P. 95 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9401-9500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	snopfsupp 9401	A singleton containing an ordered pair is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {⟨𝑋, 𝑌⟩} finSupp 𝑍)
Theorem	funsnfsupp 9402	Finite support for a function extended by a singleton. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by AV, 19-Jul-2019.)
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (Fun 𝐹 ∧ 𝑋 ∉ dom 𝐹)) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) finSupp 𝑍 ↔ 𝐹 finSupp 𝑍))
Theorem	fsuppres 9403	The restriction of a finitely supported function is finitely supported. (Contributed by AV, 14-Jul-2019.)
⊢ (𝜑 → 𝐹 finSupp 𝑍)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍)
Theorem	fmptssfisupp 9404*	The restriction of a mapping function has finite support if that function has finite support. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) finSupp 𝑍)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐵) finSupp 𝑍)
Theorem	ressuppfi 9405	If the support of the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finite, the support of the function itself is finite. (Contributed by AV, 22-Apr-2019.)
⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵))    &   ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
Theorem	resfsupp 9406	If the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finitely supported, the function itself is finitely supported. (Contributed by AV, 27-May-2019.)
⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑊)    &   ⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵))    &   ⊢ (𝜑 → 𝐺 finSupp 𝑍)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
Theorem	resfifsupp 9407	The restriction of a function to a finite set is finitely supported. (Contributed by AV, 12-Dec-2019.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝑋) finSupp 𝑍)
Theorem	ffsuppbi 9408	Two ways of saying that a function with known codomain is finitely supported. (Contributed by AV, 8-Jul-2019.)
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 finSupp 𝑍 ↔ (◡𝐹 “ (𝑆 ∖ {𝑍})) ∈ Fin)))
Theorem	fsuppmptif 9409*	A function mapping an argument to either a value of a finitely supported function or zero is finitely supported. (Contributed by AV, 6-Jun-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑍)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍)
Theorem	sniffsupp 9410*	A function mapping all but one arguments to zero is finitely supported. (Contributed by AV, 8-Jul-2019.)
⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 0 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 ))    ⇒   ⊢ (𝜑 → 𝐹 finSupp 0 )
Theorem	fsuppcolem 9411	Lemma for fsuppco 9412. Formula building theorem for finite supports: rearranging the index set. (Contributed by Stefan O'Rear, 21-Mar-2015.)
⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin)    &   ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌)    ⇒   ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin)
Theorem	fsuppco 9412	The composition of a 1-1 function with a finitely supported function is finitely supported. (Contributed by AV, 28-May-2019.)
⊢ (𝜑 → 𝐹 finSupp 𝑍)    &   ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍)
Theorem	fsuppco2 9413	The composition of a function which maps the zero to zero with a finitely supported function is finitely supported. This is not only a special case of fsuppcor 9414 because it does not require that the "zero" is an element of the range of the finitely supported function. (Contributed by AV, 6-Jun-2019.)
⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑍)    &   ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍)
Theorem	fsuppcor 9414	The composition of a function which maps the zero of the range of a finitely supported function to the zero of its range with this finitely supported function is finitely supported. (Contributed by AV, 6-Jun-2019.)
⊢ (𝜑 → 0 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑍)    &   ⊢ (𝜑 → (𝐺‘𝑍) = 0 )    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 )
Theorem	mapfienlem1 9415*	Lemma 1 for mapfien 9418. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   ⊢ 𝑊 = (𝐺‘𝑍)    &   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊)
Theorem	mapfienlem2 9416*	Lemma 2 for mapfien 9418. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   ⊢ 𝑊 = (𝐺‘𝑍)    &   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) finSupp 𝑍)
Theorem	mapfienlem3 9417*	Lemma 3 for mapfien 9418. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   ⊢ 𝑊 = (𝐺‘𝑍)    &   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ 𝑆)
Theorem	mapfien 9418*	A bijection of the base sets induces a bijection on the set of finitely supported functions. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   ⊢ 𝑊 = (𝐺‘𝑍)    &   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹))):𝑆–1-1-onto→𝑇)
Theorem	mapfien2 9419*	Equinumerousity relation for sets of finitely supported functions. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 0 }    &   ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   ⊢ (𝜑 → 𝐴 ≈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≈ 𝐷)    &   ⊢ (𝜑 → 0 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐷)    ⇒   ⊢ (𝜑 → 𝑆 ≈ 𝑇)
2.4.36  Finite intersections
Syntax	cfi 9420	Extend class notation with the function whose value is the class of finite intersections of the elements of a given set.
class fi
Definition	df-fi 9421*	Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 9424). (Contributed by FL, 27-Apr-2008.)
⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
Theorem	fival 9422*	The set of all the finite intersections of the elements of 𝐴. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥})
Theorem	elfi 9423*	Specific properties of an element of (fi‘𝐵). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥))
Theorem	elfi2 9424*	The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥))
Theorem	elfir 9425	Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.)
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ∈ (fi‘𝐵))
Theorem	intrnfi 9426	Sufficient condition for the intersection of the range of a function to be in the set of finite intersections. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ ran 𝐹 ∈ (fi‘𝐵))
Theorem	iinfi 9427*	An indexed intersection of elements of 𝐶 is an element of the finite intersections of 𝐶. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ ((𝐶 ∈ 𝑉 ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (fi‘𝐶))
Theorem	inelfi 9428	The intersection of two sets is a finite intersection. (Contributed by Thierry Arnoux, 6-Jan-2017.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∩ 𝐵) ∈ (fi‘𝑋))
Theorem	ssfii 9429	Any element of a set 𝐴 is the intersection of a finite subset of 𝐴. (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴))
Theorem	fi0 9430	The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ (fi‘∅) = ∅
Theorem	fieq0 9431	A set is empty iff the class of all the finite intersections of that set is empty. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
Theorem	fiin 9432	The elements of (fi‘𝐶) are closed under finite intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)
⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴 ∩ 𝐵) ∈ (fi‘𝐶))
Theorem	dffi2 9433*	The set of finite intersections is the smallest set that contains 𝐴 and is closed under pairwise intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)
⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∩ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)})
Theorem	fiss 9434	Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵))
Theorem	inficl 9435*	A set which is closed under pairwise intersection is closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
Theorem	fipwuni 9436	The set of finite intersections of a set is contained in the powerset of the union of the elements of 𝐴. (Contributed by Mario Carneiro, 24-Nov-2013.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
⊢ (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴
Theorem	fisn 9437	A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (fi‘{𝐴}) = {𝐴}
Theorem	fiuni 9438	The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴))
Theorem	fipwss 9439	If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋)
Theorem	elfiun 9440*	A finite intersection of elements taken from a union of collections. (Contributed by Jeff Hankins, 15-Nov-2009.) (Proof shortened by Mario Carneiro, 26-Nov-2013.)
⊢ ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐾) → (𝐴 ∈ (fi‘(𝐵 ∪ 𝐶)) ↔ (𝐴 ∈ (fi‘𝐵) ∨ 𝐴 ∈ (fi‘𝐶) ∨ ∃𝑥 ∈ (fi‘𝐵)∃𝑦 ∈ (fi‘𝐶)𝐴 = (𝑥 ∩ 𝑦))))
Theorem	dffi3 9441*	The set of finite intersections can be "constructed" inductively by iterating binary intersection ω-many times. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ 𝑅 = (𝑢 ∈ V ↦ ran (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧)))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∪ (rec(𝑅, 𝐴) “ ω))
Theorem	fifo 9442*	Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)
⊢ 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
2.4.37  Hall's marriage theorem
Theorem	marypha1lem 9443*	Core induction for Philip Hall's marriage theorem. (Contributed by Stefan O'Rear, 19-Feb-2015.)
⊢ (𝐴 ∈ Fin → (𝑏 ∈ Fin → ∀𝑐 ∈ 𝒫 (𝐴 × 𝑏)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑒 ∈ 𝒫 𝑐𝑒:𝐴–1-1→V)))
Theorem	marypha1 9444*	(Philip) Hall's marriage theorem, sufficiency: a finite relation contains an injection if there is no subset of its domain which would be forced to violate the pigeonhole principle. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ⊆ (𝐴 × 𝐵))    &   ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑))    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→𝐵)
Theorem	marypha2lem1 9445*	Lemma for marypha2 9449. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))    ⇒   ⊢ 𝑇 ⊆ (𝐴 × ∪ ran 𝐹)
Theorem	marypha2lem2 9446*	Lemma for marypha2 9449. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))    ⇒   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
Theorem	marypha2lem3 9447*	Lemma for marypha2 9449. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐺 ⊆ 𝑇 ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
Theorem	marypha2lem4 9448*	Lemma for marypha2 9449. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ (𝐹 “ 𝑋))
Theorem	marypha2 9449*	Version of marypha1 9444 using a functional family of sets instead of a relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶Fin)    &   ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ ∪ (𝐹 “ 𝑑))    ⇒   ⊢ (𝜑 → ∃𝑔(𝑔:𝐴–1-1→V ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
2.4.38  Supremum and infimum
Syntax	csup 9450	Extend class notation to include supremum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class sup(𝐴, 𝐵, 𝑅)
Syntax	cinf 9451	Extend class notation to include infimum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class inf(𝐴, 𝐵, 𝑅)
Definition	df-sup 9452*	Define the supremum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the supremum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals whose square is less than 2; in this case the supremum is defined as the square root of 2 per sqrtval 15254. See dfsup2 9454 for alternate definition not requiring dummy variables. (Contributed by NM, 22-May-1999.)
⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
Definition	df-inf 9453	Define the infimum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the infimum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)
Theorem	dfsup2 9454	Quantifier-free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ sup(𝐵, 𝐴, 𝑅) = ∪ (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
Theorem	supeq1 9455	Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
Theorem	supeq1d 9456	Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
Theorem	supeq1i 9457	Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
Theorem	supeq2 9458	Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
Theorem	supeq3 9459	Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))
Theorem	supeq123d 9460	Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ (𝜑 → 𝐴 = 𝐷)    &   ⊢ (𝜑 → 𝐵 = 𝐸)    &   ⊢ (𝜑 → 𝐶 = 𝐹)    ⇒   ⊢ (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
Theorem	nfsup 9461	Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝑅    ⇒   ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅)
Theorem	supmo 9462*	Any class 𝐵 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by NM, 5-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
Theorem	supexd 9463	A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V)
Theorem	supeu 9464*	A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
Theorem	supval2 9465*	Alternate expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Thierry Arnoux, 24-Sep-2017.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
Theorem	eqsup 9466*	Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶))
Theorem	eqsupd 9467*	Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	supcl 9468*	A supremum belongs to its base class (closure law). See also supub 9469 and suplub 9470. (Contributed by NM, 12-Oct-2004.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
Theorem	supub 9469*	A supremum is an upper bound. See also supcl 9468 and suplub 9470. This proof demonstrates how to expand an iota-based definition (df-iota 6483) using riotacl2 7376. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
Theorem	suplub 9470*	A supremum is the least upper bound. See also supcl 9468 and supub 9469. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
Theorem	suplub2 9471*	Bidirectional form of suplub 9470. (Contributed by Mario Carneiro, 6-Sep-2014.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
Theorem	supnub 9472*	An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝐶𝑅𝑧) → ¬ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
Theorem	supssd 9473*	Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅))
Theorem	supex 9474	A supremum is a set. (Contributed by NM, 22-May-1999.)
⊢ 𝑅 Or 𝐴    ⇒   ⊢ sup(𝐵, 𝐴, 𝑅) ∈ V
Theorem	sup00 9475	The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
⊢ sup(𝐵, ∅, 𝑅) = ∅
Theorem	sup0riota 9476*	The supremum of an empty set is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)
⊢ (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥))
Theorem	sup0 9477*	The supremum of an empty set under a base set which has a unique smallest element is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)
⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋)
Theorem	supmax 9478*	The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof shortened by OpenAI, 30-Mar-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	fisup2g 9479*	A finite set satisfies the conditions to have a supremum. (Contributed by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
Theorem	fisupcl 9480	A nonempty finite set contains its supremum. (Contributed by Jeff Madsen, 9-May-2011.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
Theorem	supgtoreq 9481	The supremum of a finite set is greater than or equal to all the elements of the set. (Contributed by AV, 1-Oct-2019.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, 𝑅))    ⇒   ⊢ (𝜑 → (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))
Theorem	suppr 9482	The supremum of a pair. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → sup({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐶𝑅𝐵, 𝐵, 𝐶))
Theorem	supsn 9483	The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵)
Theorem	supisolem 9484*	Lemma for supiso 9486. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐴) → ((∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐷 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
Theorem	supisoex 9485*	Lemma for supiso 9486. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
Theorem	supiso 9486*	Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))    &   ⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
Theorem	infeq1 9487	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
Theorem	infeq1d 9488	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
Theorem	infeq1i 9489	Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)
Theorem	infeq2 9490	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))
Theorem	infeq3 9491	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))
Theorem	infeq123d 9492	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝐴 = 𝐷)    &   ⊢ (𝜑 → 𝐵 = 𝐸)    &   ⊢ (𝜑 → 𝐶 = 𝐹)    ⇒   ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))
Theorem	nfinf 9493	Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝑅    ⇒   ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅)
Theorem	infexd 9494	An infimum is a set. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V)
Theorem	eqinf 9495*	Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))
Theorem	eqinfd 9496*	Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	infval 9497*	Alternate expression for the infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))
Theorem	infcllem 9498*	Lemma for infcl 9499, inflb 9500, infglb 9501, etc. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
Theorem	infcl 9499*	An infimum belongs to its base class (closure law). See also inflb 9500 and infglb 9501. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)
Theorem	inflb 9500*	An infimum is a lower bound. See also infcl 9499 and infglb 9501. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))