Theorem dfid4 5594

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 2017	. . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
2		vex 3492	. . . . 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 5233	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
6		df-id 5593	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
7		df-mpt 5250	. 2 ⊢ (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
8	5, 6, 7	3eqtr4i 2778	1 ⊢ I = (𝑥 ∈ V ↦ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  {copab 5228   ↦ cmpt 5249   I cid 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-opab 5229  df-mpt 5250  df-id 5593

This theorem is referenced by:  dfid5  15076  dfid6  15077  dfid7  43574