran Singleton | ||
Definition | df-image 35308 | Define the image functor. This function takes a set 𝐴 to a function 𝑥 ↦ (𝐴 “ 𝑥), 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))) | ||
Definition | df-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))) | ||
Definition | df-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) | ||
Definition | df-domain 35311 | Define the domain function. See brdomain 35377 for its value. (Contributed by Scott Fenton, 11-Apr-2014.) |
⊢ Domain = Image(1st ↾ (V × V)) | ||
Definition | df-range 35312 | Define the range function. See brrange 35378 for its value. (Contributed by Scott Fenton, 11-Apr-2014.) |
⊢ Range = Image(2nd ↾ (V × V)) | ||
Definition | df-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))) | ||
Definition | df-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))) | ||
Definition | df-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 ))))) | ||
Definition | df-succf 35316 | Define the successor function. See brsuccf 35385 for its value. (Contributed by Scott Fenton, 14-Apr-2014.) |
⊢ Succ = (Cup ∘ ( I ⊗ Singleton)) | ||
Definition | df-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)))) | ||
Definition | df-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 ))) | ||
Definition | df-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) × {∅})) | ||
Definition | df-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 )) | ||
Definition | df-lb 35321 | Define the lower bound relationship functor. See brlb 35399 for value. (Contributed by Scott Fenton, 3-May-2018.) |
⊢ LB𝑅 = UB◡𝑅 | ||
Theorem | txpss3v 35322 | A tail Cartesian product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V)) | ||
Theorem | txprel 35323 | A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ Rel (𝐴 ⊗ 𝐵) | ||
Theorem | brtxp 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(𝐴 ⊗ 𝐵)〈𝑌, 𝑍〉 ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍)) | ||
Theorem | brtxp2 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(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)) | ||
Theorem | dfpprod2 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))))) | ||
Theorem | pprodcnveq 35327 | A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.) |
⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆) | ||
Theorem | pprodss4v 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)) | ||
Theorem | brpprod 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〉 ↔ (𝑋𝐴𝑍 ∧ 𝑌𝐵𝑊)) | ||
Theorem | brpprod3a 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 ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) | ||
Theorem | brpprod3b 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〉 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) | ||
Theorem | relsset 35332 | The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ Rel SSet | ||
Theorem | brsset 35333 | For sets, the SSet binary relation is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 SSet 𝐵 ↔ 𝐴 ⊆ 𝐵) | ||
Theorem | idsset 35334 | I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ I = ( SSet ∩ ◡ SSet ) | ||
Theorem | eltrans 35335 | Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴) | ||
Theorem | dfon3 35336 | A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.) |
⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) | ||
Theorem | dfon4 35337 | Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.) |
⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans )) | ||
Theorem | brtxpsd 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(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) | ||
Theorem | brtxpsd2 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(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) | ||
Theorem | brtxpsd3 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 & ⊢ (𝑥 ∈ 𝑋 ↔ 𝑥𝑆𝐴) ⇒ ⊢ (𝐴𝑅𝐵 ↔ 𝐵 = 𝑋) | ||
Theorem | relbigcup 35341 | The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ Rel Bigcup | ||
Theorem | brbigcup 35342 | Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵) | ||
Theorem | dfbigcup2 35343 | Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥) | ||
Theorem | fobigcup 35344 | Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Bigcup :V–onto→V | ||
Theorem | fnbigcup 35345 | Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ Bigcup Fn V | ||
Theorem | fvbigcup 35346 | For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ( Bigcup ‘𝐴) = ∪ 𝐴 | ||
Theorem | elfix 35347 | Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) | ||
Theorem | elfix2 35348 | Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ Rel 𝑅 ⇒ ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) | ||
Theorem | dffix2 35349 | The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Fix 𝐴 = ran (𝐴 ∩ I ) | ||
Theorem | fixssdm 35350 | The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Fix 𝐴 ⊆ dom 𝐴 | ||
Theorem | fixssrn 35351 | The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Fix 𝐴 ⊆ ran 𝐴 | ||
Theorem | fixcnv 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 | ||
Theorem | fixun 35353 | The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵) | ||
Theorem | ellimits 35354 | Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ Limits ↔ Lim 𝐴) | ||
Theorem | limitssson 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 | ||
Theorem | dfom5b 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 ) | ||
Theorem | sscoid 35357 | A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.) |
⊢ (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴 ∧ 𝐴 ⊆ 𝐵)) | ||
Theorem | dffun10 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 𝐹 ↔ 𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹)))) | ||
Theorem | elfuns 35359 | Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐹 ∈ Funs ↔ Fun 𝐹) | ||
Theorem | elfunsg 35360 | Closed form of elfuns 35359. (Contributed by Scott Fenton, 2-May-2014.) |
⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ Funs ↔ Fun 𝐹)) | ||
Theorem | brsingle 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 ⇒ ⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴}) | ||
Theorem | elsingles 35362* | Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.) |
⊢ (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥}) | ||
Theorem | fnsingle 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 | ||
Theorem | fvsingle 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} | ||
Theorem | dfsingles2 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}} | ||
Theorem | snelsingles 35366 | A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝐴} ∈ Singletons | ||
Theorem | dfiota3 35367 | A definition of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.) |
⊢ (℩𝑥𝜑) = ∪ ∪ ({{𝑥 ∣ 𝜑}} ∩ Singletons ) | ||
Theorem | dffv5 35368 | Another quantifier-free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.) |
⊢ (𝐹‘𝐴) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons ) | ||
Theorem | unisnif 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, ∅) | ||
Theorem | brimage 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 ⇒ ⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)) | ||
Theorem | brimageg 35371 | Closed form of brimage 35370. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴))) | ||
Theorem | funimage 35372 | Image𝐴 is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ Fun Image𝐴 | ||
Theorem | fnimage 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 ∣ (𝑅 “ 𝑥) ∈ V} | ||
Theorem | imageval 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 ↦ (𝑅 “ 𝑥)) | ||
Theorem | fvimage 35375 | Value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅 “ 𝐴) ∈ 𝑊) → (Image𝑅‘𝐴) = (𝑅 “ 𝐴)) | ||
Theorem | brcart 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〉Cart𝐶 ↔ 𝐶 = (𝐴 × 𝐵)) | ||
Theorem | brdomain 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 ⇒ ⊢ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴) | ||
Theorem | brrange 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 ⇒ ⊢ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴) | ||
Theorem | brdomaing 35379 | Closed form of brdomain 35377. (Contributed by Scott Fenton, 2-May-2014.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)) | ||
Theorem | brrangeg 35380 | Closed form of brrange 35378. (Contributed by Scott Fenton, 3-May-2014.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)) | ||
Theorem | brimg 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〉Img𝐶 ↔ 𝐶 = (𝐴 “ 𝐵)) | ||
Theorem | brapply 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〉Apply𝐶 ↔ 𝐶 = (𝐴‘𝐵)) | ||
Theorem | brcup 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〉Cup𝐶 ↔ 𝐶 = (𝐴 ∪ 𝐵)) | ||
Theorem | brcap 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〉Cap𝐶 ↔ 𝐶 = (𝐴 ∩ 𝐵)) | ||
Theorem | brsuccf 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 ⇒ ⊢ (𝐴Succ𝐵 ↔ 𝐵 = suc 𝐴) | ||
Theorem | funpartlem 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}) | ||
Theorem | funpartfun 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 | ||
Theorem | funpartss 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.) |
⊢ Funpart𝐹 ⊆ 𝐹 | ||
Theorem | funpartfv 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.) |
⊢ (Funpart𝐹‘𝐴) = (𝐹‘𝐴) | ||
Theorem | fullfunfnv 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 | ||
Theorem | fullfunfv 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.) |
⊢ (FullFun𝐹‘𝐴) = (𝐹‘𝐴) | ||
Theorem | brfullfun 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 ⇒ ⊢ (𝐴FullFun𝐹𝐵 ↔ 𝐵 = (𝐹‘𝐴)) | ||
Theorem | brrestrict 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〉Restrict𝐶 ↔ 𝐶 = (𝐴 ↾ 𝐵)) | ||
Theorem | dfrecs2 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)))) | ||
Theorem | dfrdg4 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))))))) | ||
Theorem | dfint3 35396 | Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.) |
⊢ ∩ 𝐴 = (V ∖ (◡(V ∖ E ) “ 𝐴)) | ||
Theorem | imagesset 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 | ||
Theorem | brub 35398* | Binary relation form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.) |
⊢ 𝑆 ∈ V & ⊢ 𝐴 ∈ V ⇒ ⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴) | ||
Theorem | brlb 35399* | Binary relation form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.) |
⊢ 𝑆 ∈ V & ⊢ 𝐴 ∈ V ⇒ ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) | ||
Syntax | caltop 35400 | Declare the syntax for an alternate ordered pair. |
class ⟪𝐴, 𝐵⟫ |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |