P. 365 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	wsuceq2 36401	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝐴 = 𝐵 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋))
Theorem	wsuceq3 36402	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌))
Theorem	nfwsuc 36403	Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝑋    ⇒   ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋)
Theorem	wlimeq12 36404	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))
Theorem	wlimeq1 36405	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
⊢ (𝑅 = 𝑆 → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐴))
Theorem	wlimeq2 36406	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
⊢ (𝐴 = 𝐵 → WLim(𝑅, 𝐴) = WLim(𝑅, 𝐵))
Theorem	nfwlim 36407	Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥WLim(𝑅, 𝐴)
Theorem	elwlim 36408	Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))
Theorem	wzel 36409	The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ 𝐴 ≠ ∅) → inf(𝐴, 𝐴, 𝑅) ∈ 𝐴)
Theorem	wsuclem 36410*	Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)))
Theorem	wsucex 36411	Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)
Theorem	wsuccl 36412*	If 𝑋 is a set with an 𝑅 successor in 𝐴, then its well-founded successor is a member of 𝐴. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦)    ⇒   ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)
Theorem	wsuclb 36413	A well-founded successor is a lower bound on points after 𝑋. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋𝑅𝑌)    ⇒   ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))
Theorem	wlimss 36414	The class of limit points is a subclass of the base class. (Contributed by Scott Fenton, 16-Jun-2018.)
⊢ WLim(𝑅, 𝐴) ⊆ 𝐴
21.11.15  Quantifier-free definitions
Syntax	ctxp 36415	Declare the syntax for tail Cartesian product.
class (𝐴 ⊗ 𝐵)
Syntax	cpprod 36416	Declare the syntax for the parallel product.
class pprod(𝑅, 𝑆)
Syntax	csset 36417	Declare the subset relationship class.
class SSet
Syntax	ctrans 36418	Declare the transitive set class.
class Trans
Syntax	cbigcup 36419	Declare the set union relationship.
class Bigcup
Syntax	cfix 36420	Declare the syntax for the fixpoints of a class.
class Fix 𝐴
Syntax	climits 36421	Declare the class of limit ordinals.
class Limits
Syntax	cfuns 36422	Declare the syntax for the class of all function.
class Funs
Syntax	csingle 36423	Declare the syntax for the singleton function.
class Singleton
Syntax	csingles 36424	Declare the syntax for the class of all singletons.
class Singletons
Syntax	cimage 36425	Declare the syntax for the image functor.
class Image𝐴
Syntax	ccart 36426	Declare the syntax for the cartesian function.
class Cart
Syntax	cimg 36427	Declare the syntax for the image function.
class Img
Syntax	cdomain 36428	Declare the syntax for the domain function.
class Domain
Syntax	crange 36429	Declare the syntax for the range function.
class Range
Syntax	capply 36430	Declare the syntax for the application function.
class Apply
Syntax	ccup 36431	Declare the syntax for the cup function.
class Cup
Syntax	ccap 36432	Declare the syntax for the cap function.
class Cap
Syntax	csuccf 36433	Declare the syntax for the successor function.
class Succ
Syntax	cfunpart 36434	Declare the syntax for the functional part functor.
class Funpart𝐹
Syntax	cfullfn 36435	Declare the syntax for the full function functor.
class FullFun𝐹
Syntax	crestrict 36436	Declare the syntax for the restriction function.
class Restrict
Syntax	cub 36437	Declare the syntax for the upper bound relationship functor.
class UB𝑅
Syntax	clb 36438	Declare the syntax for the lower bound relationship functor.
class LB𝑅
Definition	df-txp 36439	Define the tail cross of two classes. Membership in this class is defined by txpss3v 36463 and brtxp 36465. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
Definition	df-pprod 36440	Define the parallel product of two classes. Membership in this class is defined by pprodss4v 36469 and brpprod 36470. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
Definition	df-sset 36441	Define the subset class. For the value, see brsset 36474. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
Definition	df-trans 36442	Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
Definition	df-bigcup 36443	Define the Bigcup function, which, per fvbigcup 36487, 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 36444	Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Fix 𝐴 = dom (𝐴 ∩ I )
Definition	df-limits 36445	Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
Definition	df-funs 36446	Define the class of all functions. See elfuns 36500 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ ◡ E )))
Definition	df-singleton 36447	Define the singleton function. See brsingle 36502 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
Definition	df-singles 36448	Define the class of all singletons. See elsingles 36503 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ Singletons = ran Singleton
Definition	df-image 36449	Define the image functor. This function takes a set 𝐴 to a function 𝑥 ↦ (𝐴 “ 𝑥), providing that the latter exists. See imageval 36515 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))
Definition	df-cart 36450	Define the cartesian product function. See brcart 36517 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
Definition	df-img 36451	Define the image function. See brimg 36522 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
⊢ Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
Definition	df-domain 36452	Define the domain function. See brdomain 36518 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Domain = Image(1st ↾ (V × V))
Definition	df-range 36453	Define the range function. See brrange 36519 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
⊢ Range = Image(2nd ↾ (V × V))
Definition	df-cup 36454	Define the little cup function. See brcup 36524 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 36455	Define the little cap function. See brcap 36525 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 36456	Define the restriction function. See brrestrict 36536 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
⊢ Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
Definition	df-succf 36457	Define the successor function. See its alternate version dfsuccf2 36528. See brsuccf 36527 for its value. Cf. the equivalent df-sucmap 39218 family. (Contributed by Scott Fenton, 14-Apr-2014.)
⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
Definition	df-apply 36458	Define the application function. See brapply 36523 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 36459	Define the functional part of a class 𝐹. This is the maximal part of 𝐹 that is a function. See funpartfun 36530 and funpartfv 36532 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
Definition	df-fullfun 36460	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 36461	Define the upper bound relationship functor. See brub 36541 for value. (Contributed by Scott Fenton, 3-May-2018.)
⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))
Definition	df-lb 36462	Define the lower bound relationship functor. See brlb 36542 for value. (Contributed by Scott Fenton, 3-May-2018.)
⊢ LB𝑅 = UB◡𝑅
Theorem	txpss3v 36463	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 36464	A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Rel (𝐴 ⊗ 𝐵)
Theorem	brtxp 36465	Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 36463, 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 36466*	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 36467	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 36468	A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)
Theorem	pprodss4v 36469	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 36470	Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 36469, 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 36471*	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 36472*	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 36473	The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Rel SSet
Theorem	brsset 36474	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 36475	I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ I = ( SSet ∩ ◡ SSet )
Theorem	eltrans 36476	Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴)
Theorem	dfon3 36477	A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.)
⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
Theorem	dfon4 36478	Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))
Theorem	brtxpsd 36479*	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 36480*	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 36481*	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 36482	The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Rel Bigcup
Theorem	brbigcup 36483	Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵)
Theorem	dfbigcup2 36484	Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)
Theorem	fobigcup 36485	Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Bigcup :V–onto→V
Theorem	fnbigcup 36486	Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Bigcup Fn V
Theorem	fvbigcup 36487	For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ( Bigcup ‘𝐴) = ∪ 𝐴
Theorem	elfix 36488	Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)
Theorem	elfix2 36489	Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ Rel 𝑅    ⇒   ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)
Theorem	dffix2 36490	The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix 𝐴 = ran (𝐴 ∩ I )
Theorem	fixssdm 36491	The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix 𝐴 ⊆ dom 𝐴
Theorem	fixssrn 36492	The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix 𝐴 ⊆ ran 𝐴
Theorem	fixcnv 36493	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 36494	The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵)
Theorem	ellimits 36495	Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Limits ↔ Lim 𝐴)
Theorem	limitssson 36496	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 36497	A quantifier-free definition of ω that does not depend on ax-inf 9621. (Note: label was changed from dfom5 9633 to dfom5b 36497 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ ω = (On ∩ ∩ Limits )
Theorem	sscoid 36498	A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
⊢ (𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴 ∧ 𝐴 ⊆ 𝐵))
Theorem	dffun10 36499	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 36500	Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝐹 ∈ Funs ↔ Fun 𝐹)