P. 414 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 41301-41400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iuneq2df 41301	Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
Theorem	nnfoctb 41302*	There exists a mapping from ℕ onto any (nonempty) countable set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto→𝐴)
Theorem	ssinss1d 41303	Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
Theorem	elpwinss 41304	An element of the powerset of 𝐵 intersected with anything, is a subset of 𝐵. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐴 ∈ (𝒫 𝐵 ∩ 𝐶) → 𝐴 ⊆ 𝐵)
Theorem	unidmex 41305	If 𝐹 is a set, then ∪ dom 𝐹 is a set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ 𝑋 = ∪ dom 𝐹    ⇒   ⊢ (𝜑 → 𝑋 ∈ V)
Theorem	ndisj2 41306*	A non-disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
Theorem	zenom 41307	The set of integer numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ℤ ≈ ω
Theorem	uzwo4 41308*	Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑗𝜓    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))
Theorem	unisn0 41309	The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ∪ {∅} = ∅
Theorem	ssin0 41310	If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐶 ∩ 𝐷) = ∅)
Theorem	inabs3 41311	Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ 𝐶))
Theorem	pwpwuni 41312	Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))
Theorem	disjiun2 41313*	In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶))    &   ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐸)    ⇒   ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸) = ∅)
Theorem	0pwfi 41314	The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin)
Theorem	ssinss2d 41315	Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
Theorem	zct 41316	The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ℤ ≼ ω
Theorem	pwfin0 41317	A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅
Theorem	uzct 41318	An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝑍 = (ℤ≥‘𝑁)    ⇒   ⊢ 𝑍 ≼ ω
Theorem	iunxsnf 41319*	A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝐶    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶
Theorem	fiiuncl 41320*	If a set is closed under the union of two sets, then it is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (𝑦 ∪ 𝑧) ∈ 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐷)
Theorem	iunp1 41321*	The addition of the next set to a union indexed by a finite set of sequential integers. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝐵    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ∪ 𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (∪ 𝑘 ∈ (𝑀...𝑁)𝐴 ∪ 𝐵))
Theorem	fiunicl 41322*	If a set is closed under the union of two sets, then it is closed under finite union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∪ 𝑦) ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴)
Theorem	ixpeq2d 41323	Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)
Theorem	disjxp1 41324*	The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 (𝐵 × 𝐶))
Theorem	disjsnxp 41325*	The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
Theorem	eliind 41326*	Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐵)    &   ⊢ (𝑥 = 𝐾 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐷)
Theorem	rspcef 41327	Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)
Theorem	inn0f 41328	A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	ixpssmapc 41329*	An infinite Cartesian product is a subset of set exponentiation. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐶 ↑m 𝐴))
Theorem	inn0 41330*	A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	elintd 41331*	Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)
Theorem	ssdf 41332*	A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	brneqtrd 41333	Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → ¬ 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐴𝑅𝐶)
Theorem	ssnct 41334	A set containing an uncountable set is itself uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → ¬ 𝐴 ≼ ω)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐵 ≼ ω)
Theorem	ssuniint 41335*	Sufficient condition for being a subclass of the union of an intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ ∪ ∩ 𝐵)
Theorem	elintdv 41336*	Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)
Theorem	ssd 41337*	A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	ralimralim 41338	Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑))
Theorem	snelmap 41339	Membership of the element in the range of a constant map. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → (𝐴 × {𝑥}) ∈ (𝐵 ↑m 𝐴))    ⇒   ⊢ (𝜑 → 𝑥 ∈ 𝐵)
Theorem	xrnmnfpnf 41340	An extended real that is neither real nor minus infinity, is plus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ -∞)    ⇒   ⊢ (𝜑 → 𝐴 = +∞)
Theorem	nelrnmpt 41341*	Non-membership in the range of a function in maps-to notaion. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ ran 𝐹)
Theorem	snn0d 41342	The singleton of a set is not empty. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → {𝐴} ≠ ∅)
Theorem	iuneq1i 41343*	Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶
Theorem	nssrex 41344*	Negation of subclass relationship. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
Theorem	iunssf 41345	Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
Theorem	ssinc 41346*	Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀..^𝑁)) → (𝐹‘𝑚) ⊆ (𝐹‘(𝑚 + 1)))    ⇒   ⊢ (𝜑 → (𝐹‘𝑀) ⊆ (𝐹‘𝑁))
Theorem	ssdec 41347*	Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑚 + 1)) ⊆ (𝐹‘𝑚))    ⇒   ⊢ (𝜑 → (𝐹‘𝑁) ⊆ (𝐹‘𝑀))
Theorem	elixpconstg 41348*	Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ 𝐹:𝐴⟶𝐵))
Theorem	iineq1d 41349*	Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)
Theorem	metpsmet 41350	A metric is a pseudometric. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))
Theorem	ixpssixp 41351	Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)
Theorem	ballss3 41352*	A sufficient condition for a ball being a subset. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑋))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)
Theorem	iunincfi 41353*	Given a sequence of increasing sets, the union of a finite subsequence, is its last element. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))    ⇒   ⊢ (𝜑 → ∪ 𝑛 ∈ (𝑀...𝑁)(𝐹‘𝑛) = (𝐹‘𝑁))
Theorem	nsstr 41354	If it's not a subclass, it's not a subclass of a smaller one. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ ((¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐶)
Theorem	rexanuz3 41355*	Combine two different upper integer properties into one, for a single integer. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑗𝜑    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜒)    &   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)    &   ⊢ (𝑘 = 𝑗 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑘 = 𝑗 → (𝜓 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝜃 ∧ 𝜏))
Theorem	cbvmpo2 41356*	Rule to change the second bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑤𝐴    &   ⊢ Ⅎ𝑤𝐶    &   ⊢ Ⅎ𝑦𝐸    &   ⊢ (𝑦 = 𝑤 → 𝐶 = 𝐸)    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸)
Theorem	cbvmpo1 41357*	Rule to change the first bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑧𝐵    &   ⊢ Ⅎ𝑧𝐶    &   ⊢ Ⅎ𝑥𝐸    &   ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸)
Theorem	eliuniin 41358*	Indexed union of indexed intersections. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐴 = ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷    ⇒   ⊢ (𝑍 ∈ 𝑉 → (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
Theorem	ssabf 41359	Subclass of a class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
Theorem	pssnssi 41360	A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐴 ⊊ 𝐵    ⇒   ⊢ ¬ 𝐵 ⊆ 𝐴
Theorem	rabidim2 41361	Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑)
Theorem	eluni2f 41362*	Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)
Theorem	eliin2f 41363*	Membership in indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
Theorem	nssd 41364	Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐴 ⊆ 𝐵)
Theorem	iineq12dv 41365*	Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)
Theorem	supxrcld 41366	The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ*)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) ∈ ℝ*)
Theorem	elrestd 41367	A sufficient condition for being an open set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐽 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐽)    &   ⊢ 𝐴 = (𝑋 ∩ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐵))
Theorem	eliuniincex 41368*	Counterexample to show that the additional conditions in eliuniin 41358 and eliuniin2 41379 are actually needed. Notice that the definition of 𝐴 is not even needed (it can be any class). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐵 = {∅}    &   ⊢ 𝐶 = ∅    &   ⊢ 𝐷 = ∅    &   ⊢ 𝑍 = V    ⇒   ⊢ ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)
Theorem	eliincex 41369*	Counterexample to show that the additional conditions in eliin 4917 and eliin2 41375 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐴 = V    &   ⊢ 𝐵 = ∅    ⇒   ⊢ ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
Theorem	eliinid 41370*	Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶)
Theorem	abssf 41371	Class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴))
Theorem	fexd 41372	If the domain of a mapping is a set, the function is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐹 ∈ V)
Theorem	supxrubd 41373	A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ 𝑆 = sup(𝐴, ℝ*, < )    ⇒   ⊢ (𝜑 → 𝐵 ≤ 𝑆)
Theorem	ssrabf 41374	Subclass of a restricted class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑))
Theorem	eliin2 41375*	Membership in indexed intersection. See eliincex 41369 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4917 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 41363). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
Theorem	ssrab2f 41376	Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
Theorem	restuni3 41377	The underlying set of a subspace induced by the subspace operator ↾t. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))
Theorem	rabssf 41378	Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵))
Theorem	eliuniin2 41379*	Indexed union of indexed intersections. See eliincex 41369 for a counterexample showing that the precondition 𝐶 ≠ ∅ cannot be simply dropped. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐶    &   ⊢ 𝐴 = ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷    ⇒   ⊢ (𝐶 ≠ ∅ → (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
Theorem	restuni4 41380	The underlying set of a subspace induced by the ↾t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴)    ⇒   ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵)
Theorem	restuni6 41381	The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))
Theorem	restuni5 41382	The underlying set of a subspace induced by the ↾t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
Theorem	unirestss 41383	The union of an elementwise intersection is a subset of the underlying set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) ⊆ ∪ 𝐴)
Theorem	iniin1 41384*	Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐴 ≠ ∅ → (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵))
Theorem	iniin2 41385*	Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐴 ≠ ∅ → (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶))
Theorem	cbvrabv2 41386*	A more general version of cbvrabv 3492. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
Theorem	iinssiin 41387	Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶)
Theorem	eliind2 41388*	Membership in indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
Theorem	iinssd 41389*	Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
Theorem	ralrimia 41390	Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	rabbida2 41391	Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	iinexd 41392*	The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)
Theorem	rabexf 41393	Separation Scheme in terms of a restricted class abstraction. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐴    &   ⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V
Theorem	rabbida3 41394	Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	resexd 41395	The restriction of a set is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ V)
Theorem	r19.36vf 41396	Restricted quantifier version of one direction of 19.36 2227. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓))
Theorem	raleqd 41397	Equality deduction for restricted universal quantifier. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
Theorem	ralimda 41398	Deduction quantifying both antecedent and consequent. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
Theorem	iinssf 41399	Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
Theorem	iinssdf 41400	Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝑋    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)