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Mirrors > Home > MPE Home > Th. List > df-iota | Structured version Visualization version GIF version |
Description: Define Russell's
definition description binder, which can be read as
"the unique 𝑥 such that 𝜑", where 𝜑
ordinarily contains
𝑥 as a free variable. Our definition
is meaningful only when there
is exactly one 𝑥 such that 𝜑 is true (see iotaval 6524);
otherwise, it evaluates to the empty set (see iotanul 6531). Russell used
the inverted iota symbol ℩ to represent
the binder.
Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use riotacl2 7399 (or iotacl 6539 for unbounded iota), as demonstrated in the proof of supub 9490. This can be easier than applying riotasbc 7401 or a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Ref | Expression |
---|---|
df-iota | ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | 1, 2 | cio 6503 | . 2 class (℩𝑥𝜑) |
4 | 1, 2 | cab 2705 | . . . . 5 class {𝑥 ∣ 𝜑} |
5 | vy | . . . . . . 7 setvar 𝑦 | |
6 | 5 | cv 1532 | . . . . . 6 class 𝑦 |
7 | 6 | csn 4632 | . . . . 5 class {𝑦} |
8 | 4, 7 | wceq 1533 | . . . 4 wff {𝑥 ∣ 𝜑} = {𝑦} |
9 | 8, 5 | cab 2705 | . . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} |
10 | 9 | cuni 4912 | . 2 class ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} |
11 | 3, 10 | wceq 1533 | 1 wff (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} |
Colors of variables: wff setvar class |
This definition is referenced by: dfiota2 6506 cbviotavw 6513 iotaeq 6518 iotabi 6519 iotaval2 6521 iotanul2 6523 dffv4 6899 dfiota3 35552 sn-iotalemcor 41741 reuabaiotaiota 46496 |
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