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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | adh-minimp-jarr-ax2c-lem3 43301 | Third lemma for the derivation of jarr 106 and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7 proper , from adh-minimp 43298 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 43302 | Derivation of jarr 106 (also called "syll-simp") from minimp 1622 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 43303 | Derivation of ax-1 6 from adh-minimp 43298 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 43304 | Derivation of imim1 83 ("left antimonotonicity of implication", theorem *2.06 of [WhiteheadRussell] p. 100) from adh-minimp 43298 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 43305 | Derivation of a commuted form of ax-2 7 from adh-minimp 43298 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 43306 | Fourth lemma for the derivation of ax-2 7 from adh-minimp 43298 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 43307 | Derivation of ax-2 7 from adh-minimp 43298 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 43308 | Derivation of id 22 (reflexivity of implication, PM *2.08 WhiteheadRussell p. 101) from adh-minimp-ax1 43303, adh-minimp-ax2 43307, 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 43309 | Derivation of pm2.43 56 WhiteheadRussell p. 106 (also called "hilbert" or "W") from adh-minimp-ax1 43303, adh-minimp-ax2 43307, 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)) | ||
Theorem | eusnsn 43310* | 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 43311* | 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 43312* | Another way to express existential uniqueness of a wff 𝜑: its associated class abstraction {𝑥 ∣ 𝜑} is a singleton. Variant of euabsn2 4661 using existential uniqueness for the singleton element instead of existence only. (Contributed by AV, 24-Aug-2022.) |
⊢ (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥 ∣ 𝜑} = {𝑦}) | ||
Theorem | elprneb 43313 | 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)) | ||
Theorem | oppr 43314 | 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 43315 | 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 43316* | Lemma 1 for or2expropbi 43318 and ich2exprop 43682. (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 43317* | Lemma 2 for or2expropbi 43318 and ich2exprop 43682. (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 43318* | 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)))) | ||
Theorem | eubrv 43319* | 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 43320* | 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 43321 | 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) | ||
Theorem | iota0def 43322* | 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 43323* | 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) = ∅ | ||
Theorem | fveqvfvv 43324 | If a function's value at an argument is the universal class (which can never be the case because of fvex 6683), 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 119). (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵) | ||
Theorem | fnresfnco 43325 | Composition of two functions, similar to fnco 6465. (Contributed by Alexander van der Vekens, 25-Jul-2017.) |
⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐵) | ||
Theorem | funcoressn 43326 | 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 43327 | 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 43328 | 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 43329* | 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 43330* | 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 43331* | There is exactly one value of a class which is a function restricted to a singleton, analogous to funeu 6380. 𝐴 ∈ V is required because otherwise ∃!𝑦𝐴𝐹𝑦, see brprcneu 6662. (Contributed by AV, 7-Sep-2022.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦) | ||
Syntax | caiota 43332 | Extend class notation with an alternative for Russell's definition of a description binder (inverted iota). |
class (℩'𝑥𝜑) | ||
Theorem | aiotajust 43333* | Soundness justification theorem for df-aiota 43334. (Contributed by AV, 24-Aug-2022.) |
⊢ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | ||
Definition | df-aiota 43334* |
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 43342); otherwise, it is not a set (see aiotaexb 43338), or even
more concrete, it is the universe V (see aiotavb 43339). Since this
is an alternative for df-iota 6314, 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 43338). 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 43322). Only the right to left implication would hold, see (negated) iotanul 6333. For defined cases, however, both definitions df-iota 6314 and df-aiota 43334 are equivalent, see reuaiotaiota 43337. (Proposed by BJ, 13-Aug-2022.) (Contributed by AV, 24-Aug-2022.) |
⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | ||
Theorem | dfaiota2 43335* | 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 43336* | 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 43337 | 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 43338 | 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 43339 | 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 | iotan0aiotaex 43340 | 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 43341 | 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 43342* | 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 43343* | 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 43322. (Contributed by AV, 25-Aug-2022.) |
⊢ (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅ | ||
Theorem | aiota0ndef 43344* | 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 43323, where the iota still is a set (the empty set). (Contributed by AV, 25-Aug-2022.) |
⊢ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V | ||
Theorem | r19.32 43345 | Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3340. (Contributed by Alexander van der Vekens, 29-Jun-2017.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓)) | ||
Theorem | rexsb 43346* | An equivalent expression for restricted existence, analogous to exsb 2378. (Contributed by Alexander van der Vekens, 1-Jul-2017.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | rexrsb 43347* | An equivalent expression for restricted existence, analogous to exsb 2378. (Contributed by Alexander van der Vekens, 1-Jul-2017.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑)) | ||
Theorem | 2rexsb 43348* | An equivalent expression for double restricted existence, analogous to rexsb 43346. (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 43349* | An equivalent expression for double restricted existence, analogous to 2exsb 2379. (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 43350* | Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3464. (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 43351* | Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3465. (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 | 2ralbiim 43352 | Split a biconditional and distribute 2 quantifiers, analogous to 2albiim 1891 and ralbiim 3174. (Contributed by Alexander van der Vekens, 2-Jul-2017.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ↔ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑))) | ||
Theorem | ralndv1 43353 | Example for a theorem about a restricted universal quantification in which the restricting class depends on (actually is) the bound variable: All sets containing themselves contain the universal class. (Contributed by AV, 24-Jun-2023.) |
⊢ ∀𝑥 ∈ 𝑥 V ∈ 𝑥 | ||
Theorem | ralndv2 43354 | Second example for a theorem about a restricted universal quantification in which the restricting class depends on the bound variable: all subsets of a set are sets. (Contributed by AV, 24-Jun-2023.) |
⊢ ∀𝑥 ∈ 𝒫 𝑥𝑥 ∈ V | ||
Theorem | reuf1odnf 43355* | There is exactly one element in each of two isomorphic sets. Variant of reuf1od 43356 with no distinct variable condition for 𝜒. (Contributed by AV, 19-Mar-2023.) |
⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵) & ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜃)) & ⊢ Ⅎ𝑥𝜒 ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | reuf1od 43356* | There is exactly one element in each of two isomorphic sets. (Contributed by AV, 19-Mar-2023.) |
⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵) & ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | euoreqb 43357* | There is a set which is equal to one of two other sets iff the other sets are equal. (Contributed by AV, 24-Jan-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∃!𝑥 ∈ 𝑉 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | 2reu3 43358* | Double restricted existential uniqueness, analogous to 2eu3 2738. (Contributed by Alexander van der Vekens, 29-Jun-2017.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (∃*𝑥 ∈ 𝐴 𝜑 ∨ ∃*𝑦 ∈ 𝐵 𝜑) → ((∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))) | ||
Theorem | 2reu7 43359* | Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2743. (Contributed by Alexander van der Vekens, 2-Jul-2017.) |
⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑)) | ||
Theorem | 2reu8 43360* | Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2744. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥 ∈ 𝐴∃!𝑦 ∈ 𝐵 using 2reu7 43359. (Contributed by Alexander van der Vekens, 2-Jul-2017.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑)) | ||
Theorem | 2reu8i 43361* | Implication of a double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, see also 2reu8 43360. The involved wffs depend on the setvar variables as follows: ph(x,y), ta(v,y), ch(x,w), th(v,w), et(x,b), ps(a,b), ze(a,w) (Contributed by AV, 1-Apr-2023.) |
⊢ (𝑥 = 𝑣 → (𝜑 ↔ 𝜏)) & ⊢ (𝑥 = 𝑣 → (𝜒 ↔ 𝜃)) & ⊢ (𝑦 = 𝑤 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝑏 → (𝜑 ↔ 𝜂)) & ⊢ (𝑥 = 𝑎 → (𝜒 ↔ 𝜁)) & ⊢ (((𝜒 → 𝑦 = 𝑤) ∧ 𝜁) → 𝑦 = 𝑤) & ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜂 → (𝑏 = 𝑦 ∧ (𝜓 → 𝑎 = 𝑥))))) | ||
Theorem | 2reuimp0 43362* | Implication of a double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification. The involved wffs depend on the setvar variables as follows: ph(a,b), th(a,c), ch(d,b), ta(d,c), et(a,e), ps(a,f) (Contributed by AV, 13-Mar-2023.) |
⊢ (𝑏 = 𝑐 → (𝜑 ↔ 𝜃)) & ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜒)) & ⊢ (𝑎 = 𝑑 → (𝜃 ↔ 𝜏)) & ⊢ (𝑏 = 𝑒 → (𝜑 ↔ 𝜂)) & ⊢ (𝑐 = 𝑓 → (𝜃 ↔ 𝜓)) ⇒ ⊢ (∃!𝑎 ∈ 𝑉 ∃!𝑏 ∈ 𝑉 𝜑 → ∃𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ((𝜂 ∧ ((𝜒 ∧ ∀𝑐 ∈ 𝑉 (𝜏 → 𝑏 = 𝑐)) → 𝑎 = 𝑑)) ∧ (𝜓 → 𝑒 = 𝑓))) | ||
Theorem | 2reuimp 43363* | Implication of a double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification if the class of the quantified elements is not empty. (Contributed by AV, 13-Mar-2023.) |
⊢ (𝑏 = 𝑐 → (𝜑 ↔ 𝜃)) & ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜒)) & ⊢ (𝑎 = 𝑑 → (𝜃 ↔ 𝜏)) & ⊢ (𝑏 = 𝑒 → (𝜑 ↔ 𝜂)) & ⊢ (𝑐 = 𝑓 → (𝜃 ↔ 𝜓)) ⇒ ⊢ ((𝑉 ≠ ∅ ∧ ∃!𝑎 ∈ 𝑉 ∃!𝑏 ∈ 𝑉 𝜑) → ∃𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))) | ||
The current definition of the value (𝐹‘𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6363) assures that this value is always a set, see fex 6989. This is because this definition can be applied to any classes 𝐹 and 𝐴, and evaluates to the empty set when it is not meaningful (as shown by ndmfv 6700 and fvprc 6663). Although it is very convenient for many theorems on functions and their proofs, there are some cases in which from (𝐹‘𝐴) = ∅ alone it cannot be decided/derived whether (𝐹‘𝐴) is meaningful (𝐹 is actually a function which is defined for 𝐴 and really has the function value ∅ at 𝐴) or not. Therefore, additional assumptions are required, such as ∅ ∉ ran 𝐹, ∅ ∈ ran 𝐹 or Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 (see, for example, ndmfvrcl 6701). To avoid such an ambiguity, an alternative definition (𝐹'''𝐴) (see df-afv 43368) would be possible which evaluates to the universal class ((𝐹'''𝐴) = V) if it is not meaningful (see afvnfundmuv 43387, ndmafv 43388, afvprc 43392 and nfunsnafv 43390), and which corresponds to the current definition ((𝐹‘𝐴) = (𝐹'''𝐴)) if it is (see afvfundmfveq 43386). That means (𝐹'''𝐴) = V → (𝐹‘𝐴) = ∅ (see afvpcfv0 43394), but (𝐹‘𝐴) = ∅ → (𝐹'''𝐴) = V is not generally valid. In the theory of partial functions, it is a common case that 𝐹 is not defined at 𝐴, which also would result in (𝐹'''𝐴) = V. In this context we say (𝐹'''𝐴) "is not defined" instead of "is not meaningful". With this definition the following intuitive equivalence holds: (𝐹'''𝐴) ∈ V <-> "(𝐹'''𝐴) is meaningful/defined". An interesting question would be if (𝐹‘𝐴) could be replaced by (𝐹'''𝐴) in most of the theorems based on function values. If we look at the (currently 19) proofs using the definition df-fv 6363 of (𝐹‘𝐴), we see that analogues for the following 8 theorems can be proven using the alternative definition: fveq1 6669-> afveq1 43382, fveq2 6670-> afveq2 43383, nffv 6680-> nfafv 43384, csbfv12 6713-> csbafv12g , fvres 6689-> afvres 43420, rlimdm 14908-> rlimdmafv 43425, tz6.12-1 6692-> tz6.12-1-afv 43422, fveu 6661-> afveu 43401. Three theorems proved by directly using df-fv 6363 are within a mathbox (fvsb 40833) or not used (isumclim3 15114, avril1 28242). However, the remaining 8 theorems proved by directly using df-fv 6363 are used more or less often: * fvex 6683: used in about 1750 proofs. * tz6.12-1 6692: root theorem of many theorems which have not a strict analogue, and which are used many times: fvprc 6663 (used in about 127 proofs), tz6.12i 6696 (used - indirectly via fvbr0 6697 and fvrn0 6698- in 18 proofs, and in fvclss 7001 used in fvclex 7660 used in fvresex 7661, which is not used!), dcomex 9869 (used in 4 proofs), ndmfv 6700 (used in 86 proofs) and nfunsn 6707 (used by dffv2 6756 which is not used). * fv2 6665: only used by elfv 6668, which is only used by fv3 6688, which is not used. * dffv3 6666: used by dffv4 6667 (the previous "df-fv"), which now is only used in deprecated (usage discouraged) theorems or within mathboxes (csbfv12gALTVD 41282), by shftval 14433 (itself used in 9 proofs), by dffv5 33385 (mathbox) and by fvco2 6758, which has the analogue afvco2 43424. * fvopab5 6800: used only by ajval 28638 (not used) and by adjval 29667 (used - indirectly - in 9 proofs). * zsum 15075: used (via isum 15076, sum0 15078 and fsumsers 15085) in more than 90 proofs. * isumshft 15194: used in pserdv2 25018 and (via logtayl 25243) 4 other proofs. * ovtpos 7907: used in 14 proofs. As a result of this analysis we can say that the current definition of a function value is crucial for Metamath and cannot be exchanged easily with an alternative definition. While fv2 6665, dffv3 6666, fvopab5 6800, zsum 15075, isumshft 15194 and ovtpos 7907 are not critical or are, hopefully, also valid for the alternative definition, fvex 6683 and tz6.12-1 6692 (and the theorems based on them) are essential for the current definition of function values. With the same arguments, an alternative definition of operation values ((𝐴𝑂𝐵)) could be meaningful to avoid ambiguities, see df-aov 43369. For additional details, see https://groups.google.com/g/metamath/c/cteNUppB6A4 43369. | ||
Syntax | wdfat 43364 | Extend the definition of a wff to include the "defined at" predicate. Read: "(the function) 𝐹 is defined at (the argument) 𝐴". In a previous version, the token "def@" was used. However, since the @ is used (informally) as a replacement for $ in commented out sections that may be deleted some day. While there is no violation of any standard to use the @ in a token, it could make the search for such commented-out sections slightly more difficult. (See remark of Norman Megill at https://groups.google.com/g/metamath/c/cteNUppB6A4). |
wff 𝐹 defAt 𝐴 | ||
Syntax | cafv 43365 | Extend the definition of a class to include the value of a function. Read: "the value of 𝐹 at 𝐴 " or "𝐹 of 𝐴". In a previous version, the symbol " ' " was used. However, since the similarity with the symbol ‘ used for the current definition of a function's value (see df-fv 6363), which, by the way, was intended to visualize that in many cases ‘ and " ' " are exchangeable, makes reading the theorems, especially those which uses both definitions as dfafv2 43380, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 6363 and df-ima 5568. And not three backticks ( three times ‘) since that would be annoying to escape in a comment. (See remark of Norman Megill and Gerard Lang at https://groups.google.com/g/metamath/c/cteNUppB6A4 5568). |
class (𝐹'''𝐴) | ||
Syntax | caov 43366 | 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 a pair of parentheses in contrast to the current definition, see df-ov 7159. |
class ((𝐴𝐹𝐵)) | ||
Definition | df-dfat 43367 | Definition of the predicate that determines if some class 𝐹 is defined as function for an argument 𝐴 or, in other words, if the function value for some class 𝐹 for an argument 𝐴 is defined. We say that 𝐹 is defined at 𝐴 if a 𝐹 is a function restricted to the member 𝐴 of its domain. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | ||
Definition | df-afv 43368* | Alternative definition of the value of a function, (𝐹'''𝐴), also known as function application. In contrast to (𝐹‘𝐴) = ∅ (see df-fv 6363 and ndmfv 6700), (𝐹'''𝐴) = V if F is not defined for A! (Contributed by Alexander van der Vekens, 25-May-2017.) (Revised by BJ/AV, 25-Aug-2022.) |
⊢ (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥) | ||
Definition | df-aov 43369 | Define the value of an operation. In contrast to df-ov 7159, the alternative definition for a function value (see df-afv 43368) is used. By this, the value of the operation applied to two arguments is the universal class if the operation is not defined for these two arguments. There are still 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. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | ||
Theorem | ralbinrald 43370* | Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.) |
⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝑥 ∈ 𝐴 → 𝑥 = 𝑋) & ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ 𝜃)) | ||
Theorem | nvelim 43371 | If a class is the universal class it doesn't belong to any class, generalization of nvel 5220. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵) | ||
Theorem | alneu 43372 | If a statement holds for all sets, there is not a unique set for which the statement holds. (Contributed by Alexander van der Vekens, 28-Nov-2017.) |
⊢ (∀𝑥𝜑 → ¬ ∃!𝑥𝜑) | ||
Theorem | eu2ndop1stv 43373* | If there is a unique second component in an ordered pair contained in a given set, the first component must be a set. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
⊢ (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → 𝐴 ∈ V) | ||
Theorem | dfateq12d 43374 | Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) | ||
Theorem | nfdfat 43375 | Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g., for Fun/Rel, dom, ⊆, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 | ||
Theorem | dfdfat2 43376* | Alternate definition of the predicate "defined at" not using the Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦)) | ||
Theorem | fundmdfat 43377 | A function is defined at any element of its domain. (Contributed by AV, 2-Sep-2022.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴) | ||
Theorem | dfatprc 43378 | A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.) |
⊢ (¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴) | ||
Theorem | dfatelrn 43379 | The value of a function 𝐹 at a set 𝐴 is in the range of the function 𝐹 if 𝐹 is defined at 𝐴. (Contributed by AV, 1-Sep-2022.) |
⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) ∈ ran 𝐹) | ||
Theorem | dfafv2 43380 | Alternative definition of (𝐹'''𝐴) using (𝐹‘𝐴) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.) (Revised by AV, 25-Aug-2022.) |
⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | ||
Theorem | afveq12d 43381 | Equality deduction for function value, analogous to fveq12d 6677. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵)) | ||
Theorem | afveq1 43382 | Equality theorem for function value, analogous to fveq1 6669. (Contributed by Alexander van der Vekens, 22-Jul-2017.) |
⊢ (𝐹 = 𝐺 → (𝐹'''𝐴) = (𝐺'''𝐴)) | ||
Theorem | afveq2 43383 | Equality theorem for function value, analogous to fveq1 6669. (Contributed by Alexander van der Vekens, 22-Jul-2017.) |
⊢ (𝐴 = 𝐵 → (𝐹'''𝐴) = (𝐹'''𝐵)) | ||
Theorem | nfafv 43384 | Bound-variable hypothesis builder for function value, analogous to nffv 6680. To prove a deduction version of this analogous to nffvd 6682 is not easily possible because a deduction version of nfdfat 43375 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥(𝐹'''𝐴) | ||
Theorem | csbafv12g 43385 | Move class substitution in and out of a function value, analogous to csbfv12 6713, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7198. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵)) | ||
Theorem | afvfundmfveq 43386 | If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴)) | ||
Theorem | afvnfundmuv 43387 | If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V) | ||
Theorem | ndmafv 43388 | The value of a class outside its domain is the universe, compare with ndmfv 6700. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V) | ||
Theorem | afvvdm 43389 | If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹'''𝐴) ∈ 𝐵 → 𝐴 ∈ dom 𝐹) | ||
Theorem | nfunsnafv 43390 | If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6707. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V) | ||
Theorem | afvvfunressn 43391 | If the function value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹'''𝐴) ∈ 𝐵 → Fun (𝐹 ↾ {𝐴})) | ||
Theorem | afvprc 43392 | A function's value at a proper class is the universe, compare with fvprc 6663. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (¬ 𝐴 ∈ V → (𝐹'''𝐴) = V) | ||
Theorem | afvvv 43393 | If a function's value at an argument is a set, the argument is also a set. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹'''𝐴) ∈ 𝐵 → 𝐴 ∈ V) | ||
Theorem | afvpcfv0 43394 | If the value of the alternative function at an argument is the universe, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹'''𝐴) = V → (𝐹‘𝐴) = ∅) | ||
Theorem | afvnufveq 43395 | The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹‘𝐴)) | ||
Theorem | afvvfveq 43396 | The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹'''𝐴) ∈ 𝐵 → (𝐹'''𝐴) = (𝐹‘𝐴)) | ||
Theorem | afv0fv0 43397 | If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹'''𝐴) = ∅ → (𝐹‘𝐴) = ∅) | ||
Theorem | afvfvn0fveq 43398 | If the function's value at an argument is not the empty set, it equals the value of the alternative function at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹'''𝐴) = (𝐹‘𝐴)) | ||
Theorem | afv0nbfvbi 43399 | The function's value at an argument is an element of a set if and only if the value of the alternative function at this argument is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (∅ ∉ 𝐵 → ((𝐹'''𝐴) ∈ 𝐵 ↔ (𝐹‘𝐴) ∈ 𝐵)) | ||
Theorem | afvfv0bi 43400 | The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V)) |
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