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Theorem dfid4 4991
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 1942 . . . 4 (𝑥 = 𝑦𝑦 = 𝑥)
2 vex 3189 . . . . 5 𝑥 ∈ V
32biantrur 527 . . . 4 (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
41, 3bitri 264 . . 3 (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥))
54opabbii 4679 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
6 df-id 4989 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
7 df-mpt 4675 . 2 (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)}
85, 6, 73eqtr4i 2653 1 I = (𝑥 ∈ V ↦ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  {copab 4672  cmpt 4673   I cid 4984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3188  df-opab 4674  df-mpt 4675  df-id 4989
This theorem is referenced by:  dfid5  13701  dfid6  13702  dfid7  37400
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