P. 459 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45801-45900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iotan0aiotaex 45801	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 45802	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 45803*	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 45804*	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 45748. (Contributed by AV, 25-Aug-2022.)
⊢ (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅
Theorem	aiota0ndef 45805*	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 45749, where the iota still is a set (the empty set). (Contributed by AV, 25-Aug-2022.)
⊢ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V
21.45.3  Double restricted existential uniqueness
21.45.3.1  Restricted quantification (extension)
Theorem	r19.32 45806	Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3192. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	rexsb 45807*	An equivalent expression for restricted existence, analogous to exsb 2356. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	rexrsb 45808*	An equivalent expression for restricted existence, analogous to exsb 2356. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑))
Theorem	2rexsb 45809*	An equivalent expression for double restricted existence, analogous to rexsb 45807. (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 45810*	An equivalent expression for double restricted existence, analogous to 2exsb 2357. (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 45811*	Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3365. (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 45812*	Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3366. (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 45813	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 45814	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
21.45.3.2  Restricted uniqueness and "at most one" quantification
Theorem	reuf1odnf 45815*	There is exactly one element in each of two isomorphic sets. Variant of reuf1od 45816 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 45816*	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 45817*	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))
21.45.3.3  Analogs to Existential uniqueness (double quantification)
Theorem	2reu3 45818*	Double restricted existential uniqueness, analogous to 2eu3 2650. (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 45819*	Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2654. (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 45820*	Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2655. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥 ∈ 𝐴∃!𝑦 ∈ 𝐵 using 2reu7 45819. (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))
21.45.3.4  Additional theorems for double restricted existential uniqueness
Theorem	2reu8i 45821*	Implication of a double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, see also 2reu8 45820. 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 45822*	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 45823*	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)))))
21.45.4  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 6552) assures that this value is always a set, see fex 7228. 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 6927 and fvprc 6884). 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 6928). To avoid such an ambiguity, an alternative definition (𝐹'''𝐴) (see df-afv 45828) would be possible which evaluates to the universal class ((𝐹'''𝐴) = V) if it is not meaningful (see afvnfundmuv 45847, ndmafv 45848, afvprc 45852 and nfunsnafv 45850), and which corresponds to the current definition ((𝐹‘𝐴) = (𝐹'''𝐴)) if it is (see afvfundmfveq 45846). That means (𝐹'''𝐴) = V → (𝐹‘𝐴) = ∅ (see afvpcfv0 45854), 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 6552 of (𝐹‘𝐴), we see that analogues for the following 8 theorems can be proven using the alternative definition: fveq1 6891-> afveq1 45842, fveq2 6892-> afveq2 45843, nffv 6902-> nfafv 45844, csbfv12 6940-> csbafv12g , fvres 6911-> afvres 45880, rlimdm 15495-> rlimdmafv 45885, tz6.12-1 6915-> tz6.12-1-afv 45882, fveu 6881-> afveu 45861. Three theorems proved by directly using df-fv 6552 are within a mathbox (fvsb 43211) or not used (isumclim3 15705, avril1 29716). However, the remaining 8 theorems proved by directly using df-fv 6552 are used more or less often: * fvex 6905: used in about 1750 proofs. * tz6.12-1 6915: root theorem of many theorems which have not a strict analogue, and which are used many times: fvprc 6884 (used in about 127 proofs), tz6.12i 6920 (used - indirectly via fvbr0 6921 and fvrn0 6922- in 18 proofs, and in fvclss 7241 used in fvclex 7945 used in fvresex 7946, which is not used!), dcomex 10442 (used in 4 proofs), ndmfv 6927 (used in 86 proofs) and nfunsn 6934 (used by dffv2 6987 which is not used). * fv2 6887: only used by elfv 6890, which is only used by fv3 6910, which is not used. * dffv3 6888: used by dffv4 6889 (the previous "df-fv"), which now is only used in deprecated (usage discouraged) theorems or within mathboxes (csbfv12gALTVD 43660), by shftval 15021 (itself used in 9 proofs), by dffv5 34896 (mathbox) and by fvco2 6989, which has the analogue afvco2 45884. * fvopab5 7031: used only by ajval 30114 (not used) and by adjval 31143 (used - indirectly - in 9 proofs). * zsum 15664: used (via isum 15665, sum0 15667 and fsumsers 15674) in more than 90 proofs. * isumshft 15785: used in pserdv2 25942 and (via logtayl 26168) 4 other proofs. * ovtpos 8226: 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 6887, dffv3 6888, fvopab5 7031, zsum 15664, isumshft 15785 and ovtpos 8226 are not critical or are, hopefully, also valid for the alternative definition, fvex 6905 and tz6.12-1 6915 (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 45829. For additional details, see https://groups.google.com/g/metamath/c/cteNUppB6A4 45829.
Syntax	wdfat 45824	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 45825	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 6552), 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 45840, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 6552 and df-ima 5690. 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 5690).
class (𝐹'''𝐴)
Syntax	caov 45826	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 7412.
class ((𝐴𝐹𝐵))
Definition	df-dfat 45827	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 45828*	Alternative definition of the value of a function, (𝐹'''𝐴), also known as function application. In contrast to (𝐹‘𝐴) = ∅ (see df-fv 6552 and ndmfv 6927), (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 45829	Define the value of an operation. In contrast to df-ov 7412, the alternative definition for a function value (see df-afv 45828) 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⟩)
21.45.4.1  Restricted quantification (extension)
Theorem	ralbinrald 45830*	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))
21.45.4.2  The universal class (extension)
Theorem	nvelim 45831	If a class is the universal class it doesn't belong to any class, generalization of nvel 5317. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵)
21.45.4.3  Introduce the Axiom of Power Sets (extension)
Theorem	alneu 45832	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 45833*	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)
21.45.4.4  Predicate "defined at"
Theorem	dfateq12d 45834	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 45835	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 45836*	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 45837	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 45838	A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.)
⊢ (¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)
Theorem	dfatelrn 45839	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)
21.45.4.5  Alternative definition of the value of a function
Theorem	dfafv2 45840	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 45841	Equality deduction for function value, analogous to fveq12d 6899. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))
Theorem	afveq1 45842	Equality theorem for function value, analogous to fveq1 6891. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
⊢ (𝐹 = 𝐺 → (𝐹'''𝐴) = (𝐺'''𝐴))
Theorem	afveq2 45843	Equality theorem for function value, analogous to fveq1 6891. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
⊢ (𝐴 = 𝐵 → (𝐹'''𝐴) = (𝐹'''𝐵))
Theorem	nfafv 45844	Bound-variable hypothesis builder for function value, analogous to nffv 6902. To prove a deduction version of this analogous to nffvd 6904 is not easily possible because a deduction version of nfdfat 45835 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(𝐹'''𝐴)
Theorem	csbafv12g 45845	Move class substitution in and out of a function value, analogous to csbfv12 6940, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7451. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵))
Theorem	afvfundmfveq 45846	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 45847	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 45848	The value of a class outside its domain is the universe, compare with ndmfv 6927. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)
Theorem	afvvdm 45849	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 45850	If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6934. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)
Theorem	afvvfunressn 45851	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 45852	A function's value at a proper class is the universe, compare with fvprc 6884. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (¬ 𝐴 ∈ V → (𝐹'''𝐴) = V)
Theorem	afvvv 45853	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 45854	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 45855	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 45856	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 45857	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 45858	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 45859	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 45860	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))
Theorem	afveu 45861*	The value of a function at a unique point, analogous to fveu 6881. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
Theorem	fnbrafvb 45862	Equivalence of function value and binary relation, analogous to fnbrfvb 6945. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))
Theorem	fnopafvb 45863	Equivalence of function value and ordered pair membership, analogous to fnopfvb 6946. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
Theorem	funbrafvb 45864	Equivalence of function value and binary relation, analogous to funbrfvb 6947. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	funopafvb 45865	Equivalence of function value and ordered pair membership, analogous to funopfvb 6948. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Theorem	funbrafv 45866	The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6943. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))
Theorem	funbrafv2b 45867	Function value in terms of a binary relation, analogous to funbrfv2b 6950. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵)))
Theorem	dfafn5a 45868*	Representation of a function in terms of its values, analogous to dffn5 6951 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))
Theorem	dfafn5b 45869*	Representation of a function in terms of its values, analogous to dffn5 6951 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))))
Theorem	fnrnafv 45870*	The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6952. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)})
Theorem	afvelrnb 45871*	A member of a function's range is a value of the function, analogous to fvelrnb 6953 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
Theorem	afvelrnb0 45872*	A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6953. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
Theorem	dfaimafn 45873*	Alternate definition of the image of a function, analogous to dfimafn 6955. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦})
Theorem	dfaimafn2 45874*	Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6956. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)})
Theorem	afvelima 45875*	Function value in an image, analogous to fvelima 6958. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹'''𝑥) = 𝐴)
Theorem	afvelrn 45876	A function's value belongs to its range, analogous to fvelrn 7079. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹)
Theorem	fnafvelrn 45877	A function's value belongs to its range, analogous to fnfvelrn 7083. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)
Theorem	fafvelcdm 45878	A function's value belongs to its codomain, analogous to ffvelcdm 7084. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ 𝐵)
Theorem	ffnafv 45879*	A function maps to a class to which all values belong, analogous to ffnfv 7118. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵))
Theorem	afvres 45880	The value of a restricted function, analogous to fvres 6911. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))
Theorem	tz6.12-afv 45881*	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6917. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)
Theorem	tz6.12-1-afv 45882*	Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6915. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦)
Theorem	dmfcoafv 45883	Domains of a function composition, analogous to dmfco 6988. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))
Theorem	afvco2 45884	Value of a function composition, analogous to fvco2 6989. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))
Theorem	rlimdmafv 45885	Two ways to express that a function has a limit, analogous to rlimdm 15495. (Contributed by Alexander van der Vekens, 27-Nov-2017.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝𝑟 ↔ 𝐹 ⇝𝑟 ( ⇝𝑟 '''𝐹)))
21.45.4.6  Alternative definition of the value of an operation
Theorem	aoveq123d 45886	Equality deduction for operation value, analogous to oveq123d 7430. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )
Theorem	nfaov 45887	Bound-variable hypothesis builder for operation value, analogous to nfov 7439. To prove a deduction version of this analogous to nfovd 7438 is not quickly possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of alternative operation values is based on are not available (see nfafv 45844). (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵))
Theorem	csbaovg 45888	Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌ ((𝐵𝐹𝐶)) = ((⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)) )
Theorem	aovfundmoveq 45889	If a class is a function restricted to an ordered pair of its domain, then the value of the operation on this pair is equal for both definitions. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝐹 defAt ⟨𝐴, 𝐵⟩ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
Theorem	aovnfundmuv 45890	If an ordered pair is not in the domain of a class or the class is not a function restricted to the ordered pair, then the operation value for this pair is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (¬ 𝐹 defAt ⟨𝐴, 𝐵⟩ → ((𝐴𝐹𝐵)) = V)
Theorem	ndmaov 45891	The value of an operation outside its domain, analogous to ndmafv 45848. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
Theorem	ndmaovg 45892	The value of an operation outside its domain, analogous to ndmovg 7590. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V)
Theorem	aovvdm 45893	If the operation value of a class for an ordered pair is a set, the ordered pair is contained in the domain of the class. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
Theorem	nfunsnaov 45894	If the restriction of a class to a singleton is not a function, its operation value is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (¬ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}) → ((𝐴𝐹𝐵)) = V)
Theorem	aovvfunressn 45895	If the operation 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, 26-May-2017.)
⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}))
Theorem	aovprc 45896	The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7447. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Rel dom 𝐹    ⇒   ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)
Theorem	aovrcl 45897	Reverse closure for an operation value, analogous to afvvv 45853. In contrast to ovrcl 7450, elementhood of the operation's value in a set is required, not containing an element. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Rel dom 𝐹    ⇒   ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	aovpcov0 45898	If the alternative value of the operation on an ordered pair is the universal class, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ( ((𝐴𝐹𝐵)) = V → (𝐴𝐹𝐵) = ∅)
Theorem	aovnuoveq 45899	The alternative value of the operation on an ordered pair equals the operation's value at this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ( ((𝐴𝐹𝐵)) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
Theorem	aovvoveq 45900	The alternative value of the operation on an ordered pair equals the operation's value on this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))