Theorem sn-iotaval

Description: Version of iotaval using df-iota instead of dfiota2 . (Contributed by SN, 6-Nov-2024)

	Ref	Expression
Assertion	sn-iotaval	⊢ ( { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( ℩ 𝑥 𝜑 ) = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		df-iota	⊢ ( ℩ 𝑥 𝜑 ) = ∪ { 𝑤 ∣ { 𝑥 ∣ 𝜑 } = { 𝑤 } }
2		eqeq1	⊢ ( { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( { 𝑥 ∣ 𝜑 } = { 𝑤 } ↔ { 𝑦 } = { 𝑤 } ) )
3		sneqbg	⊢ ( 𝑦 ∈ V → ( { 𝑦 } = { 𝑤 } ↔ 𝑦 = 𝑤 ) )
4	3	elv	⊢ ( { 𝑦 } = { 𝑤 } ↔ 𝑦 = 𝑤 )
5		equcom	⊢ ( 𝑦 = 𝑤 ↔ 𝑤 = 𝑦 )
6	4 5	bitri	⊢ ( { 𝑦 } = { 𝑤 } ↔ 𝑤 = 𝑦 )
7	2 6	bitrdi	⊢ ( { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( { 𝑥 ∣ 𝜑 } = { 𝑤 } ↔ 𝑤 = 𝑦 ) )
8	7	alrimiv	⊢ ( { 𝑥 ∣ 𝜑 } = { 𝑦 } → ∀ 𝑤 ( { 𝑥 ∣ 𝜑 } = { 𝑤 } ↔ 𝑤 = 𝑦 ) )
9		uniabio	⊢ ( ∀ 𝑤 ( { 𝑥 ∣ 𝜑 } = { 𝑤 } ↔ 𝑤 = 𝑦 ) → ∪ { 𝑤 ∣ { 𝑥 ∣ 𝜑 } = { 𝑤 } } = 𝑦 )
10	8 9	syl	⊢ ( { 𝑥 ∣ 𝜑 } = { 𝑦 } → ∪ { 𝑤 ∣ { 𝑥 ∣ 𝜑 } = { 𝑤 } } = 𝑦 )
11	1 10	eqtrid	⊢ ( { 𝑥 ∣ 𝜑 } = { 𝑦 } → ( ℩ 𝑥 𝜑 ) = 𝑦 )