Theorem dfid4 5566

Description: The identity function expressed 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 2013	. . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
2		vex 3470	. . . . 5 ⊢ 𝑥 ∈ V
3	2	biantrur 530	. . . 4 ⊢ (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
4	1, 3	bitri 275	. . 3 ⊢ (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
5	4	opabbii 5206	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
6		df-id 5565	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
7		df-mpt 5223	. 2 ⊢ (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
8	5, 6, 7	3eqtr4i 2762	1 ⊢ I = (𝑥 ∈ V ↦ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466  {copab 5201   ↦ cmpt 5222   I cid 5564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-opab 5202  df-mpt 5223  df-id 5565

This theorem is referenced by:  dfid5  14976  dfid6  14977  dfid7  42912