P. 462 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 46101-46200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iotan0aiotaex 46101	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 46102	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 46103*	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 46104*	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 46048. (Contributed by AV, 25-Aug-2022.)
⊢ (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅
Theorem	aiota0ndef 46105*	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 46049, 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 46106	Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3190. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	rexsb 46107*	An equivalent expression for restricted existence, analogous to exsb 2354. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	rexrsb 46108*	An equivalent expression for restricted existence, analogous to exsb 2354. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑))
Theorem	2rexsb 46109*	An equivalent expression for double restricted existence, analogous to rexsb 46107. (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 46110*	An equivalent expression for double restricted existence, analogous to 2exsb 2355. (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 46111*	Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3363. (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 46112*	Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3364. (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 46113	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 46114	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 46115*	There is exactly one element in each of two isomorphic sets. Variant of reuf1od 46116 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 46116*	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 46117*	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 46118*	Double restricted existential uniqueness, analogous to 2eu3 2648. (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 46119*	Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2652. (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 46120*	Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2653. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥 ∈ 𝐴∃!𝑦 ∈ 𝐵 using 2reu7 46119. (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 46121*	Implication of a double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, see also 2reu8 46120. 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 46122*	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 46123*	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 7231. 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 46128) would be possible which evaluates to the universal class ((𝐹'''𝐴) = V) if it is not meaningful (see afvnfundmuv 46147, ndmafv 46148, afvprc 46152 and nfunsnafv 46150), and which corresponds to the current definition ((𝐹‘𝐴) = (𝐹'''𝐴)) if it is (see afvfundmfveq 46146). That means (𝐹'''𝐴) = V → (𝐹‘𝐴) = ∅ (see afvpcfv0 46154), 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 46142, fveq2 6892-> afveq2 46143, nffv 6902-> nfafv 46144, csbfv12 6940-> csbafv12g , fvres 6911-> afvres 46180, rlimdm 15500-> rlimdmafv 46185, tz6.12-1 6915-> tz6.12-1-afv 46182, fveu 6881-> afveu 46161. Three theorems proved by directly using df-fv 6552 are within a mathbox (fvsb 43514) or not used (isumclim3 15710, avril1 29980). 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 7244 used in fvclex 7948 used in fvresex 7949, which is not used!), dcomex 10445 (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 43963), by shftval 15026 (itself used in 9 proofs), by dffv5 35197 (mathbox) and by fvco2 6989, which has the analogue afvco2 46184. * fvopab5 7031: used only by ajval 30378 (not used) and by adjval 31407 (used - indirectly - in 9 proofs). * zsum 15669: used (via isum 15670, sum0 15672 and fsumsers 15679) in more than 90 proofs. * isumshft 15790: used in pserdv2 26175 and (via logtayl 26401) 4 other proofs. * ovtpos 8229: 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 15669, isumshft 15790 and ovtpos 8229 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 46129. For additional details, see https://groups.google.com/g/metamath/c/cteNUppB6A4 46129.
Syntax	wdfat 46124	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 46125	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 46140, 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 46126	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 7415.
class ((𝐴𝐹𝐵))
Definition	df-dfat 46127	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 46128*	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 46129	Define the value of an operation. In contrast to df-ov 7415, the alternative definition for a function value (see df-afv 46128) 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 46130*	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 46131	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 46132	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 46133*	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 46134	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 46135	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 46136*	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 46137	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 46138	A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.)
⊢ (¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)
Theorem	dfatelrn 46139	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 46140	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 46141	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 46142	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 46143	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 46144	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 46135 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(𝐹'''𝐴)
Theorem	csbafv12g 46145	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 7454. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵))
Theorem	afvfundmfveq 46146	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 46147	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 46148	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 46149	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 46150	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 46151	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 46152	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 46153	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 46154	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 46155	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 46156	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 46157	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 46158	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 46159	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 46160	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 46161*	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 46162	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 46163	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 46164	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 46165	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 46166	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 46167	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 46168*	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 46169*	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 46170*	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 46171*	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 46172*	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 46173*	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 46174*	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 46175*	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 46176	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 46177	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 46178	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 46179*	A function maps to a class to which all values belong, analogous to ffnfv 7121. (Contributed by Alexander van der Vekens, 25-May-2017.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵))
Theorem	afvres 46180	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 46181*	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 46182*	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 46183	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 46184	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 46185	Two ways to express that a function has a limit, analogous to rlimdm 15500. (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 46186	Equality deduction for operation value, analogous to oveq123d 7433. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )
Theorem	nfaov 46187	Bound-variable hypothesis builder for operation value, analogous to nfov 7442. To prove a deduction version of this analogous to nfovd 7441 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 46144). (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵))
Theorem	csbaovg 46188	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 46189	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 46190	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 46191	The value of an operation outside its domain, analogous to ndmafv 46148. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
Theorem	ndmaovg 46192	The value of an operation outside its domain, analogous to ndmovg 7593. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V)
Theorem	aovvdm 46193	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 46194	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 46195	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 46196	The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7450. (Contributed by Alexander van der Vekens, 26-May-2017.)
⊢ Rel dom 𝐹    ⇒   ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)
Theorem	aovrcl 46197	Reverse closure for an operation value, analogous to afvvv 46153. In contrast to ovrcl 7453, 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 46198	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 46199	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 46200	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))