Theorem dfid4

Description: The identity function expressed using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017)

	Ref	Expression
Assertion	dfid4	⊢ I = ( 𝑥 ∈ V ↦ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		equcom	⊢ ( 𝑥 = 𝑦 ↔ 𝑦 = 𝑥 )
2		vex	⊢ 𝑥 ∈ V
3	2	biantrur	⊢ ( 𝑦 = 𝑥 ↔ ( 𝑥 ∈ V ∧ 𝑦 = 𝑥 ) )
4	1 3	bitri	⊢ ( 𝑥 = 𝑦 ↔ ( 𝑥 ∈ V ∧ 𝑦 = 𝑥 ) )
5	4	opabbii	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 = 𝑦 } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ V ∧ 𝑦 = 𝑥 ) }
6		df-id	⊢ I = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 = 𝑦 }
7		df-mpt	⊢ ( 𝑥 ∈ V ↦ 𝑥 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ V ∧ 𝑦 = 𝑥 ) }
8	5 6 7	3eqtr4i	⊢ I = ( 𝑥 ∈ V ↦ 𝑥 )