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Theorem dfid4 5577
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.
StepHypRef Expression
1 equcom 2014 . . . 4 (𝑥 = 𝑦𝑦 = 𝑥)
2 vex 3475 . . . . 5 𝑥 ∈ V
32biantrur 530 . . . 4 (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
41, 3bitri 275 . . 3 (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
54opabbii 5215 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
6 df-id 5576 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
7 df-mpt 5232 . 2 (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
85, 6, 73eqtr4i 2766 1 I = (𝑥 ∈ V ↦ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wcel 2099  Vcvv 3471  {copab 5210  cmpt 5231   I cid 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-opab 5211  df-mpt 5232  df-id 5576
This theorem is referenced by:  dfid5  15006  dfid6  15007  dfid7  43042
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