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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | isoid 5501 | 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 5502 | 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 5503 | 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 5504 | 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 5505 | 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 5506 | 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 5507 | 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 | isoini 5508 | 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 5509 | 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 5510* | Lemma for isose 5511. (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 5511 | 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 5512 | Lemma for isopo 5513. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴)) | ||
Theorem | isopo 5513 | 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 5514 | Lemma for isoso 5515. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴)) | ||
Theorem | isoso 5515 | 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 5516* | 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 5517* | 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)) | ||
Syntax | crio 5518 | Extend class notation with restricted description binder. |
class (℩𝑥 ∈ 𝐴 𝜑) | ||
Definition | df-riota 5519 | 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 4917. (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 5520* | Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | riotabidv 5521* | Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | riotaeqbidv 5522* | 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 5523* | Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.) |
⊢ (𝐴 ∈ 𝑉 → (℩𝑥 ∈ 𝐴 𝜓) ∈ V) | ||
Theorem | riotav 5524 | An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.) |
⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥𝜑) | ||
Theorem | riotauni 5525 | Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑}) | ||
Theorem | nfriota1 5526* | 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 5527* | Deduction version of nfriota 5528. (Contributed by Jim Kingdon, 12-Jan-2019.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓)) | ||
Theorem | nfriota 5528* | 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 5529* | 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 5530* | 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 5531* | 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 5532 |
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 5533* | Closure of restricted iota. (Contributed by NM, 21-Aug-2011.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | ||
Theorem | riotasbc 5534 | 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 5535* | Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2598 analog.) (Contributed by NM, 17-Jan-2012.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | riotabiia 5536 | Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2596 analog.) (Contributed by NM, 16-Jan-2012.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓) | ||
Theorem | riota1 5537* | Property of restricted iota. Compare iota1 4931. (Contributed by Mario Carneiro, 15-Oct-2016.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝑥)) | ||
Theorem | riota1a 5538 | Property of iota. (Contributed by NM, 23-Aug-2011.) |
⊢ ((𝑥 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥)) | ||
Theorem | riota2df 5539* | A deduction version of riota2f 5540. (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 5540* | 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 5541* | 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 | riotaprop 5542* | Properties of a restricted definite description operator. Todo (df-riota 5519 update): can some uses of riota2f 5540 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 5543* | 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 5544* | 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 5545* | 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 5546* | 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 5547* | 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 5548 | 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 5549* | 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 5550* | 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 5551* | 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 5552* | 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 5553* | 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 5554* | Lemma for acexmid 5562. (Contributed by Jim Kingdon, 6-Aug-2019.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} & ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} & ⊢ 𝐶 = {𝐴, 𝐵} ⇒ ⊢ ({∅} ∈ 𝐴 → 𝜑) | ||
Theorem | acexmidlemb 5555* | Lemma for acexmid 5562. (Contributed by Jim Kingdon, 6-Aug-2019.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} & ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} & ⊢ 𝐶 = {𝐴, 𝐵} ⇒ ⊢ (∅ ∈ 𝐵 → 𝜑) | ||
Theorem | acexmidlemph 5556* | Lemma for acexmid 5562. (Contributed by Jim Kingdon, 6-Aug-2019.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} & ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} & ⊢ 𝐶 = {𝐴, 𝐵} ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | acexmidlemab 5557* | Lemma for acexmid 5562. (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 5558* |
Lemma for acexmid 5562. 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 4954. 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 5559* | Lemma for acexmid 5562. List the cases identified in acexmidlemcase 5558 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 5560* |
Lemma for acexmid 5562. This builds on acexmidlem1 5559 by noting that every
element of 𝐶 is inhabited.
(Note that 𝑦 is not quite a function in the df-fun 4954 sense because it uses ordered pairs as described in opthreg 4327 rather than df-op 3425). The set 𝐴 is also found in onsucelsucexmidlem 4300. (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 5561* |
Lemma for acexmid 5562.
This is acexmid 5562 with additional distinct variable constraints, 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 5562* |
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 non-empty 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). (Contributed by Jim Kingdon, 4-Aug-2019.) |
⊢ ∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Syntax | co 5563 | 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 5564 | Extend class notation to include class abstraction (class builder) of nested ordered pairs. |
class {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | ||
Syntax | cmpt2 5565 | 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 5566 | 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 5592 and ovprc2 5593. 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 5567. (Contributed by NM, 28-Feb-1995.) |
⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | ||
Definition | df-oprab 5567* | 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 5566 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 ovmpt2 5687. (Contributed by NM, 12-Mar-1995.) |
⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} | ||
Definition | df-mpt2 5568* | 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 3861 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 5569 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) | ||
Theorem | oveq1 5570 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
⊢ (𝐴 = 𝐵 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶)) | ||
Theorem | oveq2 5571 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
⊢ (𝐴 = 𝐵 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵)) | ||
Theorem | oveq12 5572 | Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) | ||
Theorem | oveq1i 5573 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴𝐹𝐶) = (𝐵𝐹𝐶) | ||
Theorem | oveq2i 5574 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶𝐹𝐴) = (𝐶𝐹𝐵) | ||
Theorem | oveq12i 5575 | 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 5576 | Equality inference for operation value. (Contributed by NM, 24-Nov-2007.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶𝐴𝐷) = (𝐶𝐵𝐷) | ||
Theorem | oveq123i 5577 | Equality inference for operation value. (Contributed by FL, 11-Jul-2010.) |
⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 & ⊢ 𝐹 = 𝐺 ⇒ ⊢ (𝐴𝐹𝐵) = (𝐶𝐺𝐷) | ||
Theorem | oveq1d 5578 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶)) | ||
Theorem | oveq2d 5579 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵)) | ||
Theorem | oveqd 5580 | Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷)) | ||
Theorem | oveq12d 5581 | 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 5582 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) | ||
Theorem | oveqan12rd 5583 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) | ||
Theorem | oveq123d 5584 | Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷)) | ||
Theorem | nfovd 5585 | Deduction version of bound-variable hypothesis builder nfov 5586. (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 5586 | Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴𝐹𝐵) | ||
Theorem | oprabidlem 5587* | Slight elaboration of exdistrfor 1723. A lemma for oprabid 5588. (Contributed by Jim Kingdon, 15-Jan-2019.) |
⊢ (∃𝑥∃𝑦(𝑥 = 𝑧 ∧ 𝜓) → ∃𝑥(𝑥 = 𝑧 ∧ ∃𝑦𝜓)) | ||
Theorem | oprabid 5588 | The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable constraint between 𝑥, 𝑦, and 𝑧, we use ax-bndl 1440 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.) |
⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜑) | ||
Theorem | fnovex 5589 | The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.) |
⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝐹𝐵) ∈ V) | ||
Theorem | ovexg 5590 | Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐹𝐵) ∈ V) | ||
Theorem | ovprc 5591 | 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 5592 | 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 5593 | 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 | csbov123g 5594 | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | csbov12g 5595* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | csbov1g 5596* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹𝐶)) | ||
Theorem | csbov2g 5597* | Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵𝐹⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | rspceov 5598* | A frequently used special case of rspc2ev 2723 for operation values. (Contributed by NM, 21-Mar-2007.) |
⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = (𝐶𝐹𝐷)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = (𝑥𝐹𝑦)) | ||
Theorem | fnotovb 5599 | Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5267. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹)) | ||
Theorem | opabbrex 5600* | A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) |
⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝 → 𝜃)) & ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {〈𝑓, 𝑝〉 ∣ 𝜃} ∈ V) ⇒ ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {〈𝑓, 𝑝〉 ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ∈ V) |
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