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Theorem List for Metamath Proof Explorer - 42101-42200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeuabsneu 42101* Another way to express existential uniqueness of a wff 𝜑: its associated class abstraction {𝑥𝜑} is a singleton. Variant of euabsn2 4492 using existential uniqueness for the singleton element instead of existence only. (Contributed by AV, 24-Aug-2022.)
(∃!𝑥𝜑 ↔ ∃!𝑦{𝑥𝜑} = {𝑦})

20.36.1.3  Unordered and ordered pairs - extension for unordered pairs

Theoremelprneb 42102 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} ∧ 𝐵𝐶) → (𝐴 = 𝐵𝐴𝐶))

20.36.1.4  Relations - extension

Theoremeubrv 42103* 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 𝐴𝑅𝑏𝐴 ∈ V)

Theoremeubrdm 42104* 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 𝐴𝑅𝑏𝐴 ∈ dom 𝑅)

Theoremeldmressn 42105 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)

20.36.1.5  Definite description binder (inverted iota) - extension

Theoremiota0def 42106* Example for a defined iota being the empty set, i.e., 𝑦𝑥𝑦 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.)
(℩𝑥𝑦 𝑥𝑦) = ∅

Theoremiota0ndef 42107* Example for an undefined iota being the empty set, i.e., 𝑦𝑦𝑥 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.)
(℩𝑥𝑦 𝑦𝑥) = ∅

20.36.1.6  Functions - extension

Theoremfveqvfvv 42108 If a function's value at an argument is the universal class (which can never be the case because of fvex 6459), 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 117). (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐹𝐴) = V → (𝐹𝐴) = 𝐵)

Theoremfnresfnco 42109 Composition of two functions, similar to fnco 6245. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
(((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → (𝐹𝐺) Fn 𝐵)

Theoremfuncoressn 42110 A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))

Theoremfunressnfv 42111 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 (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun (𝐹 ↾ {(𝐺𝑋)}))

Theoremfunressndmfvrn 42112 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) → (𝐹𝐴) ∈ ran 𝐹)

Theoremfunressnvmo 42113* 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)

TheoremfunressnvmoOLD 42114* Old proof of funressnvmo 42113. Obsolete as of 9-Oct-2022. (Contributed by AV, 2-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)

Theoremfunressnmo 42115* A function restricted to a singleton has at most one value for the singleton element as argument. (Contributed by AV, 2-Sep-2022.)
((𝐴𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦)

Theoremfunressneu 42116* There is exactly one value of a class which is a function restricted to a singleton, analogous to funeu 6160. 𝐴 ∈ V is required because otherwise ∃!𝑦𝐴𝐹𝑦, see brprcneu 6438. (Contributed by AV, 7-Sep-2022.)
(((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)

20.36.2  Alternative for Russell's definition of a description binder

Syntaxcaiota 42117 Extend class notation with an alternative for Russell's definition of a description binder (inverted iota).
class (℩'𝑥𝜑)

Theoremaiotajust 42118* Soundness justification theorem for df-aiota 42119. (Contributed by AV, 24-Aug-2022.)
{𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}

Definitiondf-aiota 42119* 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 42127); otherwise, it is not a set (see aiotaexb 42123), or even more concrete, it is the universe V (see aiotavb 42124). Since this is an alternative for df-iota 6099, 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 42123). 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 42106). Only the right to left implication would hold, see (negated) iotanul 6114. For defined cases, however, both definitions df-iota 6099 and df-aiota 42119 are equivalent, see reuaiotaiota 42122. (Proposed by BJ, 13-Aug-2022.) (Contributed by AV, 24-Aug-2022.)

(℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}

Theoremdfaiota2 42120* 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(𝜑𝑥 = 𝑦)}

Theoremreuabaiotaiota 42121* The iota and the alternate iota over a wff 𝜑 are equal iff there is a unique satisfying value of {𝑥𝜑} = {𝑦}. (Contributed by AV, 25-Aug-2022.)
(∃!𝑦{𝑥𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Theoremreuaiotaiota 42122 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))

Theoremaiotaexb 42123 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)

Theoremaiotavb 42124 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)

Theoremiotan0aiotaex 42125 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)

Theoremaiotaexaiotaiota 42126 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))

Theoremaiotaval 42127* Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of (alternate) iota. (Contributed by AV, 24-Aug-2022.)
(∀𝑥(𝜑𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦)

Theoremaiota0def 42128* Example for a defined alternate iota being the empty set, i.e., 𝑦𝑥𝑦 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 42106. (Contributed by AV, 25-Aug-2022.)
(℩'𝑥𝑦 𝑥𝑦) = ∅

Theoremaiota0ndef 42129* Example for an undefined alternate iota being no set, i.e., 𝑦𝑦𝑥 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 42107, where the iota still is a set (the empty set). (Contributed by AV, 25-Aug-2022.)
(℩'𝑥𝑦 𝑦𝑥) ∉ V

20.36.3  Double restricted existential uniqueness

20.36.3.1  Restricted quantification (extension)

Theoremr19.32 42130 Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3269. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
𝑥𝜑       (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))

Theoremrexsb 42131* An equivalent expression for restricted existence, analogous to exsb 2327. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥(𝑥 = 𝑦𝜑))

Theoremrexrsb 42132* An equivalent expression for restricted existence, analogous to exsb 2327. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝑥 = 𝑦𝜑))

Theorem2rexsb 42133* An equivalent expression for double restricted existence, analogous to rexsb 42131. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))

Theorem2rexrsb 42134* An equivalent expression for double restricted existence, analogous to 2exsb 2328. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))

Theoremcbvral2 42135* Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3375. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
𝑧𝜑    &   𝑥𝜒    &   𝑤𝜒    &   𝑦𝜓    &   (𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)

Theoremcbvrex2 42136* Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3376. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
𝑧𝜑    &   𝑥𝜒    &   𝑤𝜒    &   𝑦𝜓    &   (𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)

Theorem2ralbiim 42137 Split a biconditional and distribute 2 quantifiers, analogous to 2albiim 1936 and ralbiim 3255. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
(∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ ∀𝑥𝐴𝑦𝐵 (𝜓𝜑)))

20.36.3.2  Restricted uniqueness and "at most one" quantification

Theoremrmoimi 42138 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝜑𝜓)       (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑)

Theorem2reu5a 42139 Double restricted existential uniqueness in terms of restricted existence and restricted "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑) ∧ ∃*𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑)))

Theoremreuimrmo 42140 Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2651. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∀𝑥𝐴 (𝜑𝜓) → (∃!𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))

Theoremrmoanim 42141* Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2655. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
𝑥𝜑       (∃*𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝐴 𝜓))

Theoremreuan 42142* Introduction of a conjunct into restricted unique existential quantifier, analogous to euan 2656. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
𝑥𝜑       (∃!𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝐴 𝜓))

Theoremreuf1odnf 42143* There is exactly one element in each of two isomorphic sets. Variant of reuf1od 42144 with no distinct variable condition for 𝜒. (Contributed by AV, 19-Mar-2023.)
(𝜑𝐹:𝐶1-1-onto𝐵)    &   ((𝜑𝑥 = (𝐹𝑦)) → (𝜓𝜒))    &   (𝑥 = 𝑧 → (𝜓𝜃))    &   𝑥𝜒       (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))

Theoremreuf1od 42144* There is exactly one element in each of two isomorphic sets. (Contributed by AV, 19-Mar-2023.)
(𝜑𝐹:𝐶1-1-onto𝐵)    &   ((𝜑𝑥 = (𝐹𝑦)) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))

Theoremeuoreqb 42145* 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.)
((𝐴𝑉𝐵𝑉) → (∃!𝑥𝑉 (𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴 = 𝐵))

20.36.3.3  Analogs to Existential uniqueness (double quantification)

Theorem2reurex 42146* Double restricted quantification with existential uniqueness, analogous to 2euex 2671. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
(∃!𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)

Theorem2reurmo 42147* Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2672. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
(∃!𝑥𝐴 ∃*𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)

Theorem2reu2rex 42148* Double restricted existential uniqueness, analogous to 2eu2ex 2673. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)

Theorem2rmoswap 42149* A condition allowing swap of restricted "at most one" and restricted existential quantifiers, analogous to 2moswap 2674. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑦𝐵𝑥𝐴 𝜑))

Theorem2rexreu 42150* Double restricted existential uniqueness implies double restricted unique existential quantification, analogous to 2exeu 2676. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)

Theorem2reu1 42151* Double restricted existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one, analogous to 2eu1 2680. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑)))

Theorem2reu2 42152* Double restricted existential uniqueness, analogous to 2eu2 2682. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
(∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃!𝑥𝐴𝑦𝐵 𝜑))

Theorem2reu3 42153* Double restricted existential uniqueness, analogous to 2eu3 2683. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
(∀𝑥𝐴𝑦𝐵 (∃*𝑥𝐴 𝜑 ∨ ∃*𝑦𝐵 𝜑) → ((∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑)))

Theorem2reu4a 42154* Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2685 with the additional requirement that the restricting classes are not empty (which is not necessary as shown in 2reu4 42155). (Contributed by Alexander van der Vekens, 1-Jul-2017.)
((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))))

Theorem2reu4 42155* Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2685. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

Theorem2reu7 42156* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2688. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))

Theorem2reu8 42157* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2689. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥𝐴∃!𝑦𝐵 using 2reu7 42156. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))

20.36.3.4  Additional theorems for double restricted existential uniqueness

Theorem2reu8i 42158* Implication of a double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, see also 2reu8 42157. 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 → (𝜑𝜏))    &   (𝑥 = 𝑣 → (𝜒𝜃))    &   (𝑦 = 𝑤 → (𝜑𝜒))    &   (𝑦 = 𝑏 → (𝜑𝜂))    &   (𝑥 = 𝑎 → (𝜒𝜁))    &   (((𝜒𝑦 = 𝑤) ∧ 𝜁) → 𝑦 = 𝑤)    &   ((𝑥 = 𝑎𝑦 = 𝑏) → (𝜑𝜓))       (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 (𝜑 ∧ ∀𝑎𝐴𝑏𝐵 (𝜂 → (𝑏 = 𝑦 ∧ (𝜓𝑎 = 𝑥)))))

Theorem2reuimp0 42159* Implication of a double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification. The involved wwfs 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 → (𝜑𝜃))    &   (𝑎 = 𝑑 → (𝜑𝜒))    &   (𝑎 = 𝑑 → (𝜃𝜏))    &   (𝑏 = 𝑒 → (𝜑𝜂))    &   (𝑐 = 𝑓 → (𝜃𝜓))       (∃!𝑎𝑉 ∃!𝑏𝑉 𝜑 → ∃𝑎𝑉𝑑𝑉𝑏𝑉𝑒𝑉𝑓𝑉 ((𝜂 ∧ ((𝜒 ∧ ∀𝑐𝑉 (𝜏𝑏 = 𝑐)) → 𝑎 = 𝑑)) ∧ (𝜓𝑒 = 𝑓)))

Theorem2reuimp 42160* 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 → (𝜑𝜃))    &   (𝑎 = 𝑑 → (𝜑𝜒))    &   (𝑎 = 𝑑 → (𝜃𝜏))    &   (𝑏 = 𝑒 → (𝜑𝜂))    &   (𝑐 = 𝑓 → (𝜃𝜓))       ((𝑉 ≠ ∅ ∧ ∃!𝑎𝑉 ∃!𝑏𝑉 𝜑) → ∃𝑎𝑉𝑑𝑉𝑏𝑉𝑒𝑉𝑓𝑉𝑐𝑉 ((𝜒 ∧ (𝜏𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑𝑒 = 𝑓)))))

20.36.4  Alternative definitions of function and operation values

The current definition of the value (𝐹𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6143) assures that this value is always a set, see fex 6761. 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 6476 and fvprc 6439).

Although it is very convenient for many theorems on functions and their proofs, there are some cases in which from (𝐹𝐴) = ∅ alone it cannot be decided/derived whether (𝐹𝐴) 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 𝐹𝐴 ∈ dom 𝐹 (see, for example, ndmfvrcl 6477).

To avoid such an ambiguity, an alternative definition (𝐹'''𝐴) (see df-afv 42165) would be possible which evaluates to the universal class ((𝐹'''𝐴) = V) if it is not meaningful (see afvnfundmuv 42184, ndmafv 42185, afvprc 42189 and nfunsnafv 42187), and which corresponds to the current definition ((𝐹𝐴) = (𝐹'''𝐴)) if it is (see afvfundmfveq 42183). That means (𝐹'''𝐴) = V → (𝐹𝐴) = ∅ (see afvpcfv0 42191), but (𝐹𝐴) = ∅ → (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 (𝐹𝐴) 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 6143 of (𝐹𝐴), we see that analogues for the following 8 theorems can be proven using the alternative definition: fveq1 6445-> afveq1 42179, fveq2 6446-> afveq2 42180, nffv 6456-> nfafv 42181, csbfv12 6490-> csbafv12g , fvres 6465-> afvres 42217, rlimdm 14690-> rlimdmafv 42222, tz6.12-1 6468-> tz6.12-1-afv 42219, fveu 6437-> afveu 42198.

Three theorems proved by directly using df-fv 6143 are within a mathbox (fvsb 39614) or not used (isumclim3 14895, avril1 27894).

However, the remaining 8 theorems proved by directly using df-fv 6143 are used more or less often:

* fvex 6459: used in about 1750 proofs.

* tz6.12-1 6468: root theorem of many theorems which have not a strict analogue, and which are used many times: fvprc 6439 (used in about 127 proofs), tz6.12i 6472 (used - indirectly via fvbr0 6473 and fvrn0 6474- in 18 proofs, and in fvclss 6772 used in fvclex 7417 used in fvresex 7418, which is not used!), dcomex 9604 (used in 4 proofs), ndmfv 6476 (used in 86 proofs) and nfunsn 6484 (used by dffv2 6531 which is not used).

* fv2 6441: only used by elfv 6444, which is only used by fv3 6464, which is not used.

* dffv3 6442: used by dffv4 6443 (the previous "df-fv"), which now is only used in deprecated (usage discouraged) theorems or within mathboxes (csbfv12gALTVD 40072), by shftval 14221 (itself used in 9 proofs), by dffv5 32620 (mathbox) and by fvco2 6533, which has the analogue afvco2 42221.

* fvopab5 6572: used only by ajval 28289 (not used) and by adjval 29321 ( used - indirectly - in 9 proofs).

* zsum 14856: used (via isum 14857, sum0 14859 and fsumsers 14866) in more than 90 proofs.

* isumshft 14975: used in pserdv2 24621 and (via logtayl 24843) 4 other proofs.

* ovtpos 7649: 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 6441, dffv3 6442, fvopab5 6572, zsum 14856, isumshft 14975 and ovtpos 7649 are not critical or are, hopefully, also valid for the alternative definition, fvex 6459 and tz6.12-1 6468 (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 42166.

Syntaxwdfat 42161 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/forum/#!topic/metamath/cteNUppB6A4).
wff 𝐹 defAt 𝐴

Syntaxcafv 42162 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 6143), 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 42177, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 6143 and df-ima 5368. 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/forum/#!topic/metamath/cteNUppB6A4).
class (𝐹'''𝐴)

Syntaxcaov 42163 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 6925.
class ((𝐴𝐹𝐵))

Definitiondf-dfat 42164 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})))

Definitiondf-afv 42165* Alternative definition of the value of a function, (𝐹'''𝐴), also known as function application. In contrast to (𝐹𝐴) = ∅ (see df-fv 6143 and ndmfv 6476), (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)

Definitiondf-aov 42166 Define the value of an operation. In contrast to df-ov 6925, the alternative definition for a function value (see df-afv 42165) 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⟩)

20.36.4.1  Restricted quantification (extension)

Theoremralbinrald 42167* Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)
(𝜑𝑋𝐴)    &   (𝑥𝐴𝑥 = 𝑋)    &   (𝑥 = 𝑋 → (𝜓𝜃))       (𝜑 → (∀𝑥𝐴 𝜓𝜃))

20.36.4.2  The universal class (extension)

Theoremnvelim 42168 If a class is the universal class it doesn't belong to any class, generalization of nvel 5035. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐴 = V → ¬ 𝐴𝐵)

20.36.4.3  Introduce the Axiom of Power Sets (extension)

Theoremalneu 42169 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)

Theoremeu2ndop1stv 42170* 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.)
(∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V)

20.36.4.4  Predicate "defined at"

Theoremdfateq12d 42171 Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))

Theoremnfdfat 42172 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, C_, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐹    &   𝑥𝐴       𝑥 𝐹 defAt 𝐴

Theoremdfdfat2 42173* 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))

Theoremfundmdfat 42174 A function is defined at any element of its domain. (Contributed by AV, 2-Sep-2022.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)

Theoremdfatprc 42175 A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.)
𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)

Theoremdfatelrn 42176 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 → (𝐹𝐴) ∈ ran 𝐹)

20.36.4.5  Alternative definition of the value of a function

Theoremdfafv2 42177 Alternative definition of (𝐹'''𝐴) using (𝐹𝐴) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.) (Revised by AV, 25-Aug-2022.)
(𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)

Theoremafveq12d 42178 Equality deduction for function value, analogous to fveq12d 6453. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))

Theoremafveq1 42179 Equality theorem for function value, analogous to fveq1 6445. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹 = 𝐺 → (𝐹'''𝐴) = (𝐺'''𝐴))

Theoremafveq2 42180 Equality theorem for function value, analogous to fveq1 6445. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴 = 𝐵 → (𝐹'''𝐴) = (𝐹'''𝐵))

Theoremnfafv 42181 Bound-variable hypothesis builder for function value, analogous to nffv 6456. To prove a deduction version of this analogous to nffvd 6458 is not easily possible because a deduction version of nfdfat 42172 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹'''𝐴)

Theoremcsbafv12g 42182 Move class substitution in and out of a function value, analogous to csbfv12 6490, with a direct proof proposed by Mario Carneiro, analogous to csbov123 6963. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))

Theoremafvfundmfveq 42183 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) = (𝐹𝐴))

Theoremafvnfundmuv 42184 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)

Theoremndmafv 42185 The value of a class outside its domain is the universe, compare with ndmfv 6476. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)

Theoremafvvdm 42186 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) ∈ 𝐵𝐴 ∈ dom 𝐹)

Theoremnfunsnafv 42187 If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6484. (Contributed by Alexander van der Vekens, 25-May-2017.)
(¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)

Theoremafvvfunressn 42188 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}))

Theoremafvprc 42189 A function's value at a proper class is the universe, compare with fvprc 6439. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ V → (𝐹'''𝐴) = V)

Theoremafvvv 42190 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) ∈ 𝐵𝐴 ∈ V)

Theoremafvpcfv0 42191 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 → (𝐹𝐴) = ∅)

Theoremafvnufveq 42192 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) = (𝐹𝐴))

Theoremafvvfveq 42193 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) = (𝐹𝐴))

Theoremafv0fv0 42194 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) = ∅ → (𝐹𝐴) = ∅)

Theoremafvfvn0fveq 42195 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.)
((𝐹𝐴) ≠ ∅ → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafv0nbfvbi 42196 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 ↔ (𝐹𝐴) ∈ 𝐵))

Theoremafvfv0bi 42197 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.)
((𝐹𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))

Theoremafveu 42198* The value of a function at a unique point, analogous to fveu 6437. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})

Theoremfnbrafvb 42199 Equivalence of function value and binary relation, analogous to fnbrfvb 6495. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶𝐵𝐹𝐶))

Theoremfnopafvb 42200 Equivalence of function value and ordered pair membership, analogous to fnopfvb 6496. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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