Theorem dfid4 5559

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 2051	. . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
2		vex 3461	. . . . 5 ⊢ 𝑥 ∈ V
3	2	biantrur 540	. . . 4 ⊢ (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
4	1, 3	bitri 278	. . 3 ⊢ (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
5	4	opabbii 5180	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
6		df-id 5558	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
7		df-mpt 5195	. 2 ⊢ (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
8	5, 6, 7	3eqtr4i 2798	1 ⊢ I = (𝑥 ∈ V ↦ 𝑥)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  {copab 5175   ↦ cmpt 5194   I cid 5557

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-opab 5176  df-mpt 5195  df-id 5558

This theorem is used by:  dfid5  15084  dfid6  15085  dfid7  44371