P. 74 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 7301-7400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	weisoeq 7301	Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 7917. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)
Theorem	weisoeq2 7302	Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 7918. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)
Theorem	knatar 7303*	The Knaster-Tarski theorem says that every monotone function over a complete lattice has a (least) fixpoint. Here we specialize this theorem to the case when the lattice is the powerset lattice 𝒫 𝐴. (Contributed by Mario Carneiro, 11-Jun-2015.)
⊢ 𝑋 = ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝑋 ⊆ 𝐴 ∧ (𝐹‘𝑋) = 𝑋))
Theorem	fvresval 7304	The value of a restricted function at a class is either the empty set or the value of the unrestricted function at that class. (Contributed by Scott Fenton, 4-Sep-2011.)
⊢ (((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴) ∨ ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
Theorem	funeldmb 7305	If ∅ is not part of the range of a function 𝐹, then 𝐴 is in the domain of 𝐹 iff (𝐹‘𝐴) ≠ ∅. (Contributed by Scott Fenton, 7-Dec-2021.)
⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹‘𝐴) ≠ ∅))
Theorem	eqfunresadj 7306	Law for adjoining an element to restrictions of functions. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌})))
Theorem	eqfunressuc 7307	Law for equality of restriction to successors. This is primarily useful when 𝑋 is an ordinal, but it does not require that. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ (𝐹‘𝑋) = (𝐺‘𝑋))) → (𝐹 ↾ suc 𝑋) = (𝐺 ↾ suc 𝑋))
Theorem	fnssintima 7308*	Condition for subset of an intersection of an image. (Contributed by Scott Fenton, 16-Aug-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥)))
Theorem	imaeqsexvOLD 7309*	Obsolete version of rexima 7184 as of 14-Aug-2025. Duplicate version of rexima 7184. (Contributed by Scott Fenton, 27-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓))
Theorem	imaeqsalvOLD 7310*	Obsolete version of ralima 7183 as of 14-Aug-2025. Duplicate version of ralima 7183. (Contributed by Scott Fenton, 27-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))
Theorem	fnimasnd 7311	The image of a function by a singleton whose element is in the domain of the function. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)})
2.3.17  Cantor's Theorem
Theorem	canth 7312	No set 𝐴 is equinumerous to its power set (Cantor's theorem), i.e., no function can map 𝐴 onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 9058. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 7313 for a counterexample. (Use nex 1801 if you want the form ¬ ∃𝑓𝑓:𝐴–onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ¬ 𝐹:𝐴–onto→𝒫 𝐴
Theorem	ncanth 7313	Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5260). Specifically, the identity function maps the universe onto its power class. Compare canth 7312 that works for sets. This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3738): 𝒫 V, being a class, cannot contain proper classes, so it is no larger than V, which is why the identity function "succeeds" in being surjective onto 𝒫 V (see pwv 4860). See also the remark in ru 3738 about NF, in which Cantor's theorem fails for sets that are "too large". This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set. (Contributed by NM, 29-Jun-2004.) (Proof shortened by BJ, 29-Dec-2023.)
⊢ I :V–onto→𝒫 V
2.3.18  Restricted iota (description binder)
Syntax	crio 7314	Extend class notation with restricted description binder.
class (℩𝑥 ∈ 𝐴 𝜑)
Definition	df-riota 7315	Define restricted description binder. In case there is no unique 𝑥 such that (𝑥 ∈ 𝐴 ∧ 𝜑) holds, it evaluates to the empty set. See also comments for df-iota 6448. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)
⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	riotaeqdv 7316*	Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜓))
Theorem	riotabidv 7317*	Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))
Theorem	riotaeqbidv 7318*	Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒))
Theorem	riotaex 7319	Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)
⊢ (℩𝑥 ∈ 𝐴 𝜓) ∈ V
Theorem	riotav 7320	An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥𝜑)
Theorem	riotauni 7321	Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑})
Theorem	nfriota1 7322*	The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑)
Theorem	nfriotadw 7323*	Deduction version of nfriota 7327 with a disjoint variable condition, which contrary to nfriotad 7326 does not require ax-13 2376. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2376. (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))
Theorem	cbvriotaw 7324*	Change bound variable in a restricted description binder. Version of cbvriota 7328 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by NM, 18-Mar-2013.) Avoid ax-13 2376. (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
Theorem	cbvriotavw 7325*	Change bound variable in a restricted description binder. Version of cbvriotav 7329 with a disjoint variable condition, which requires fewer axioms . (Contributed by NM, 18-Mar-2013.) (Revised by GG, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
Theorem	nfriotad 7326	Deduction version of nfriota 7327. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker nfriotadw 7323 when possible. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))
Theorem	nfriota 7327*	A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑)
Theorem	cbvriota 7328*	Change bound variable in a restricted description binder. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker cbvriotaw 7324 when possible. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
Theorem	cbvriotav 7329*	Change bound variable in a restricted description binder. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker cbvriotavw 7325 when possible. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
Theorem	csbriota 7330*	Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 2-Sep-2018.)
⊢ ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)
Theorem	riotacl2 7331	Membership law for "the unique element in 𝐴 such that 𝜑". (Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
Theorem	riotacl 7332*	Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
Theorem	riotasbc 7333	Substitution law for descriptions. Compare iotasbc 44656. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
Theorem	riotabidva 7334*	Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3405 analog.) (Contributed by NM, 17-Jan-2012.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))
Theorem	riotabiia 7335	Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3403 analog.) (Contributed by NM, 16-Jan-2012.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓)
Theorem	riota1 7336*	Property of restricted iota. Compare iota1 6471. (Contributed by Mario Carneiro, 15-Oct-2016.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝑥))
Theorem	riota1a 7337	Property of iota. (Contributed by NM, 23-Aug-2011.)
⊢ ((𝑥 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥))
Theorem	riota2df 7338*	A deduction version of riota2f 7339. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝐵))
Theorem	riota2f 7339*	This theorem shows a condition that allows to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))
Theorem	riota2 7340*	This theorem shows a condition that allows to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.)
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))
Theorem	riotaeqimp 7341*	If two restricted iota descriptors for an equality are equal, then the terms of the equality are equal. (Contributed by AV, 6-Dec-2020.)
⊢ 𝐼 = (℩𝑎 ∈ 𝑉 𝑋 = 𝐴)    &   ⊢ 𝐽 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)    &   ⊢ (𝜑 → ∃!𝑎 ∈ 𝑉 𝑋 = 𝐴)    &   ⊢ (𝜑 → ∃!𝑎 ∈ 𝑉 𝑌 = 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → 𝑋 = 𝑌)
Theorem	riotaprop 7342*	Properties of a restricted definite description operator. (Contributed by NM, 23-Nov-2013.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐵 = (℩𝑥 ∈ 𝐴 𝜑)    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜓))
Theorem	riota5f 7343*	A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)
Theorem	riota5 7344*	A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)
⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)
Theorem	riotass2 7345*	Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓))
Theorem	riotass 7346*	Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑))
Theorem	moriotass 7347*	Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑))
Theorem	snriota 7348	A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {(℩𝑥 ∈ 𝐴 𝜑)})
Theorem	riotaxfrd 7349*	Change the variable 𝑥 in the expression for "the unique 𝑥 such that 𝜓 " to another variable 𝑦 contained in expression 𝐵. Use reuhypd 5364 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑦𝐶    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴) → 𝐶 ∈ 𝐴)    &   ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = (℩𝑦 ∈ 𝐴 𝜒) → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐴 𝑥 = 𝐵)    ⇒   ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐶)
Theorem	eusvobj2 7350*	Specify the same property in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
Theorem	eusvobj1 7351*	Specify the same object in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (℩𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) = (℩𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
Theorem	f1ofveu 7352*	There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶)
Theorem	f1ocnvfv3 7353*	Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) = (℩𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶))
Theorem	riotaund 7354*	Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.)
⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅)
Theorem	riotassuni 7355*	The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (℩𝑥 ∈ 𝐴 𝜑) ⊆ (𝒫 ∪ 𝐴 ∪ ∪ 𝐴)
Theorem	riotaclb 7356*	Bidirectional closure of restricted iota when domain is not empty. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) (Revised by NM, 13-Sep-2018.)
⊢ (¬ ∅ ∈ 𝐴 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))
Theorem	riotarab 7357*	Restricted iota of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒) = (℩𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒))
2.3.19  Operations
Syntax	co 7358	Extend class notation to include the value of an operation 𝐹 (such as +) for two arguments 𝐴 and 𝐵. Note that the syntax is simply three class symbols in a row surrounded by parentheses. Since operation values are the only possible class expressions consisting of three class expressions in a row surrounded by parentheses, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 11365.)
class (𝐴𝐹𝐵)
Syntax	coprab 7359	Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Syntax	cmpo 7360	Extend the definition of a class to include maps-to notation for defining an operation via a rule.
class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
Definition	df-ov 7361	Define the value of an operation. Definition of operation value in [Enderton] p. 79. Note that the syntax is simply three class expressions in a row bracketed by parentheses. There are no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation 𝐹 and its arguments 𝐴 and 𝐵- will be useful for proving meaningful theorems. For example, if class 𝐹 is the operation + and arguments 𝐴 and 𝐵 are 3 and 2, the expression (3 + 2) can be proved to equal 5 (see 3p2e5 12291). This definition is well-defined, although not very meaningful, when classes 𝐴 and/or 𝐵 are proper classes (i.e. are not sets); see ovprc1 7397 and ovprc2 7398. On the other hand, we often find uses for this definition when 𝐹 is a proper class, such as +o in oav 8438. 𝐹 is normally equal to a class of nested ordered pairs of the form defined by df-oprab 7362. (Contributed by NM, 28-Feb-1995.)
⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
Definition	df-oprab 7362*	Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally 𝑥, 𝑦, and 𝑧 are distinct, although the definition doesn't strictly require it. See df-ov 7361 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of an operation given by a class abstraction is given by ovmpo 7518. (Contributed by NM, 12-Mar-1995.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
Definition	df-mpo 7363*	Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from 𝑥, 𝑦 (in 𝐴 × 𝐵) to 𝐶(𝑥, 𝑦)". An extension of df-mpt 5180 for two arguments. (Contributed by NM, 17-Feb-2008.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
Theorem	oveq 7364	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
Theorem	oveq1 7365	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ (𝐴 = 𝐵 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
Theorem	oveq2 7366	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ (𝐴 = 𝐵 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
Theorem	oveq12 7367	Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
Theorem	oveq1i 7368	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴𝐹𝐶) = (𝐵𝐹𝐶)
Theorem	oveq2i 7369	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶𝐹𝐴) = (𝐶𝐹𝐵)
Theorem	oveq12i 7370	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴𝐹𝐶) = (𝐵𝐹𝐷)
Theorem	oveqi 7371	Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶𝐴𝐷) = (𝐶𝐵𝐷)
Theorem	oveq123i 7372	Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    &   ⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐴𝐹𝐵) = (𝐶𝐺𝐷)
Theorem	oveq1d 7373	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
Theorem	oveq2d 7374	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
Theorem	oveqd 7375	Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷))
Theorem	oveq12d 7376	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
Theorem	oveqan12d 7377	Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
Theorem	oveqan12rd 7378	Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
Theorem	oveq123d 7379	Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷))
Theorem	fvoveq1d 7380	Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶)))
Theorem	fvoveq1 7381	Equality theorem for nested function and operation value. Closed form of fvoveq1d 7380. (Contributed by AV, 23-Jul-2022.)
⊢ (𝐴 = 𝐵 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶)))
Theorem	ovanraleqv 7382*	Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
⊢ (𝐵 = 𝑋 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐵 = 𝑋 → (∀𝑥 ∈ 𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥 ∈ 𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
Theorem	imbrov2fvoveq 7383	Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.)
⊢ (𝑋 = 𝑌 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴)))
Theorem	ovrspc2v 7384*	If an operation value is an element of a class for all operands of two classes, then the operation value is an element of the class for specific operands of the two classes. (Contributed by Mario Carneiro, 6-Dec-2014.)
⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶) → (𝑋𝐹𝑌) ∈ 𝐶)
Theorem	oveqrspc2v 7385*	Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌))
Theorem	oveqdr 7386	Equality of two operations for any two operands. Useful in proofs using *propd theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)
⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
Theorem	nfovd 7387	Deduction version of bound-variable hypothesis builder nfov 7388. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐹)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝐴𝐹𝐵))
Theorem	nfov 7388	Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴𝐹𝐵)
Theorem	oprabidw 7389*	The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of oprabid 7390 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by Mario Carneiro, 20-Mar-2013.) Avoid ax-13 2376. (Revised by GG, 26-Jan-2024.)
⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
Theorem	oprabid 7390	The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker oprabidw 7389 when possible. (Contributed by Mario Carneiro, 20-Mar-2013.) (New usage is discouraged.)
⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
Theorem	ovex 7391	The result of an operation is a set. (Contributed by NM, 13-Mar-1995.)
⊢ (𝐴𝐹𝐵) ∈ V
Theorem	ovexi 7392	The result of an operation is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐴 = (𝐵𝐹𝐶)    ⇒   ⊢ 𝐴 ∈ V
Theorem	ovexd 7393	The result of an operation is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → (𝐴𝐹𝐵) ∈ V)
Theorem	ovssunirn 7394	The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹
Theorem	0ov 7395	Operation value of the empty set. (Contributed by AV, 15-May-2021.)
⊢ (𝐴∅𝐵) = ∅
Theorem	ovprc 7396	The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ Rel dom 𝐹    ⇒   ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
Theorem	ovprc1 7397	The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.)
⊢ Rel dom 𝐹    ⇒   ⊢ (¬ 𝐴 ∈ V → (𝐴𝐹𝐵) = ∅)
Theorem	ovprc2 7398	The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ Rel dom 𝐹    ⇒   ⊢ (¬ 𝐵 ∈ V → (𝐴𝐹𝐵) = ∅)
Theorem	ovrcl 7399	Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.)
⊢ Rel dom 𝐹    ⇒   ⊢ (𝐶 ∈ (𝐴𝐹𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	elfvov1 7400	Utility theorem: reverse closure for any operation that results in a function. (Contributed by SN, 18-May-2025.)
⊢ Rel dom 𝑂    &   ⊢ 𝑆 = (𝐼𝑂𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑆‘𝑌))    ⇒   ⊢ (𝜑 → 𝐼 ∈ V)