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Theorem opabresid 5490
Description: The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
Assertion
Ref Expression
opabresid {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ( I ↾ 𝐴)
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem opabresid
StepHypRef Expression
1 resopab 5481 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
2 equcom 1991 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
32opabbii 4750 . . . 4 {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
4 dfid3 5054 . . . 4 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
53, 4eqtr4i 2676 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} = I
65reseq1i 5424 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = ( I ↾ 𝐴)
71, 6eqtr3i 2675 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ( I ↾ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wcel 2030  {copab 4745   I cid 5052  cres 5145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-res 5155
This theorem is referenced by:  mptresid  5491  pospo  17020  opsrtoslem1  19532  tgphaus  21967  relexp0eq  38310
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