P. 61 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6001-6100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isoeq1 6001	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
Theorem	isoeq2 6002	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵)))
Theorem	isoeq3 6003	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑇 (𝐴, 𝐵)))
Theorem	isoeq4 6004	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵)))
Theorem	isoeq5 6005	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝐵 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶)))
Theorem	nfiso 6006	Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ Ⅎ𝑥𝐻    &   ⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝑆    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
Theorem	isof1o 6007	An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
Theorem	isorel 6008	An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))
Theorem	isoresbr 6009*	A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)
⊢ ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
Theorem	isoid 6010	Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
⊢ ( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴)
Theorem	isocnv 6011	Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
Theorem	isocnv2 6012	Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵))
Theorem	isores2 6013	An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
Theorem	isores1 6014	An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
Theorem	isores3 6015	Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾 ⊆ 𝐴 ∧ 𝑋 = (𝐻 “ 𝐾)) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))
Theorem	isotr 6016	Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑆, 𝑇 (𝐵, 𝐶)) → (𝐺 ∘ 𝐻) Isom 𝑅, 𝑇 (𝐴, 𝐶))
Theorem	iso0 6017	The empty set is an 𝑅, 𝑆 isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
⊢ ∅ Isom 𝑅, 𝑆 (∅, ∅)
Theorem	isoini 6018	Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝐷}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)})))
Theorem	isoini2 6019	Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)
⊢ 𝐶 = (𝐴 ∩ (◡𝑅 “ {𝑋}))    &   ⊢ 𝐷 = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑋)}))    ⇒   ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))
Theorem	isoselem 6020*	Lemma for isose 6021. (Contributed by Mario Carneiro, 23-Jun-2015.)
⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V)    ⇒   ⊢ (𝜑 → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))
Theorem	isose 6021	An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))
Theorem	isopolem 6022	Lemma for isopo 6023. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))
Theorem	isopo 6023	An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐵))
Theorem	isosolem 6024	Lemma for isoso 6025. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))
Theorem	isoso 6025	An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))
Theorem	f1oiso 6026*	Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)
⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝑧 = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Theorem	f1oiso2 6027*	Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. (Contributed by Mario Carneiro, 9-Mar-2013.)
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))}    ⇒   ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Theorem	fdmrn 6028	A different way to write 𝐹 is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
Theorem	rinvf1o 6029	Sufficient conditions for the restriction of an involution to be a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
⊢ Fun 𝐹    &   ⊢ ◡𝐹 = 𝐹    &   ⊢ (𝐹 “ 𝐴) ⊆ 𝐵    &   ⊢ (𝐹 “ 𝐵) ⊆ 𝐴    &   ⊢ 𝐴 ⊆ dom 𝐹    &   ⊢ 𝐵 ⊆ dom 𝐹    ⇒   ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵
2.6.9  Cantor's Theorem
Theorem	canth 6030	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. (Use nex 1553 if you want the form ¬ ∃𝑓𝑓:𝐴–onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ¬ 𝐹:𝐴–onto→𝒫 𝐴
2.6.10  Restricted iota (description binder)
Syntax	crio 6031	Extend class notation with restricted description binder.
class (℩𝑥 ∈ 𝐴 𝜑)
Definition	df-riota 6032	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 5335. (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 6033*	Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜓))
Theorem	riotabidv 6034*	Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))
Theorem	riotaeqbidv 6035*	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	riotaexg 6036*	Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
⊢ (𝐴 ∈ 𝑉 → (℩𝑥 ∈ 𝐴 𝜓) ∈ V)
Theorem	iotaexel 6037*	Set existence of an iota expression in which all values are contained within a set. (Contributed by Jim Kingdon, 28-Jun-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥𝜑) ∈ V)
Theorem	riotav 6038	An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥𝜑)
Theorem	riotauni 6039	Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑})
Theorem	nfriota1 6040*	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	nfriotadxy 6041*	Deduction version of nfriota 6042. (Contributed by Jim Kingdon, 12-Jan-2019.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))
Theorem	nfriota 6042*	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	cbvriotavw 6043*	Change bound variable in a restricted description binder. Version of cbvriotav 6045 with a disjoint variable condition. (Contributed by NM, 18-Mar-2013.) (Revised by GG, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
Theorem	cbvriota 6044*	Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
Theorem	cbvriotav 6045*	Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
Theorem	csbriotag 6046*	Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
Theorem	riotacl2 6047	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 6048*	Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
Theorem	riotasbc 6049	Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
Theorem	riotabidva 6050*	Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2809 analog.) (Contributed by NM, 17-Jan-2012.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))
Theorem	riotabiia 6051	Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2807 analog.) (Contributed by NM, 16-Jan-2012.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓)
Theorem	riota1 6052*	Property of restricted iota. Compare iota1 5350. (Contributed by Mario Carneiro, 15-Oct-2016.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝑥))
Theorem	riota1a 6053	Property of iota. (Contributed by NM, 23-Aug-2011.)
⊢ ((𝑥 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥))
Theorem	riota2df 6054*	A deduction version of riota2f 6055. (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 6055*	This theorem shows a condition that allows us 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 6056*	This theorem shows a condition that allows us 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 6057*	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 6058*	Properties of a restricted definite description operator. Todo (df-riota 6032 update): can some uses of riota2f 6055 be shortened with this? (Contributed by NM, 23-Nov-2013.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐵 = (℩𝑥 ∈ 𝐴 𝜑)    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜓))
Theorem	riota5f 6059*	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 6060*	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 6061*	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 6062*	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 6063*	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 6064	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	eusvobj2 6065*	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 6066*	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 6067*	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 6068*	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 6069*	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	acexmidlema 6070*	Lemma for acexmid 6078. (Contributed by Jim Kingdon, 6-Aug-2019.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   ⊢ 𝐶 = {𝐴, 𝐵}    ⇒   ⊢ ({∅} ∈ 𝐴 → 𝜑)
Theorem	acexmidlemb 6071*	Lemma for acexmid 6078. (Contributed by Jim Kingdon, 6-Aug-2019.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   ⊢ 𝐶 = {𝐴, 𝐵}    ⇒   ⊢ (∅ ∈ 𝐵 → 𝜑)
Theorem	acexmidlemph 6072*	Lemma for acexmid 6078. (Contributed by Jim Kingdon, 6-Aug-2019.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   ⊢ 𝐶 = {𝐴, 𝐵}    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	acexmidlemab 6073*	Lemma for acexmid 6078. (Contributed by Jim Kingdon, 6-Aug-2019.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   ⊢ 𝐶 = {𝐴, 𝐵}    ⇒   ⊢ (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → ¬ 𝜑)
Theorem	acexmidlemcase 6074*	Lemma for acexmid 6078. Here we divide the proof into cases (based on the disjunction implicit in an unordered pair, not the sort of case elimination which relies on excluded middle). The cases are (1) the choice function evaluated at 𝐴 equals {∅}, (2) the choice function evaluated at 𝐵 equals ∅, and (3) the choice function evaluated at 𝐴 equals ∅ and the choice function evaluated at 𝐵 equals {∅}. Because of the way we represent the choice function 𝑦, the choice function evaluated at 𝐴 is (℩𝑣 ∈ 𝐴∃𝑢 ∈ 𝑦(𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) and the choice function evaluated at 𝐵 is (℩𝑣 ∈ 𝐵∃𝑢 ∈ 𝑦(𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)). Other than the difference in notation these work just as (𝑦‘𝐴) and (𝑦‘𝐵) would if 𝑦 were a function as defined by df-fun 5377. Although it isn't exactly about the division into cases, it is also convenient for this lemma to also include the step that if the choice function evaluated at 𝐴 equals {∅}, then {∅} ∈ 𝐴 and likewise for 𝐵. (Contributed by Jim Kingdon, 7-Aug-2019.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   ⊢ 𝐶 = {𝐴, 𝐵}    ⇒   ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})))
Theorem	acexmidlem1 6075*	Lemma for acexmid 6078. List the cases identified in acexmidlemcase 6074 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   ⊢ 𝐶 = {𝐴, 𝐵}    ⇒   ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑))
Theorem	acexmidlem2 6076*	Lemma for acexmid 6078. This builds on acexmidlem1 6075 by noting that every element of 𝐶 is inhabited. (Note that 𝑦 is not quite a function in the df-fun 5377 sense because it uses ordered pairs as described in opthreg 4701 rather than df-op 3717). The set 𝐴 is also found in onsucelsucexmidlem 4674. (Contributed by Jim Kingdon, 5-Aug-2019.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   ⊢ 𝐶 = {𝐴, 𝐵}    ⇒   ⊢ (∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑))
Theorem	acexmidlemv 6077*	Lemma for acexmid 6078. This is acexmid 6078 with additional disjoint variable conditions, most notably between 𝜑 and 𝑥. (Contributed by Jim Kingdon, 6-Aug-2019.)
⊢ ∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	acexmid 6078*	The axiom of choice implies excluded middle. Theorem 1.3 in [Bauer] p. 483. The statement of the axiom of choice given here is ac2 in the Metamath Proof Explorer (version of 3-Aug-2019). In particular, note that the choice function 𝑦 provides a value when 𝑧 is inhabited (as opposed to nonempty as in some statements of the axiom of choice). Essentially the same proof can also be found at "The axiom of choice implies instances of EM", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill theorem, after Radu Diaconescu who discovered it in 1975 in the framework of topos theory and N. D. Goodman and John Myhill in 1978 in the framework of set theory (although it already appeared as an exercise in Errett Bishop's book Foundations of Constructive Analysis from 1967). For this theorem stated using the df-ac 7556 and df-exmid 4330 syntaxes, see exmidac 7559. (Contributed by Jim Kingdon, 4-Aug-2019.)
⊢ ∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
2.6.11  Operations
Syntax	co 6079	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.
class (𝐴𝐹𝐵)
Syntax	coprab 6080	Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Syntax	cmpo 6081	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 6082	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 . 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 6116 and ovprc2 6117. On the other hand, we often find uses for this definition when 𝐹 is a proper class. 𝐹 is normally equal to a class of nested ordered pairs of the form defined by df-oprab 6083. (Contributed by NM, 28-Feb-1995.)
⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
Definition	df-oprab 6083*	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 6082 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 the most common operation class builder is given by ovmpo 6218. (Contributed by NM, 12-Mar-1995.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
Definition	df-mpo 6084*	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 𝐵(𝑥, 𝑦)". An extension of df-mpt 4192 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 6085	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
Theorem	oveq1 6086	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ (𝐴 = 𝐵 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
Theorem	oveq2 6087	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ (𝐴 = 𝐵 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
Theorem	oveq12 6088	Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
Theorem	oveq1i 6089	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴𝐹𝐶) = (𝐵𝐹𝐶)
Theorem	oveq2i 6090	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶𝐹𝐴) = (𝐶𝐹𝐵)
Theorem	oveq12i 6091	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 6092	Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶𝐴𝐷) = (𝐶𝐵𝐷)
Theorem	oveq123i 6093	Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    &   ⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐴𝐹𝐵) = (𝐶𝐺𝐷)
Theorem	oveq1d 6094	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
Theorem	oveq2d 6095	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
Theorem	oveqd 6096	Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷))
Theorem	oveq12d 6097	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 6098	Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
Theorem	oveqan12rd 6099	Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
Theorem	oveq123d 6100	Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷))