P. 452 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45101-45200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rexanuz3 45101*	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 45102*	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 45103*	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 45104*	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 45105	Subclass of a class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
Theorem	pssnssi 45106	A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐴 ⊊ 𝐵    ⇒   ⊢ ¬ 𝐵 ⊆ 𝐴
Theorem	rabidim2 45107	Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑)
Theorem	eluni2f 45108*	Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)
Theorem	eliin2f 45109*	Membership in indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
Theorem	nssd 45110	Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐴 ⊆ 𝐵)
Theorem	iineq12dv 45111*	Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)
Theorem	supxrcld 45112	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 45113	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 45114*	Counterexample to show that the additional conditions in eliuniin 45104 and eliuniin2 45125 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 45115*	Counterexample to show that the additional conditions in eliin 4996 and eliin2 45121 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐴 = V    &   ⊢ 𝐵 = ∅    ⇒   ⊢ ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
Theorem	eliinid 45116*	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 45117	Class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴))
Theorem	supxrubd 45118	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 45119	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	ssrabdf 45120	Subclass of a restricted class abstraction (deduction form). (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})
Theorem	eliin2 45121*	Membership in indexed intersection. See eliincex 45115 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4996 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 45109). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
Theorem	ssrab2f 45122	Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
Theorem	restuni3 45123	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 45124	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 45125*	Indexed union of indexed intersections. See eliincex 45115 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 45126	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 45127	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 45128	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 45129	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 45130*	Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐴 ≠ ∅ → (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵))
Theorem	iniin2 45131*	Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐴 ≠ ∅ → (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶))
Theorem	cbvrabv2 45132*	A more general version of cbvrabv 3447. Usage of this theorem is discouraged because it depends on ax-13 2377. Use of cbvrabv2w 45133 is preferred. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
Theorem	cbvrabv2w 45133*	A more general version of cbvrabv 3447. Version of cbvrabv2 45132 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Revised by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
Theorem	iinssiin 45134	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 45135*	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 45136*	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	rabbida2 45137	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 45138*	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 45139	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 45140	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	r19.36vf 45141	Restricted quantifier version of one direction of 19.36 2230. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓))
Theorem	raleqd 45142	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	iinssf 45143	Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
Theorem	iinssdf 45144	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)
Theorem	resabs2i 45145	Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵)
Theorem	ssdf2 45146	A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	rabssd 45147	Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) → 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ 𝐵)
Theorem	rexnegd 45148	Minus a real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → -𝑒𝐴 = -𝐴)
Theorem	rexlimd3 45149	* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
Theorem	nel1nelini 45150	Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ ¬ 𝐴 ∈ 𝐵    ⇒   ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)
Theorem	nel2nelini 45151	Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ ¬ 𝐴 ∈ 𝐶    ⇒   ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)
Theorem	eliunid 45152*	Membership in indexed union. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	reximdd 45153	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)
Theorem	inopnd 45154	The intersection of two open sets of a topology is an open set. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐽)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐽)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝐽)
Theorem	ss2rabdf 45155	Deduction of restricted abstraction subclass from implication. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	restopn3 45156	If 𝐴 is open, then 𝐴 is open in the restriction to itself. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ∈ (𝐽 ↾t 𝐴))
Theorem	restopnssd 45157	A topology restricted to an open set is a subset of the original topology. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐽)    ⇒   ⊢ (𝜑 → (𝐽 ↾t 𝐴) ⊆ 𝐽)
Theorem	restsubel 45158	A subset belongs in the space it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ (𝜑 → 𝐽 ∈ 𝑉)    &   ⊢ (𝜑 → ∪ 𝐽 ∈ 𝐽)    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴))
Theorem	toprestsubel 45159	A subset is open in the topology it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴))
Theorem	rabidd 45160	An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	iunssdf 45161	Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
Theorem	iinss2d 45162	Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
Theorem	r19.3rzf 45163	Restricted quantification of wff not containing quantified variable. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
Theorem	r19.28zf 45164	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
Theorem	iindif2f 45165	Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws". (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶))
Theorem	ralfal 45166	Two ways of expressing empty set. (Contributed by Glauco Siliprandi, 24-Jan-2024.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 ⊥)
Theorem	archd 45167*	Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by Glauco Siliprandi, 1-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
Theorem	nimnbi 45168	If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ ¬ (𝜑 → 𝜓)    ⇒   ⊢ ¬ (𝜑 ↔ 𝜓)
Theorem	nimnbi2 45169	If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ ¬ (𝜓 → 𝜑)    ⇒   ⊢ ¬ (𝜑 ↔ 𝜓)
Theorem	notbicom 45170	Commutative law for the negation of a biconditional. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ ¬ (𝜑 ↔ 𝜓)    ⇒   ⊢ ¬ (𝜓 ↔ 𝜑)
Theorem	rexeqif 45171	Equality inference for restricted existential quantifier. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)
Theorem	rspced 45172	Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
21.43.2  Functions
Theorem	fnresdmss 45173	A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐹 ↾ 𝐵) = 𝐹)
Theorem	fmptsnxp 45174*	Maps-to notation and Cartesian product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) = ({𝐴} × {𝐵}))
Theorem	fvmpt2bd 45175*	Value of a function given by the maps-to notation. Deduction version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝑥) = 𝐵)
Theorem	rnmptfi 45176*	The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (𝐵 ∈ Fin → ran 𝐴 ∈ Fin)
Theorem	fresin2 45177	Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ 𝐶))
Theorem	ffi 45178	A function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)
Theorem	suprnmpt 45179*	An explicit bound for the range of a bounded function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ 𝐶 = sup(ran 𝐹, ℝ, < )    ⇒   ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))
Theorem	rnffi 45180	The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin) → ran 𝐹 ∈ Fin)
Theorem	mptelpm 45181*	A function in maps-to notation is a partial map . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐶 ↑pm 𝐷))
Theorem	rnmptpr 45182*	Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)    &   ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)    ⇒   ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸})
Theorem	resmpti 45183*	Restriction of the mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝐵 ⊆ 𝐴    ⇒   ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)
Theorem	founiiun 45184*	Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
Theorem	rnresun 45185	Distribution law for range of a restriction over a union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵))
Theorem	elrnmptf 45186	The range of a function in maps-to notation. Same as elrnmpt 5969, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝐶    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
Theorem	rnmptssrn 45187*	Inclusion relation for two ranges expressed in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐶 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑦 ∈ 𝐶 ↦ 𝐷))
Theorem	disjf1 45188*	A 1 to 1 mapping built from disjoint, nonempty sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → 𝐹:𝐴–1-1→𝑉)
Theorem	rnsnf 45189	The range of a function whose domain is a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵)    ⇒   ⊢ (𝜑 → ran 𝐹 = {(𝐹‘𝐴)})
Theorem	wessf1ornlem 45190*	Given a function 𝐹 on a well-ordered domain 𝐴 there exists a subset of 𝐴 such that 𝐹 restricted to such subset is injective and onto the range of 𝐹 (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ 𝐺 = (𝑦 ∈ ran 𝐹 ↦ (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹)
Theorem	wessf1orn 45191*	Given a function 𝐹 on a well-ordered domain 𝐴 there exists a subset of 𝐴 such that 𝐹 restricted to such subset is injective and onto the range of 𝐹 (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹)
Theorem	nelrnres 45192	If 𝐴 is not in the range, it is not in the range of any restriction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (¬ 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶))
Theorem	disjrnmpt2 45193*	Disjointness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran 𝐹 𝑦)
Theorem	elrnmpt1sf 45194*	Elementhood in an image set. Same as elrnmpt1s 5970, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝐶    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)
Theorem	founiiun0 45195*	Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
Theorem	disjf1o 45196*	A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅}    &   ⊢ 𝐷 = (ran 𝐹 ∖ {∅})    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶–1-1-onto→𝐷)
Theorem	disjinfi 45197*	Only a finite number of disjoint sets can have a nonempty intersection with a finite set 𝐶. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ Fin)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐵 ∩ 𝐶) ≠ ∅} ∈ Fin)
Theorem	fvovco 45198	Value of the composition of an operator, with a given function. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐹:𝑋⟶(𝑉 × 𝑊))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = ((1st ‘(𝐹‘𝑌))𝑂(2nd ‘(𝐹‘𝑌))))
Theorem	ssnnf1octb 45199*	There exists a bijection between a subset of ℕ and a given nonempty countable set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴))
Theorem	nnf1oxpnn 45200	There is a bijection between the set of positive integers and the Cartesian product of the set of positive integers with itself. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ ∃𝑓 𝑓:ℕ–1-1-onto→(ℕ × ℕ)