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Theorem opabresid 6070
Description: The restricted identity relation expressed as an ordered-pair class abstraction. (Contributed by FL, 25-Apr-2012.)
Assertion
Ref Expression
opabresid ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem opabresid
StepHypRef Expression
1 df-id 5583 . . . 4 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2 equcom 2015 . . . . 5 (𝑥 = 𝑦𝑦 = 𝑥)
32opabbii 5215 . . . 4 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
41, 3eqtri 2763 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
54reseq1i 5996 . 2 ( I ↾ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴)
6 resopab 6054 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
75, 6eqtri 2763 1 ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  {copab 5210   I cid 5582  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-res 5701
This theorem is referenced by:  mptresid  6071  iresn0n0  6074  pospo  18403  eqg0subg  19227  opsrtoslem1  22097  tgphaus  24141  relexp0eq  43691
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