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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 6520);
otherwise, it evaluates to the empty set (see iotanul 6527). 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 7392 (or iotacl 6535 for unbounded iota), as demonstrated in the proof of supub 9484. This can be easier than applying riotasbc 7394 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 6499 | . 2 class (℩𝑥𝜑) |
4 | 1, 2 | cab 2702 | . . . . 5 class {𝑥 ∣ 𝜑} |
5 | vy | . . . . . . 7 setvar 𝑦 | |
6 | 5 | cv 1532 | . . . . . 6 class 𝑦 |
7 | 6 | csn 4630 | . . . . 5 class {𝑦} |
8 | 4, 7 | wceq 1533 | . . . 4 wff {𝑥 ∣ 𝜑} = {𝑦} |
9 | 8, 5 | cab 2702 | . . 3 class {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} |
10 | 9 | cuni 4909 | . 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 6502 cbviotavw 6509 iotaeq 6514 iotabi 6515 iotaval2 6517 iotanul2 6519 dffv4 6893 dfiota3 35650 sn-iotalemcor 41844 reuabaiotaiota 46605 |
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