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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | aiotaexb 47101 | 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 47102 | 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 47103 | This is to df-aiota 47097 what iotauni 6536 is to df-iota 6514 (it uses intersection like df-aiota 47097, similar to iotauni 6536 using union like df-iota 6514; we could also prove an analogous result using union here too, in the same way that we have iotaint 6537). (Contributed by BJ, 31-Aug-2024.) |
| ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) | ||
| Theorem | dfaiota3 47104 | Alternate definition of ℩': this is to df-aiota 47097 what dfiota4 6553 is to df-iota 6514. operation using the if operator. It is simpler than df-aiota 47097 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 47105 | 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 47106 | 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 47107* | 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 47108* | 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 47050. (Contributed by AV, 25-Aug-2022.) |
| ⊢ (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅ | ||
| Theorem | aiota0ndef 47109* | 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 47051, where the iota still is a set (the empty set). (Contributed by AV, 25-Aug-2022.) |
| ⊢ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V | ||
| Theorem | r19.32 47110 | 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 47111* | An equivalent expression for restricted existence, analogous to exsb 2362. (Contributed by Alexander van der Vekens, 1-Jul-2017.) |
| ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
| Theorem | rexrsb 47112* | An equivalent expression for restricted existence, analogous to exsb 2362. (Contributed by Alexander van der Vekens, 1-Jul-2017.) |
| ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 → 𝜑)) | ||
| Theorem | 2rexsb 47113* | An equivalent expression for double restricted existence, analogous to rexsb 47111. (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 47114* | An equivalent expression for double restricted existence, analogous to 2exsb 2363. (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 47115* | Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3368. (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 47116* | Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3369. (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 47117 | 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 47118 | Second example for a theorem about a restricted universal quantification in which the restricting class depends on the bound variable: all subsets of a set are sets. (Contributed by AV, 24-Jun-2023.) |
| ⊢ ∀𝑥 ∈ 𝒫 𝑥𝑥 ∈ V | ||
| Theorem | reuf1odnf 47119* | There is exactly one element in each of two isomorphic sets. Variant of reuf1od 47120 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 47120* | 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 47121* | There is a set which is equal to one of two other sets iff the other sets are equal. (Contributed by AV, 24-Jan-2023.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∃!𝑥 ∈ 𝑉 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ 𝐴 = 𝐵)) | ||
| Theorem | 2reu3 47122* | Double restricted existential uniqueness, analogous to 2eu3 2654. (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 47123* | Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2658. (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 47124* | Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2659. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥 ∈ 𝐴∃!𝑦 ∈ 𝐵 using 2reu7 47123. (Contributed by Alexander van der Vekens, 2-Jul-2017.) |
| ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑)) | ||
| Theorem | 2reu8i 47125* | Implication of a double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, see also 2reu8 47124. 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 47126* | 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 47127* | Implication of a double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification if the class of the quantified elements is not empty. (Contributed by AV, 13-Mar-2023.) |
| ⊢ (𝑏 = 𝑐 → (𝜑 ↔ 𝜃)) & ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜒)) & ⊢ (𝑎 = 𝑑 → (𝜃 ↔ 𝜏)) & ⊢ (𝑏 = 𝑒 → (𝜑 ↔ 𝜂)) & ⊢ (𝑐 = 𝑓 → (𝜃 ↔ 𝜓)) ⇒ ⊢ ((𝑉 ≠ ∅ ∧ ∃!𝑎 ∈ 𝑉 ∃!𝑏 ∈ 𝑉 𝜑) → ∃𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ∃𝑒 ∈ 𝑉 ∀𝑓 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝜒 ∧ (𝜏 → 𝑏 = 𝑐)) → (𝜓 → (𝜂 ∧ (𝑎 = 𝑑 ∧ 𝑒 = 𝑓))))) | ||
The current definition of the value (𝐹‘𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6569) assures that this value is always a set, see fex 7246. 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 6941 and fvprc 6898). 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 6942). To avoid such an ambiguity, an alternative definition (𝐹'''𝐴) (see df-afv 47132) would be possible which evaluates to the universal class ((𝐹'''𝐴) = V) if it is not meaningful (see afvnfundmuv 47151, ndmafv 47152, afvprc 47156 and nfunsnafv 47154), and which corresponds to the current definition ((𝐹‘𝐴) = (𝐹'''𝐴)) if it is (see afvfundmfveq 47150). That means (𝐹'''𝐴) = V → (𝐹‘𝐴) = ∅ (see afvpcfv0 47158), 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 6569 of (𝐹‘𝐴), we see that analogues for the following 8 theorems can be proven using the alternative definition: fveq1 6905-> afveq1 47146, fveq2 6906-> afveq2 47147, nffv 6916-> nfafv 47148, csbfv12 6954-> csbafv12g , fvres 6925-> afvres 47184, rlimdm 15587-> rlimdmafv 47189, tz6.12-1 6929-> tz6.12-1-afv 47186, fveu 6895-> afveu 47165. Three theorems proved by directly using df-fv 6569 are within a mathbox (fvsb 44471) or not used (isumclim3 15795, avril1 30482). However, the remaining 8 theorems proved by directly using df-fv 6569 are used more or less often: * fvex 6919: used in about 1750 proofs. * tz6.12-1 6929: root theorem of many theorems which have not a strict analogue, and which are used many times: fvprc 6898 (used in about 127 proofs), tz6.12i 6934 (used - indirectly via fvbr0 6935 and fvrn0 6936- in 18 proofs, and in fvclss 7261 used in fvclex 7983 used in fvresex 7984, which is not used!), dcomex 10487 (used in 4 proofs), ndmfv 6941 (used in 86 proofs) and nfunsn 6948 (used by dffv2 7004 which is not used). * fv2 6901: only used by elfv 6904, which is only used by fv3 6924, which is not used. * dffv3 6902: used by dffv4 6903 (the previous "df-fv"), which now is only used in deprecated (usage discouraged) theorems or within mathboxes (csbfv12gALTVD 44919), by shftval 15113 (itself used in 9 proofs), by dffv5 35925 (mathbox) and by fvco2 7006, which has the analogue afvco2 47188. * fvopab5 7049: used only by ajval 30880 (not used) and by adjval 31909 (used - indirectly - in 9 proofs). * zsum 15754: used (via isum 15755, sum0 15757 and fsumsers 15764) in more than 90 proofs. * isumshft 15875: used in pserdv2 26474 and (via logtayl 26702) 4 other proofs. * ovtpos 8266: 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 6901, dffv3 6902, fvopab5 7049, zsum 15754, isumshft 15875 and ovtpos 8266 are not critical or are, hopefully, also valid for the alternative definition, fvex 6919 and tz6.12-1 6929 (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 47133. For additional details, see https://groups.google.com/g/metamath/c/cteNUppB6A4 47133. | ||
| Syntax | wdfat 47128 | 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 47129 | 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 6569), 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 47144, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 6569 and df-ima 5698. 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 5698). |
| class (𝐹'''𝐴) | ||
| Syntax | caov 47130 | 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 7434. |
| class ((𝐴𝐹𝐵)) | ||
| Definition | df-dfat 47131 | 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 47132* | Alternative definition of the value of a function, (𝐹'''𝐴), also known as function application. In contrast to (𝐹‘𝐴) = ∅ (see df-fv 6569 and ndmfv 6941), (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 47133 | Define the value of an operation. In contrast to df-ov 7434, the alternative definition for a function value (see df-afv 47132) is used. By this, the value of the operation applied to two arguments is the universal class if the operation is not defined for these two arguments. There are still no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation 𝐹 and its arguments 𝐴 and 𝐵- will be useful for proving meaningful theorems. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | ||
| Theorem | ralbinrald 47134* | Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.) |
| ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝑥 ∈ 𝐴 → 𝑥 = 𝑋) & ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ 𝜃)) | ||
| Theorem | nvelim 47135 | If a class is the universal class it doesn't belong to any class, generalization of nvel 5316. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵) | ||
| Theorem | alneu 47136 | 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 47137* | If there is a unique second component in an ordered pair contained in a given set, the first component must be a set. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
| ⊢ (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → 𝐴 ∈ V) | ||
| Theorem | dfateq12d 47138 | 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 47139 | 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 47140* | 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 47141 | 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 47142 | A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.) |
| ⊢ (¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴) | ||
| Theorem | dfatelrn 47143 | The value of a function 𝐹 at a set 𝐴 is in the range of the function 𝐹 if 𝐹 is defined at 𝐴. (Contributed by AV, 1-Sep-2022.) |
| ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) ∈ ran 𝐹) | ||
| Theorem | dfafv2 47144 | 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 47145 | Equality deduction for function value, analogous to fveq12d 6913. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| ⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵)) | ||
| Theorem | afveq1 47146 | Equality theorem for function value, analogous to fveq1 6905. (Contributed by Alexander van der Vekens, 22-Jul-2017.) |
| ⊢ (𝐹 = 𝐺 → (𝐹'''𝐴) = (𝐺'''𝐴)) | ||
| Theorem | afveq2 47147 | Equality theorem for function value, analogous to fveq1 6905. (Contributed by Alexander van der Vekens, 22-Jul-2017.) |
| ⊢ (𝐴 = 𝐵 → (𝐹'''𝐴) = (𝐹'''𝐵)) | ||
| Theorem | nfafv 47148 | Bound-variable hypothesis builder for function value, analogous to nffv 6916. To prove a deduction version of this analogous to nffvd 6918 is not easily possible because a deduction version of nfdfat 47139 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥(𝐹'''𝐴) | ||
| Theorem | csbafv12g 47149 | Move class substitution in and out of a function value, analogous to csbfv12 6954, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7475. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
| ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵)) | ||
| Theorem | afvfundmfveq 47150 | 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 47151 | 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 47152 | The value of a class outside its domain is the universe, compare with ndmfv 6941. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V) | ||
| Theorem | afvvdm 47153 | 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 47154 | If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6948. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V) | ||
| Theorem | afvvfunressn 47155 | 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 47156 | A function's value at a proper class is the universe, compare with fvprc 6898. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ (¬ 𝐴 ∈ V → (𝐹'''𝐴) = V) | ||
| Theorem | afvvv 47157 | 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 47158 | 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 47159 | 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 47160 | 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 47161 | 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 47162 | 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 47163 | 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 47164 | 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 47165* | The value of a function at a unique point, analogous to fveu 6895. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
| ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) | ||
| Theorem | fnbrafvb 47166 | Equivalence of function value and binary relation, analogous to fnbrfvb 6959. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) | ||
| Theorem | fnopafvb 47167 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6960. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐹)) | ||
| Theorem | funbrafvb 47168 | Equivalence of function value and binary relation, analogous to funbrfvb 6962. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | ||
| Theorem | funopafvb 47169 | Equivalence of function value and ordered pair membership, analogous to funopfvb 6963. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | ||
| Theorem | funbrafv 47170 | The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6957. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)) | ||
| Theorem | funbrafv2b 47171 | Function value in terms of a binary relation, analogous to funbrfv2b 6966. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵))) | ||
| Theorem | dfafn5a 47172* | Representation of a function in terms of its values, analogous to dffn5 6967 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))) | ||
| Theorem | dfafn5b 47173* | Representation of a function in terms of its values, analogous to dffn5 6967 (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 47174* | The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6968. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)}) | ||
| Theorem | afvelrnb 47175* | A member of a function's range is a value of the function, analogous to fvelrnb 6969 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 47176* | A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6969. (Contributed by Alexander van der Vekens, 1-Jun-2017.) |
| ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵)) | ||
| Theorem | dfaimafn 47177* | Alternate definition of the image of a function, analogous to dfimafn 6971. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦}) | ||
| Theorem | dfaimafn2 47178* | Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6972. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)}) | ||
| Theorem | afvelima 47179* | Function value in an image, analogous to fvelima 6974. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹'''𝑥) = 𝐴) | ||
| Theorem | afvelrn 47180 | A function's value belongs to its range, analogous to fvelrn 7096. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹) | ||
| Theorem | fnafvelrn 47181 | A function's value belongs to its range, analogous to fnfvelrn 7100. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) ∈ ran 𝐹) | ||
| Theorem | fafvelcdm 47182 | A function's value belongs to its codomain, analogous to ffvelcdm 7101. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ 𝐵) | ||
| Theorem | ffnafv 47183* | A function maps to a class to which all values belong, analogous to ffnfv 7139. (Contributed by Alexander van der Vekens, 25-May-2017.) |
| ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵)) | ||
| Theorem | afvres 47184 | The value of a restricted function, analogous to fvres 6925. (Contributed by Alexander van der Vekens, 22-Jul-2017.) |
| ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴)) | ||
| Theorem | tz6.12-afv 47185* | Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6931. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
| ⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹'''𝐴) = 𝑦) | ||
| Theorem | tz6.12-1-afv 47186* | Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6929. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
| ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦) | ||
| Theorem | dmfcoafv 47187 | Domains of a function composition, analogous to dmfco 7005. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
| ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹)) | ||
| Theorem | afvco2 47188 | Value of a function composition, analogous to fvco2 7006. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
| ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋))) | ||
| Theorem | rlimdmafv 47189 | Two ways to express that a function has a limit, analogous to rlimdm 15587. (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))) | ||
| Theorem | aoveq123d 47190 | Equality deduction for operation value, analogous to oveq123d 7452. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| ⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) ) | ||
| Theorem | nfaov 47191 | Bound-variable hypothesis builder for operation value, analogous to nfov 7461. To prove a deduction version of this analogous to nfovd 7460 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 47148). (Contributed by Alexander van der Vekens, 26-May-2017.) |
| ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵)) | ||
| Theorem | csbaovg 47192 | 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 47193 | 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 47194 | 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 47195 | The value of an operation outside its domain, analogous to ndmafv 47152. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V) | ||
| Theorem | ndmaovg 47196 | The value of an operation outside its domain, analogous to ndmovg 7616. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V) | ||
| Theorem | aovvdm 47197 | 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 47198 | 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 47199 | 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 47200 | The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7469. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| ⊢ Rel dom 𝐹 ⇒ ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V) | ||
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