P. 480 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47901-48000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	adh-minim-ax1-ax2-lem4 47901	Fourth lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 47897 and ax-mp 5. Polish prefix notation: CCCpqrCCqCrsCqs . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜓 → (𝜒 → 𝜃)) → (𝜓 → 𝜃)))
Theorem	adh-minim-ax1 47902	Derivation of ax-1 6 from adh-minim 47897 and ax-mp 5. Carew Arthur Meredith derived ax-1 6 in A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, here we follow the shortened derivation by Ivo Thomas, On Meredith's sole positive axiom, Notre Dame Journal of Formal Logic, volume XV, number 3, July 1974, page 477. Polish prefix notation: CpCqp . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	adh-minim-ax2-lem5 47903	Fifth lemma for the derivation of ax-2 7 from adh-minim 47897 and ax-mp 5. Polish prefix notation: CpCCCqrsCCrCstCrt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜃) → ((𝜒 → (𝜃 → 𝜏)) → (𝜒 → 𝜏))))
Theorem	adh-minim-ax2-lem6 47904	Sixth lemma for the derivation of ax-2 7 from adh-minim 47897 and ax-mp 5. Polish prefix notation: CCpCCCCqrsCCrCstCrtuCpu . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → ((((𝜓 → 𝜒) → 𝜃) → ((𝜒 → (𝜃 → 𝜏)) → (𝜒 → 𝜏))) → 𝜂)) → (𝜑 → 𝜂))
Theorem	adh-minim-ax2c 47905	Derivation of a commuted form of ax-2 7 from adh-minim 47897 and ax-mp 5. Polish prefix notation: CCpqCCpCqrCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
Theorem	adh-minim-ax2 47906	Derivation of ax-2 7 from adh-minim 47897 and ax-mp 5. Carew Arthur Meredith derived ax-2 7 in A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, here we follow the shortened derivation by Ivo Thomas, On Meredith's sole positive axiom, Notre Dame Journal of Formal Logic, volume XV, number 3, July 1974, page 477. Polish prefix notation: CCpCqrCCpqCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
Theorem	adh-minim-idALT 47907	Derivation of id 23 (reflexivity of implication, PM *2.08 WhiteheadRussell p. 101) from adh-minim-ax1 47902, adh-minim-ax2 47906, and ax-mp 5. It uses the derivation written DD211 in D-notation. (See head comment for an explanation.) Polish prefix notation: Cpp . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜑)
Theorem	adh-minim-pm2.43 47908	Derivation of pm2.43 57 WhiteheadRussell p. 106 (also called "hilbert" or "W") from adh-minim-ax1 47902, adh-minim-ax2 47906, and ax-mp 5. It uses the derivation written DD22D21 in D-notation. (See head comment for an explanation.) (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
Theorem	adh-minimp 47909	Another single axiom for minimal implicational calculus, due to Meredith. Other single axioms of the same length are known, but it is thought to be the minimal length. Among single axioms of this length, it is the one with simplest antecedents (i.e., in the corresponding ordering of binary trees which first compares left subtrees, it is the first one). Known as "HI-2" on Dolph Edward "Ted" Ulrich's web page. In the next 4 lemmas and 5 theorems, ax-1 6 and ax-2 7 are derived from this other single axiom in 20 detachments (instances of ax-mp 5) in total. Polish prefix notation: CpCCqrCCCsqCrtCqt ; or CtCCpqCCCspCqrCpr in Carew Arthur Meredith and Arthur Norman Prior, Notes on the axiomatics of the propositional calculus, Notre Dame Journal of Formal Logic, volume IV, number 3, July 1963, pages 171--187, on page 180. (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.)
⊢ (𝜑 → ((𝜓 → 𝜒) → (((𝜃 → 𝜓) → (𝜒 → 𝜏)) → (𝜓 → 𝜏))))
Theorem	adh-minimp-jarr-imim1-ax2c-lem1 47910	First lemma for the derivation of jarr 107, imim1 84, and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7, from adh-minimp 47909 and ax-mp 5. Polish prefix notation: CCpqCCCrpCqsCps . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃)))
Theorem	adh-minimp-jarr-lem2 47911	Second lemma for the derivation of jarr 107, and indirectly ax-1 6, a commuted form of ax-2 7, and ax-2 7 proper, from adh-minimp 47909 and ax-mp 5. Polish prefix notation: CCCpqCCCrsCCCtrCsuCruvCqv . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → (((𝜒 → 𝜃) → (((𝜏 → 𝜒) → (𝜃 → 𝜂)) → (𝜒 → 𝜂))) → 𝜁)) → (𝜓 → 𝜁))
Theorem	adh-minimp-jarr-ax2c-lem3 47912	Third lemma for the derivation of jarr 107 and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7 proper , from adh-minimp 47909 and ax-mp 5. Polish prefix notation: CCCCpqCCCrpCqsCpstt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃))) → 𝜏) → 𝜏)
Theorem	adh-minimp-sylsimp 47913	Derivation of jarr 107 (also called "syll-simp") from minimp 1654 and ax-mp 5. Polish prefix notation: CCCpqrCqr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))
Theorem	adh-minimp-ax1 47914	Derivation of ax-1 6 from adh-minimp 47909 and ax-mp 5. Polish prefix notation: CpCqp . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	adh-minimp-imim1 47915	Derivation of imim1 84 ("left antimonotonicity of implication", theorem *2.06 of [WhiteheadRussell] p. 100) from adh-minimp 47909 and ax-mp 5. Polish prefix notation: CCpqCCqrCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	adh-minimp-ax2c 47916	Derivation of a commuted form of ax-2 7 from adh-minimp 47909 and ax-mp 5. Polish prefix notation: CCpqCCpCqrCpr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
Theorem	adh-minimp-ax2-lem4 47917	Fourth lemma for the derivation of ax-2 7 from adh-minimp 47909 and ax-mp 5. Polish prefix notation: CpCCqCprCqr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ((𝜓 → (𝜑 → 𝜒)) → (𝜓 → 𝜒)))
Theorem	adh-minimp-ax2 47918	Derivation of ax-2 7 from adh-minimp 47909 and ax-mp 5. Polish prefix notation: CCpCqrCCpqCpr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
Theorem	adh-minimp-idALT 47919	Derivation of id 23 (reflexivity of implication, PM *2.08 WhiteheadRussell p. 101) from adh-minimp-ax1 47914, adh-minimp-ax2 47918, and ax-mp 5. It uses the derivation written DD211 in D-notation. (See head comment for an explanation.) Polish prefix notation: Cpp . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜑)
Theorem	adh-minimp-pm2.43 47920	Derivation of pm2.43 57 WhiteheadRussell p. 106 (also called "hilbert" or "W") from adh-minimp-ax1 47914, adh-minimp-ax2 47918, and ax-mp 5. It uses the derivation written DD22D21 in D-notation. (See head comment for an explanation.) Polish prefix notation: CCpCpqCpq . (Contributed by BJ, 31-May-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
21.50  Mathbox for Alexander van der Vekens
21.50.1  General auxiliary theorems (1)
21.50.1.1  Unordered and ordered pairs - extension for singletons
Theorem	n0nsn2el 47921*	If a class with one element is not a singleton, there is at least another element in this class. (Contributed by AV, 6-Mar-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≠ {𝐴}) → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
Theorem	eusnsn 47922*	There is a unique element of a singleton which is equal to another singleton. (Contributed by AV, 24-Aug-2022.)
⊢ ∃!𝑥{𝑥} = {𝑦}
Theorem	absnsb 47923*	If the class abstraction {𝑥 ∣ 𝜑} associated with the wff 𝜑 is a singleton, the wff is true for the singleton element. (Contributed by AV, 24-Aug-2022.)
⊢ ({𝑥 ∣ 𝜑} = {𝑦} → [𝑦 / 𝑥]𝜑)
Theorem	euabsneu 47924*	Another way to express existential uniqueness of a wff 𝜑: its associated class abstraction {𝑥 ∣ 𝜑} is a singleton. Variant of euabsn2 4689 using existential uniqueness for the singleton element instead of existence only. (Contributed by AV, 24-Aug-2022.)
⊢ (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥 ∣ 𝜑} = {𝑦})
21.50.1.2  Unordered and ordered pairs - extension for unordered pairs
Theorem	elprneb 47925	An element of a proper unordered pair is the first element iff it is not the second element. (Contributed by AV, 18-Jun-2020.)
⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐵 ≠ 𝐶) → (𝐴 = 𝐵 ↔ 𝐴 ≠ 𝐶))
21.50.1.3  Unordered and ordered pairs - extension for ordered pairs
Theorem	oppr 47926	Equality for ordered pairs implies equality of unordered pairs with the same elements. (Contributed by AV, 9-Jul-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴, 𝐵} = {𝐶, 𝐷}))
Theorem	opprb 47927	Equality for unordered pairs corresponds to equality of unordered pairs with the same elements. (Contributed by AV, 9-Jul-2023.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∨ ⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐶⟩)))
Theorem	or2expropbilem1 47928*	Lemma 1 for or2expropbi 47930 and ich2exprop 48379. (Contributed by AV, 16-Jul-2023.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) → (𝜑 → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
Theorem	or2expropbilem2 47929*	Lemma 2 for or2expropbi 47930 and ich2exprop 48379. (Contributed by AV, 16-Jul-2023.)
⊢ (∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
Theorem	or2expropbi 47930*	If two classes are strictly ordered, there is an ordered pair of both classes fulfilling a wff iff there is an unordered pair of both classes fulfilling the wff. (Contributed by AV, 26-Aug-2023.)
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑅 Or 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴𝑅𝐵)) → (∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ (𝑎𝑅𝑏 ∧ 𝜑)) ↔ ∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝑅𝑏 ∧ 𝜑))))
21.50.1.4  Relations - extension
Theorem	eubrv 47931*	If there is a unique set which is related to a class, then the class must be a set. (Contributed by AV, 25-Aug-2022.)
⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V)
Theorem	eubrdm 47932*	If there is a unique set which is related to a class, then the class is an element of the domain of the relation. (Contributed by AV, 25-Aug-2022.)
⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ dom 𝑅)
Theorem	eldmressn 47933	Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
⊢ (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴)
21.50.1.5  Definite description binder (inverted iota) - extension
Theorem	iota0def 47934*	Example for a defined iota being the empty set, i.e., ∀𝑦𝑥 ⊆ 𝑦 is a wff satisfied by a unique value 𝑥, namely 𝑥 = ∅ (the empty set is the one and only set which is a subset of every set). (Contributed by AV, 24-Aug-2022.)
⊢ (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅
Theorem	iota0ndef 47935*	Example for an undefined iota being the empty set, i.e., ∀𝑦𝑦 ∈ 𝑥 is a wff not satisfied by a (unique) value 𝑥 (there is no set, and therefore certainly no unique set, which contains every set). (Contributed by AV, 24-Aug-2022.)
⊢ (℩𝑥∀𝑦 𝑦 ∈ 𝑥) = ∅
21.50.1.6  Functions - extension
Theorem	fveqvfvv 47936	If a function's value at an argument is the universal class (which can never be the case because of fvex 6895), the function's value at this argument is any set (especially the empty set). In short "If a function's value is a proper class, it is a set", which sounds strange/contradictory, but which is a consequence of that a contradiction implies anything (see pm2.21i 120). (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵)
Theorem	fnresfnco 47937	Composition of two functions, similar to fnco 6654. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐵)
Theorem	funcoressn 47938	A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}))
Theorem	funressnfv 47939	A restriction to a singleton with a function value is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun (𝐹 ↾ {(𝐺‘𝑋)}))
Theorem	funressndmfvrn 47940	The value of a function 𝐹 at a set 𝐴 is in the range of the function 𝐹 if 𝐴 is in the domain of the function 𝐹. It is sufficient that 𝐹 is a function at 𝐴. (Contributed by AV, 1-Sep-2022.)
⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)
Theorem	funressnvmo 47941*	A function restricted to a singleton has at most one value for the singleton element as argument. (Contributed by AV, 2-Sep-2022.)
⊢ (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)
Theorem	funressnmo 47942*	A function restricted to a singleton has at most one value for the singleton element as argument. (Contributed by AV, 2-Sep-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦)
Theorem	funressneu 47943*	There is exactly one value of a class which is a function restricted to a singleton, analogous to funeu 6562. 𝐴 ∈ V is required because otherwise ∃!𝑦𝐴𝐹𝑦, see brprcneu 6872. (Contributed by AV, 7-Sep-2022.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
Theorem	fresfo 47944	Conditions for a restriction to be an onto function. Part of fresf1o 33112. (Contributed by AV, 29-Sep-2024.)
⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–onto→𝐶)
Theorem	fsetsniunop 47945*	The class of all functions from a (proper) singleton into 𝐵 is the union of all the singletons of (proper) ordered pairs over the elements of 𝐵 as second component. (Contributed by AV, 13-Sep-2024.)
⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}})
Theorem	fsetabsnop 47946*	The class of all functions from a (proper) singleton into 𝐵 is the class of all the singletons of (proper) ordered pairs over the elements of 𝐵 as second component. (Contributed by AV, 13-Sep-2024.)
⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
Theorem	fsetsnf 47947*	The mapping of an element of a class to a singleton function is a function. (Contributed by AV, 13-Sep-2024.)
⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)
Theorem	fsetsnf1 47948*	The mapping of an element of a class to a singleton function is an injection. (Contributed by AV, 13-Sep-2024.)
⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1→𝐴)
Theorem	fsetsnfo 47949*	The mapping of an element of a class to a singleton function is a surjection. (Contributed by AV, 13-Sep-2024.)
⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–onto→𝐴)
Theorem	fsetsnf1o 47950*	The mapping of an element of a class to a singleton function is a bijection. (Contributed by AV, 13-Sep-2024.)
⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1-onto→𝐴)
Theorem	fsetsnprcnex 47951*	The class of all functions from a (proper) singleton into a proper class 𝐵 is not a set. (Contributed by AV, 13-Sep-2024.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ∉ V)
Theorem	cfsetssfset 47952	The class of constant functions is a subclass of the class of functions. (Contributed by AV, 13-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    ⇒   ⊢ 𝐹 ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}
Theorem	cfsetsnfsetfv 47953*	The function value of the mapping of the class of singleton functions into the class of constant functions. (Contributed by AV, 13-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝐻‘𝑋) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))
Theorem	cfsetsnfsetf 47954*	The mapping of the class of singleton functions into the class of constant functions is a function. (Contributed by AV, 14-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)
Theorem	cfsetsnfsetf1 47955*	The mapping of the class of singleton functions into the class of constant functions is an injection. (Contributed by AV, 14-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1→𝐹)
Theorem	cfsetsnfsetfo 47956*	The mapping of the class of singleton functions into the class of constant functions is a surjection. (Contributed by AV, 14-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–onto→𝐹)
Theorem	cfsetsnfsetf1o 47957*	The mapping of the class of singleton functions into the class of constant functions is a bijection. (Contributed by AV, 14-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1-onto→𝐹)
Theorem	fsetprcnexALT 47958*	First version of proof for fsetprcnex 8867, which was much more complicated. (Contributed by AV, 14-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)
Theorem	fcoreslem1 47959	Lemma 1 for fcores 47963. (Contributed by AV, 17-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    ⇒   ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))
Theorem	fcoreslem2 47960	Lemma 2 for fcores 47963. (Contributed by AV, 17-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    ⇒   ⊢ (𝜑 → ran 𝑋 = 𝐸)
Theorem	fcoreslem3 47961	Lemma 3 for fcores 47963. (Contributed by AV, 13-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    ⇒   ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
Theorem	fcoreslem4 47962	Lemma 4 for fcores 47963. (Contributed by AV, 17-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    ⇒   ⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃)
Theorem	fcores 47963	Every composite function (𝐺 ∘ 𝐹) can be written as composition of restrictions of the composed functions (to their minimum domains). (Contributed by GL and AV, 17-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
Theorem	fcoresf1lem 47964	Lemma for fcoresf1 47965. (Contributed by AV, 18-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    ⇒   ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = (𝑌‘(𝑋‘𝑍)))
Theorem	fcoresf1 47965	If a composition is injective, then the restrictions of its components to the minimum domains are injective. (Contributed by GL and AV, 18-Sep-2024.) (Revised by AV, 7-Oct-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    &   ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝑃–1-1→𝐷)    ⇒   ⊢ (𝜑 → (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷))
Theorem	fcoresf1b 47966	A composition is injective iff the restrictions of its components to the minimum domains are injective. (Contributed by GL and AV, 7-Oct-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    ⇒   ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))
Theorem	fcoresfo 47967	If a composition is surjective, then the restriction of its first component to the minimum domain is surjective. (Contributed by AV, 17-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    &   ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)    ⇒   ⊢ (𝜑 → 𝑌:𝐸–onto→𝐷)
Theorem	fcoresfob 47968	A composition is surjective iff the restriction of its first component to the minimum domain is surjective. (Contributed by GL and AV, 7-Oct-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    ⇒   ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))
Theorem	fcoresf1ob 47969	A composition is bijective iff the restriction of its first component to the minimum domain is bijective and the restriction of its second component to the minimum domain is injective. (Contributed by GL and AV, 7-Oct-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    ⇒   ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1-onto→𝐷)))
Theorem	f1cof1blem 47970	Lemma for f1cof1b 47973 and focofob 47976. (Contributed by AV, 18-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    &   ⊢ (𝜑 → ran 𝐹 = 𝐶)    ⇒   ⊢ (𝜑 → ((𝑃 = 𝐴 ∧ 𝐸 = 𝐶) ∧ (𝑋 = 𝐹 ∧ 𝑌 = 𝐺)))
Theorem	3f1oss1 47971	The composition of three bijections as bijection from the image of the domain onto the image of the range of the middle bijection. (Contributed by AV, 15-Aug-2025.)
⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ((𝐻 ∘ 𝐺) ∘ ◡𝐹):(𝐹 “ 𝐶)–1-1-onto→(𝐻 “ 𝐷))
Theorem	3f1oss2 47972	The composition of three bijections as bijection from the image of the converse of the domain onto the image of the converse of the range of the middle bijection. (Contributed by AV, 15-Aug-2025.)
⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷))
Theorem	f1cof1b 47973	If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is injective iff 𝐹 and 𝐺 are both injective. (Contributed by GL and AV, 19-Sep-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷)))
Theorem	funfocofob 47974	If the domain of a function 𝐺 is a subset of the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 is surjective. (Contributed by GL and AV, 29-Sep-2024.)
⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))
Theorem	fnfocofob 47975	If the domain of a function 𝐺 equals the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 is surjective. (Contributed by GL and AV, 29-Sep-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))
Theorem	focofob 47976	If the domain of a function 𝐺 equals the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 and 𝐹 as function to the domain of 𝐺 are both surjective. Symmetric version of fnfocofob 47975 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 29-Sep-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ (𝐹:𝐴–onto→𝐶 ∧ 𝐺:𝐶–onto→𝐷)))
Theorem	f1ocof1ob 47977	If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is bijective iff 𝐹 and 𝐺 are both bijective. (Contributed by GL and AV, 7-Oct-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷)))
Theorem	f1ocof1ob2 47978	If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is bijective iff 𝐹 and 𝐺 are both bijective. Symmetric version of f1ocof1ob 47977 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 7-Oct-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ (𝐹:𝐴–1-1-onto→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷)))
21.50.2  Alternative for Russell's definition of a description binder
Syntax	caiota 47979	Extend class notation with an alternative for Russell's definition of a description binder (inverted iota).
class (℩'𝑥𝜑)
Theorem	aiotajust 47980*	Soundness justification theorem for df-aiota 47981. (Contributed by AV, 24-Aug-2022.)
⊢ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
Definition	df-aiota 47981*	Alternate version of Russell's definition of a description binder, which can be read as "the unique 𝑥 such that 𝜑", where 𝜑 ordinarily contains 𝑥 as a free variable. Our definition is meaningful only when there is exactly one 𝑥 such that 𝜑 is true (see aiotaval 47991); otherwise, it is not a set (see aiotaexb 47985), or even more concrete, it is the universe V (see aiotavb 47986). Since this is an alternative for df-iota 6493, we call this symbol ℩' alternate iota in the following. The advantage of this definition is the clear distinguishability of the defined and undefined cases: the alternate iota over a wff is defined iff it is a set (see aiotaexb 47985). With the original definition, there is no corresponding theorem (∃!𝑥𝜑 ↔ (℩𝑥𝜑) ≠ ∅), because ∅ can be a valid unique set satisfying a wff (see, for example, iota0def 47934). Only the right to left implication would hold, see (negated) iotanul 6517. For defined cases, however, both definitions df-iota 6493 and df-aiota 47981 are equivalent, see reuaiotaiota 47984. (Proposed by BJ, 13-Aug-2022.) (Contributed by AV, 24-Aug-2022.)
⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
Theorem	dfaiota2 47982*	Alternate definition of the alternate version of Russell's definition of a description binder. Definition 8.18 in [Quine] p. 56. (Contributed by AV, 24-Aug-2022.)
⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
Theorem	reuabaiotaiota 47983*	The iota and the alternate iota over a wff 𝜑 are equal iff there is a unique satisfying value of {𝑥 ∣ 𝜑} = {𝑦}. (Contributed by AV, 25-Aug-2022.)
⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Theorem	reuaiotaiota 47984	The iota and the alternate iota over a wff 𝜑 are equal iff there is a unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)
⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Theorem	aiotaexb 47985	The alternate iota over a wff 𝜑 is a set iff there is a unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)
⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
Theorem	aiotavb 47986	The alternate iota over a wff 𝜑 is the universe iff there is no unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)
⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
Theorem	aiotaint 47987	This is to df-aiota 47981 what iotauni 6514 is to df-iota 6493 (it uses intersection like df-aiota 47981, similar to iotauni 6514 using union like df-iota 6493; we could also prove an analogous result using union here too, in the same way that we have iotaint 6515). (Contributed by BJ, 31-Aug-2024.)
⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
Theorem	dfaiota3 47988	Alternate definition of ℩', using the if operator: this is to df-aiota 47981 what dfiota4 6529 is to df-iota 6493. It is simpler than df-aiota 47981 and uses no dummy variables, so it would be the preferred definition if ℩' becomes the description binder used in set.mm. (Contributed by BJ, 31-Aug-2024.)
⊢ (℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V)
Theorem	iotan0aiotaex 47989	If the iota over a wff 𝜑 is not empty, the alternate iota over 𝜑 is a set. (Contributed by AV, 25-Aug-2022.)
⊢ ((℩𝑥𝜑) ≠ ∅ → (℩'𝑥𝜑) ∈ V)
Theorem	aiotaexaiotaiota 47990	The alternate iota over a wff 𝜑 is a set iff the iota and the alternate iota over 𝜑 are equal. (Contributed by AV, 25-Aug-2022.)
⊢ ((℩'𝑥𝜑) ∈ V ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Theorem	aiotaval 47991*	Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of (alternate) iota. (Contributed by AV, 24-Aug-2022.)
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦)
Theorem	aiota0def 47992*	Example for a defined alternate iota being the empty set, i.e., ∀𝑦𝑥 ⊆ 𝑦 is a wff satisfied by a unique value 𝑥, namely 𝑥 = ∅ (the empty set is the one and only set which is a subset of every set). This corresponds to iota0def 47934. (Contributed by AV, 25-Aug-2022.)
⊢ (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅
Theorem	aiota0ndef 47993*	Example for an undefined alternate iota being no set, i.e., ∀𝑦𝑦 ∈ 𝑥 is a wff not satisfied by a (unique) value 𝑥 (there is no set, and therefore certainly no unique set, which contains every set). This is different from iota0ndef 47935, where the iota still is a set (the empty set). (Contributed by AV, 25-Aug-2022.)
⊢ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V
21.50.3  Double restricted existential uniqueness
21.50.3.1  Restricted quantification (extension)
Theorem	r19.32 47994	Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3197. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	rexsb 47995*	An equivalent expression for restricted existence, analogous to exsb 2390. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	rexrsb 47996*	An equivalent expression for restricted existence, analogous to exsb 2390. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑))
Theorem	2rexsb 47997*	An equivalent expression for double restricted existence, analogous to rexsb 47995. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
Theorem	2rexrsb 47998*	An equivalent expression for double restricted existence, analogous to 2exsb 2391. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
Theorem	cbvral2 47999*	Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3355. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑤𝜒    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)
Theorem	cbvrex2 48000*	Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3356. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑤𝜒    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)