Theorem dfid4 4955

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

Assertion

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

Proof of Theorem dfid4

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equcom 1932	. . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
2		vex 3176	. . . . 5 ⊢ 𝑥 ∈ V
3	2	biantrur 526	. . . 4 ⊢ (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
4	1, 3	bitri 263	. . 3 ⊢ (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
5	4	opabbii 4649	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
6		df-id 4953	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
7		df-mpt 4645	. 2 ⊢ (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
8	5, 6, 7	3eqtr4i 2642	1 ⊢ I = (𝑥 ∈ V ↦ 𝑥)

Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {copab 4642   ↦ cmpt 4643   I cid 4948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-opab 4644  df-mpt 4645  df-id 4953

This theorem is referenced by:  dfid5  13615  dfid6  13616  dfid7  36938