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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.
StepHypRef Expression
1 equcom 2003 . . . 4 (𝑥 = 𝑦𝑦 = 𝑥)
2 vex 3439 . . . . 5 𝑥 ∈ V
32biantrur 531 . . . 4 (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
41, 3bitri 276 . . 3 (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
54opabbii 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)}
85, 6, 73eqtr4i 2828 1 I = (𝑥 ∈ V ↦ 𝑥)
 Colors of variables: wff setvar class 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
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