Theorem dfid4 5353

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 2003	. . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
2		vex 3439	. . . . 5 ⊢ 𝑥 ∈ V
3	2	biantrur 531	. . . 4 ⊢ (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
4	1, 3	bitri 276	. . 3 ⊢ (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
5	4	opabbii 5031	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
6		df-id 5351	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
7		df-mpt 5044	. 2 ⊢ (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
8	5, 6, 7	3eqtr4i 2828	1 ⊢ I = (𝑥 ∈ V ↦ 𝑥)

Syntax hints:   ∧ wa 396   = wceq 1522   ∈ wcel 2080  Vcvv 3436  {copab 5026   ↦ cmpt 5043   I cid 5350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-12 2140  ax-ext 2768

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1763  df-nf 1767  df-sb 2042  df-clab 2775  df-cleq 2787  df-clel 2862  df-v 3438  df-opab 5027  df-mpt 5044  df-id 5351

This theorem is referenced by:  dfid5  14220  dfid6  14221  cnmptid  21953  dfid7  39470