Definition df-iota

Description: Define Russell's definition description binder, which can be read as "the unique x such that ph ", where ph ordinarily contains x as a free variable. Our definition is meaningful only when there is exactly one x such that ph is true (see iotaval ); otherwise, it evaluates to the empty set (see iotanul ). Russell used the inverted iota symbol iota 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 (or iotacl for unbounded iota), as demonstrated in the proof of supub . This can be easier than applying riotasbc 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
Assertion	df-iota	⊢ ( ℩ 𝑥 𝜑 ) = ∪ { 𝑦 ∣ { 𝑥 ∣ 𝜑 } = { 𝑦 } }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		vx	⊢ 𝑥
1		wph	⊢ 𝜑
2	1 0	cio	⊢ ( ℩ 𝑥 𝜑 )
3		vy	⊢ 𝑦
4	1 0	cab	⊢ { 𝑥 ∣ 𝜑 }
5	3	cv	⊢ 𝑦
6	5	csn	⊢ { 𝑦 }
7	4 6	wceq	⊢ { 𝑥 ∣ 𝜑 } = { 𝑦 }
8	7 3	cab	⊢ { 𝑦 ∣ { 𝑥 ∣ 𝜑 } = { 𝑦 } }
9	8	cuni	⊢ ∪ { 𝑦 ∣ { 𝑥 ∣ 𝜑 } = { 𝑦 } }
10	2 9	wceq	⊢ ( ℩ 𝑥 𝜑 ) = ∪ { 𝑦 ∣ { 𝑥 ∣ 𝜑 } = { 𝑦 } }