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Theorem dfid4 5533
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 2022 . . . 4 (𝑥 = 𝑦𝑦 = 𝑥)
2 vex 3450 . . . . 5 𝑥 ∈ V
32biantrur 532 . . . 4 (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
41, 3bitri 275 . . 3 (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
54opabbii 5173 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
6 df-id 5532 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
7 df-mpt 5190 . 2 (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
85, 6, 73eqtr4i 2775 1 I = (𝑥 ∈ V ↦ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  Vcvv 3446  {copab 5168  cmpt 5189   I cid 5531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3448  df-opab 5169  df-mpt 5190  df-id 5532
This theorem is referenced by:  dfid5  14913  dfid6  14914  dfid7  41891
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