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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nfriota 5801* | 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 5802* | 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 5803* | 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 5804* | 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 5805 |
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 5806* | Closure of restricted iota. (Contributed by NM, 21-Aug-2011.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | ||
Theorem | riotasbc 5807 | 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 5808* | Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2709 analog.) (Contributed by NM, 17-Jan-2012.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | riotabiia 5809 | Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2706 analog.) (Contributed by NM, 16-Jan-2012.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓) | ||
Theorem | riota1 5810* | Property of restricted iota. Compare iota1 5161. (Contributed by Mario Carneiro, 15-Oct-2016.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝑥)) | ||
Theorem | riota1a 5811 | Property of iota. (Contributed by NM, 23-Aug-2011.) |
⊢ ((𝑥 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥)) | ||
Theorem | riota2df 5812* | A deduction version of riota2f 5813. (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 5813* | 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 5814* | 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 5815* | Properties of a restricted definite description operator. Todo (df-riota 5792 update): can some uses of riota2f 5813 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 5816* | 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 5817* | 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 5818* | 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 5819* | 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 5820* | 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 5821 | 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 5822* | 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 5823* | 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 5824* | 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 5825* | 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 5826* | 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 5827* | Lemma for acexmid 5835. (Contributed by Jim Kingdon, 6-Aug-2019.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} & ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} & ⊢ 𝐶 = {𝐴, 𝐵} ⇒ ⊢ ({∅} ∈ 𝐴 → 𝜑) | ||
Theorem | acexmidlemb 5828* | Lemma for acexmid 5835. (Contributed by Jim Kingdon, 6-Aug-2019.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} & ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} & ⊢ 𝐶 = {𝐴, 𝐵} ⇒ ⊢ (∅ ∈ 𝐵 → 𝜑) | ||
Theorem | acexmidlemph 5829* | Lemma for acexmid 5835. (Contributed by Jim Kingdon, 6-Aug-2019.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} & ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} & ⊢ 𝐶 = {𝐴, 𝐵} ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | acexmidlemab 5830* | Lemma for acexmid 5835. (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 5831* |
Lemma for acexmid 5835. 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 5184. 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 5832* | Lemma for acexmid 5835. List the cases identified in acexmidlemcase 5831 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 5833* |
Lemma for acexmid 5835. This builds on acexmidlem1 5832 by noting that every
element of 𝐶 is inhabited.
(Note that 𝑦 is not quite a function in the df-fun 5184 sense because it uses ordered pairs as described in opthreg 4527 rather than df-op 3579). The set 𝐴 is also found in onsucelsucexmidlem 4500. (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 5834* |
Lemma for acexmid 5835.
This is acexmid 5835 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 5835* |
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 7153 and df-exmid 4168 syntaxes, see exmidac 7156. (Contributed by Jim Kingdon, 4-Aug-2019.) |
⊢ ∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Syntax | co 5836 | 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 5837 | Extend class notation to include class abstraction (class builder) of nested ordered pairs. |
class {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | ||
Syntax | cmpo 5838 | 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 5839 | 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 5869 and ovprc2 5870. 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 5840. (Contributed by NM, 28-Feb-1995.) |
⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | ||
Definition | df-oprab 5840* | 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 5839 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 5968. (Contributed by NM, 12-Mar-1995.) |
⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} | ||
Definition | df-mpo 5841* | 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 4039 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 5842 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) | ||
Theorem | oveq1 5843 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
⊢ (𝐴 = 𝐵 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶)) | ||
Theorem | oveq2 5844 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
⊢ (𝐴 = 𝐵 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵)) | ||
Theorem | oveq12 5845 | Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) | ||
Theorem | oveq1i 5846 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴𝐹𝐶) = (𝐵𝐹𝐶) | ||
Theorem | oveq2i 5847 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶𝐹𝐴) = (𝐶𝐹𝐵) | ||
Theorem | oveq12i 5848 | 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 5849 | Equality inference for operation value. (Contributed by NM, 24-Nov-2007.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶𝐴𝐷) = (𝐶𝐵𝐷) | ||
Theorem | oveq123i 5850 | Equality inference for operation value. (Contributed by FL, 11-Jul-2010.) |
⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 & ⊢ 𝐹 = 𝐺 ⇒ ⊢ (𝐴𝐹𝐵) = (𝐶𝐺𝐷) | ||
Theorem | oveq1d 5851 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶)) | ||
Theorem | oveq2d 5852 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵)) | ||
Theorem | oveqd 5853 | Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷)) | ||
Theorem | oveq12d 5854 | 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 5855 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) | ||
Theorem | oveqan12rd 5856 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) | ||
Theorem | oveq123d 5857 | 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 5858 | Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶))) | ||
Theorem | fvoveq1 5859 | Equality theorem for nested function and operation value. Closed form of fvoveq1d 5858. (Contributed by AV, 23-Jul-2022.) |
⊢ (𝐴 = 𝐵 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶))) | ||
Theorem | ovanraleqv 5860* | 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 5861 | 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 | nfovd 5862 | Deduction version of bound-variable hypothesis builder nfov 5863. (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 5863 | Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴𝐹𝐵) | ||
Theorem | oprabidlem 5864* | Slight elaboration of exdistrfor 1787. A lemma for oprabid 5865. (Contributed by Jim Kingdon, 15-Jan-2019.) |
⊢ (∃𝑥∃𝑦(𝑥 = 𝑧 ∧ 𝜓) → ∃𝑥(𝑥 = 𝑧 ∧ ∃𝑦𝜓)) | ||
Theorem | oprabid 5865 | The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable condition between 𝑥, 𝑦, and 𝑧, we use ax-bndl 1496 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.) |
⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜑) | ||
Theorem | fnovex 5866 | 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 5867 | 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 5868 | 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 5869 | 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 5870 | 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 5871 | 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 5872* | 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 5873* | 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 5874* | 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 5875* | A frequently used special case of rspc2ev 2840 for operation values. (Contributed by NM, 21-Mar-2007.) |
⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = (𝐶𝐹𝐷)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = (𝑥𝐹𝑦)) | ||
Theorem | fnotovb 5876 | Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5522. (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 5877* | 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) | ||
Theorem | 0neqopab 5878 | The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) |
⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} | ||
Theorem | brabvv 5879* | If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Jim Kingdon, 16-Jan-2019.) |
⊢ (𝑋{〈𝑥, 𝑦〉 ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) | ||
Theorem | dfoprab2 5880* | Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.) |
⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | ||
Theorem | reloprab 5881* | An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.) |
⊢ Rel {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | ||
Theorem | nfoprab1 5882 | The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ Ⅎ𝑥{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | ||
Theorem | nfoprab2 5883 | The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.) |
⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | ||
Theorem | nfoprab3 5884 | The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.) |
⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | ||
Theorem | nfoprab 5885* | Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.) |
⊢ Ⅎ𝑤𝜑 ⇒ ⊢ Ⅎ𝑤{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | ||
Theorem | oprabbid 5886* | Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑧𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) | ||
Theorem | oprabbidv 5887* | Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) | ||
Theorem | oprabbii 5888* | Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} | ||
Theorem | ssoprab2 5889 | Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4247. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓}) | ||
Theorem | ssoprab2b 5890 | Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4248. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓)) | ||
Theorem | eqoprab2b 5891 | Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4251. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓)) | ||
Theorem | mpoeq123 5892* | An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.) |
⊢ ((𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)) | ||
Theorem | mpoeq12 5893* | An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸)) | ||
Theorem | mpoeq123dva 5894* | An equality deduction for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐸) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 = 𝐹) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)) | ||
Theorem | mpoeq123dv 5895* | An equality deduction for the maps-to notation. (Contributed by NM, 12-Sep-2011.) |
⊢ (𝜑 → 𝐴 = 𝐷) & ⊢ (𝜑 → 𝐵 = 𝐸) & ⊢ (𝜑 → 𝐶 = 𝐹) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)) | ||
Theorem | mpoeq123i 5896 | An equality inference for the maps-to notation. (Contributed by NM, 15-Jul-2013.) |
⊢ 𝐴 = 𝐷 & ⊢ 𝐵 = 𝐸 & ⊢ 𝐶 = 𝐹 ⇒ ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹) | ||
Theorem | mpoeq3dva 5897* | Slightly more general equality inference for the maps-to notation. (Contributed by NM, 17-Oct-2013.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)) | ||
Theorem | mpoeq3ia 5898 | An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) | ||
Theorem | mpoeq3dv 5899* | An equality deduction for the maps-to notation restricted to the value of the operation. (Contributed by SO, 16-Jul-2018.) |
⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)) | ||
Theorem | nfmpo1 5900 | Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.) |
⊢ Ⅎ𝑥(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
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