P. 479 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47801-47900   *Has distinct variable group(s)

Type	Label	Description
Statement
21.49.1.4  Relations - extension
Theorem	eubrv 47801*	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 47802*	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 47803	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)
21.49.1.5  Definite description binder (inverted iota) - extension
Theorem	iota0def 47804*	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 47805*	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) = ∅
21.49.1.6  Functions - extension
Theorem	fveqvfvv 47806	If a function's value at an argument is the universal class (which can never be the case because of fvex 6895), 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 120). (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵)
Theorem	fnresfnco 47807	Composition of two functions, similar to fnco 6654. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐵)
Theorem	funcoressn 47808	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 47809	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 47810	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 47811*	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 47812*	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 47813*	There is exactly one value of a class which is a function restricted to a singleton, analogous to funeu 6562. 𝐴 ∈ V is required because otherwise ∃!𝑦𝐴𝐹𝑦, see brprcneu 6872. (Contributed by AV, 7-Sep-2022.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
Theorem	fresfo 47814	Conditions for a restriction to be an onto function. Part of fresf1o 32986. (Contributed by AV, 29-Sep-2024.)
⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–onto→𝐶)
Theorem	fsetsniunop 47815*	The class of all functions from a (proper) singleton into 𝐵 is the union of all the singletons of (proper) ordered pairs over the elements of 𝐵 as second component. (Contributed by AV, 13-Sep-2024.)
⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}})
Theorem	fsetabsnop 47816*	The class of all functions from a (proper) singleton into 𝐵 is the class of all the singletons of (proper) ordered pairs over the elements of 𝐵 as second component. (Contributed by AV, 13-Sep-2024.)
⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
Theorem	fsetsnf 47817*	The mapping of an element of a class to a singleton function is a function. (Contributed by AV, 13-Sep-2024.)
⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)
Theorem	fsetsnf1 47818*	The mapping of an element of a class to a singleton function is an injection. (Contributed by AV, 13-Sep-2024.)
⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1→𝐴)
Theorem	fsetsnfo 47819*	The mapping of an element of a class to a singleton function is a surjection. (Contributed by AV, 13-Sep-2024.)
⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–onto→𝐴)
Theorem	fsetsnf1o 47820*	The mapping of an element of a class to a singleton function is a bijection. (Contributed by AV, 13-Sep-2024.)
⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1-onto→𝐴)
Theorem	fsetsnprcnex 47821*	The class of all functions from a (proper) singleton into a proper class 𝐵 is not a set. (Contributed by AV, 13-Sep-2024.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ∉ V)
Theorem	cfsetssfset 47822	The class of constant functions is a subclass of the class of functions. (Contributed by AV, 13-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    ⇒   ⊢ 𝐹 ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}
Theorem	cfsetsnfsetfv 47823*	The function value of the mapping of the class of singleton functions into the class of constant functions. (Contributed by AV, 13-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝐻‘𝑋) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))
Theorem	cfsetsnfsetf 47824*	The mapping of the class of singleton functions into the class of constant functions is a function. (Contributed by AV, 14-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)
Theorem	cfsetsnfsetf1 47825*	The mapping of the class of singleton functions into the class of constant functions is an injection. (Contributed by AV, 14-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1→𝐹)
Theorem	cfsetsnfsetfo 47826*	The mapping of the class of singleton functions into the class of constant functions is a surjection. (Contributed by AV, 14-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–onto→𝐹)
Theorem	cfsetsnfsetf1o 47827*	The mapping of the class of singleton functions into the class of constant functions is a bijection. (Contributed by AV, 14-Sep-2024.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1-onto→𝐹)
Theorem	fsetprcnexALT 47828*	First version of proof for fsetprcnex 8856, which was much more complicated. (Contributed by AV, 14-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)
Theorem	fcoreslem1 47829	Lemma 1 for fcores 47833. (Contributed by AV, 17-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    ⇒   ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))
Theorem	fcoreslem2 47830	Lemma 2 for fcores 47833. (Contributed by AV, 17-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    ⇒   ⊢ (𝜑 → ran 𝑋 = 𝐸)
Theorem	fcoreslem3 47831	Lemma 3 for fcores 47833. (Contributed by AV, 13-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    ⇒   ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸)
Theorem	fcoreslem4 47832	Lemma 4 for fcores 47833. (Contributed by AV, 17-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    ⇒   ⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃)
Theorem	fcores 47833	Every composite function (𝐺 ∘ 𝐹) can be written as composition of restrictions of the composed functions (to their minimum domains). (Contributed by GL and AV, 17-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
Theorem	fcoresf1lem 47834	Lemma for fcoresf1 47835. (Contributed by AV, 18-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    ⇒   ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = (𝑌‘(𝑋‘𝑍)))
Theorem	fcoresf1 47835	If a composition is injective, then the restrictions of its components to the minimum domains are injective. (Contributed by GL and AV, 18-Sep-2024.) (Revised by AV, 7-Oct-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    &   ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝑃–1-1→𝐷)    ⇒   ⊢ (𝜑 → (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷))
Theorem	fcoresf1b 47836	A composition is injective iff the restrictions of its components to the minimum domains are injective. (Contributed by GL and AV, 7-Oct-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    ⇒   ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))
Theorem	fcoresfo 47837	If a composition is surjective, then the restriction of its first component to the minimum domain is surjective. (Contributed by AV, 17-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    &   ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)    ⇒   ⊢ (𝜑 → 𝑌:𝐸–onto→𝐷)
Theorem	fcoresfob 47838	A composition is surjective iff the restriction of its first component to the minimum domain is surjective. (Contributed by GL and AV, 7-Oct-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    ⇒   ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))
Theorem	fcoresf1ob 47839	A composition is bijective iff the restriction of its first component to the minimum domain is bijective and the restriction of its second component to the minimum domain is injective. (Contributed by GL and AV, 7-Oct-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    ⇒   ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1-onto→𝐷)))
Theorem	f1cof1blem 47840	Lemma for f1cof1b 47843 and focofob 47846. (Contributed by AV, 18-Sep-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)    &   ⊢ 𝑃 = (◡𝐹 “ 𝐶)    &   ⊢ 𝑋 = (𝐹 ↾ 𝑃)    &   ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)    &   ⊢ 𝑌 = (𝐺 ↾ 𝐸)    &   ⊢ (𝜑 → ran 𝐹 = 𝐶)    ⇒   ⊢ (𝜑 → ((𝑃 = 𝐴 ∧ 𝐸 = 𝐶) ∧ (𝑋 = 𝐹 ∧ 𝑌 = 𝐺)))
Theorem	3f1oss1 47841	The composition of three bijections as bijection from the image of the domain onto the image of the range of the middle bijection. (Contributed by AV, 15-Aug-2025.)
⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ((𝐻 ∘ 𝐺) ∘ ◡𝐹):(𝐹 “ 𝐶)–1-1-onto→(𝐻 “ 𝐷))
Theorem	3f1oss2 47842	The composition of three bijections as bijection from the image of the converse of the domain onto the image of the converse of the range of the middle bijection. (Contributed by AV, 15-Aug-2025.)
⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷))
Theorem	f1cof1b 47843	If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is injective iff 𝐹 and 𝐺 are both injective. (Contributed by GL and AV, 19-Sep-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷)))
Theorem	funfocofob 47844	If the domain of a function 𝐺 is a subset of the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 is surjective. (Contributed by GL and AV, 29-Sep-2024.)
⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))
Theorem	fnfocofob 47845	If the domain of a function 𝐺 equals the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 is surjective. (Contributed by GL and AV, 29-Sep-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))
Theorem	focofob 47846	If the domain of a function 𝐺 equals the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 and 𝐹 as function to the domain of 𝐺 are both surjective. Symmetric version of fnfocofob 47845 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 29-Sep-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ (𝐹:𝐴–onto→𝐶 ∧ 𝐺:𝐶–onto→𝐷)))
Theorem	f1ocof1ob 47847	If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is bijective iff 𝐹 and 𝐺 are both bijective. (Contributed by GL and AV, 7-Oct-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷)))
Theorem	f1ocof1ob2 47848	If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is bijective iff 𝐹 and 𝐺 are both bijective. Symmetric version of f1ocof1ob 47847 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 7-Oct-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ (𝐹:𝐴–1-1-onto→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷)))
21.49.2  Alternative for Russell's definition of a description binder
Syntax	caiota 47849	Extend class notation with an alternative for Russell's definition of a description binder (inverted iota).
class (℩'𝑥𝜑)
Theorem	aiotajust 47850*	Soundness justification theorem for df-aiota 47851. (Contributed by AV, 24-Aug-2022.)
⊢ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
Definition	df-aiota 47851*	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 47861); otherwise, it is not a set (see aiotaexb 47855), or even more concrete, it is the universe V (see aiotavb 47856). Since this is an alternative for df-iota 6493, 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 47855). 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 47804). Only the right to left implication would hold, see (negated) iotanul 6517. For defined cases, however, both definitions df-iota 6493 and df-aiota 47851 are equivalent, see reuaiotaiota 47854. (Proposed by BJ, 13-Aug-2022.) (Contributed by AV, 24-Aug-2022.)
⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
Theorem	dfaiota2 47852*	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 47853*	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 47854	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 47855	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 47856	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	aiotaint 47857	This is to df-aiota 47851 what iotauni 6514 is to df-iota 6493 (it uses intersection like df-aiota 47851, similar to iotauni 6514 using union like df-iota 6493; we could also prove an analogous result using union here too, in the same way that we have iotaint 6515). (Contributed by BJ, 31-Aug-2024.)
⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
Theorem	dfaiota3 47858	Alternate definition of ℩', using the if operator: this is to df-aiota 47851 what dfiota4 6529 is to df-iota 6493. It is simpler than df-aiota 47851 and uses no dummy variables, so it would be the preferred definition if ℩' becomes the description binder used in set.mm. (Contributed by BJ, 31-Aug-2024.)
⊢ (℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V)
Theorem	iotan0aiotaex 47859	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 47860	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 47861*	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 47862*	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 47804. (Contributed by AV, 25-Aug-2022.)
⊢ (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅
Theorem	aiota0ndef 47863*	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 47805, where the iota still is a set (the empty set). (Contributed by AV, 25-Aug-2022.)
⊢ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V
21.49.3  Double restricted existential uniqueness
21.49.3.1  Restricted quantification (extension)
Theorem	r19.32 47864	Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3198. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	rexsb 47865*	An equivalent expression for restricted existence, analogous to exsb 2391. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	rexrsb 47866*	An equivalent expression for restricted existence, analogous to exsb 2391. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑))
Theorem	2rexsb 47867*	An equivalent expression for double restricted existence, analogous to rexsb 47865. (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 47868*	An equivalent expression for double restricted existence, analogous to 2exsb 2392. (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 47869*	Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3357. (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 47870*	Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3358. (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	ralndv1 47871	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 47872	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
21.49.3.2  Restricted uniqueness and "at most one" quantification
Theorem	reuf1odnf 47873*	There is exactly one element in each of two isomorphic sets. Variant of reuf1od 47874 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 47874*	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 47875*	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))
21.49.3.3  Analogs to Existential uniqueness (double quantification)
Theorem	2reu3 47876*	Double restricted existential uniqueness, analogous to 2eu3 2681. (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 47877*	Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2685. (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 47878*	Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2686. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥 ∈ 𝐴∃!𝑦 ∈ 𝐵 using 2reu7 47877. (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))
21.49.3.4  Additional theorems for double restricted existential uniqueness
Theorem	2reu8i 47879*	Implication of a double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, see also 2reu8 47878. 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 47880*	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 47881*	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)))))
21.49.4  Alternative definitions of function and operation values The current definition of the value (𝐹‘𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6545) assures that this value is always a set, see fex 7225. 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 6914 and fvprc 6874). 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 6915). To avoid such an ambiguity, an alternative definition (𝐹'''𝐴) (see df-afv 47886) would be possible which evaluates to the universal class ((𝐹'''𝐴) = V) if it is not meaningful (see afvnfundmuv 47905, ndmafv 47906, afvprc 47910 and nfunsnafv 47908), and which corresponds to the current definition ((𝐹‘𝐴) = (𝐹'''𝐴)) if it is (see afvfundmfveq 47904). That means (𝐹'''𝐴) = V → (𝐹‘𝐴) = ∅ (see afvpcfv0 47912), 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 6545 of (𝐹‘𝐴), we see that analogues for the following 8 theorems can be proven using the alternative definition: fveq1 6881-> afveq1 47900, fveq2 6882-> afveq2 47901, nffv 6892-> nfafv 47902, csbfv12 6927-> csbafv12g , fvres 6901-> afvres 47938, rlimdm 15608-> rlimdmafv 47943, tz6.12-1 6905-> tz6.12-1-afv 47940, fveu 6871-> afveu 47919. Three theorems proved by directly using df-fv 6545 are within a mathbox (fvsb 45189) or not used (isumclim3 15816, avril1 30824). However, the remaining 8 theorems proved by directly using df-fv 6545 are used more or less often: * fvex 6895: used in about 1750 proofs. * tz6.12-1 6905: root theorem of many theorems which have not a strict analogue, and which are used many times: fvprc 6874 (used in about 127 proofs), tz6.12i 6908 (used - indirectly via fvbr0 6909 and fvrn0 6910- in 18 proofs, and in fvclss 7240 used in fvclex 7953 used in fvresex 7954, which is not used!), dcomex 10436 (used in 4 proofs), ndmfv 6914 (used in 86 proofs) and nfunsn 6921 (used by dffv2 6977 which is not used). * fv2 6877: only used by elfv 6880, which is only used by fv3 6900, which is not used. * dffv3 6878: used by dffv4 6879 (the previous "df-fv"), which now is only used in deprecated (usage discouraged) theorems or within mathboxes (csbfv12gALTVD 45636), by shftval 15117 (itself used in 9 proofs), by dffv5 36423 (mathbox) and by fvco2 6979, which has the analogue afvco2 47942. * fvopab5 7024: used only by ajval 31223 (not used) and by adjval 32252 (used - indirectly - in 9 proofs). * zsum 15775: used (via isum 15776, sum0 15778 and fsumsers 15785) in more than 90 proofs. * isumshft 15899: used in pserdv2 26603 and (via logtayl 26835) 4 other proofs. * ovtpos 8234: 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 6877, dffv3 6878, fvopab5 7024, zsum 15775, isumshft 15899 and ovtpos 8234 are not critical or are, hopefully, also valid for the alternative definition, fvex 6895 and tz6.12-1 6905 (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 47887. For additional details, see https://groups.google.com/g/metamath/c/cteNUppB6A4 47887.
Syntax	wdfat 47882	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 47883	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 6545), which, by the way, was intended to visualize that in many cases ‘ and " ' " are exchangeable, makes reading the theorems, especially those which use both definitions as dfafv2 47898, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 6545 and df-ima 5674. 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 5674).
class (𝐹'''𝐴)
Syntax	caov 47884	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 7414.
class ((𝐴𝐹𝐵))
Definition	df-dfat 47885	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 47886*	Alternative definition of the value of a function, (𝐹'''𝐴), also known as function application. In contrast to (𝐹‘𝐴) = ∅ (see df-fv 6545 and ndmfv 6914), (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 47887	Define the value of an operation. In contrast to df-ov 7414, the alternative definition for a function value (see df-afv 47886) 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⟩)
21.49.4.1  Restricted quantification (extension)
Theorem	ralbinrald 47888*	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))
21.49.4.2  The universal class (extension)
Theorem	nvelim 47889	If a class is the universal class it doesn't belong to any class, generalization of nvel 5282. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵)
21.49.4.3  Introduce the Axiom of Power Sets (extension)
Theorem	alneu 47890	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 47891*	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)
21.49.4.4  Predicate "defined at"
Theorem	dfateq12d 47892	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 47893	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 47894*	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 47895	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 47896	A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.)
⊢ (¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)
Theorem	dfatelrn 47897	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)
21.49.4.5  Alternative definition of the value of a function
Theorem	dfafv2 47898	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 47899	Equality deduction for function value, analogous to fveq12d 6889. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))
Theorem	afveq1 47900	Equality theorem for function value, analogous to fveq1 6881. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
⊢ (𝐹 = 𝐺 → (𝐹'''𝐴) = (𝐺'''𝐴))