 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfid4 Structured version   Visualization version   GIF version

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.
StepHypRef Expression
1 equcom 1932 . . . 4 (𝑥 = 𝑦𝑦 = 𝑥)
2 vex 3176 . . . . 5 𝑥 ∈ V
32biantrur 526 . . . 4 (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
41, 3bitri 263 . . 3 (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
54opabbii 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)}
85, 6, 73eqtr4i 2642 1 I = (𝑥 ∈ V ↦ 𝑥)
 Colors of variables: wff setvar class 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
 Copyright terms: Public domain W3C validator