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Theorem opabresid 6068
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 5578 . . . 4 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2 equcom 2017 . . . . 5 (𝑥 = 𝑦𝑦 = 𝑥)
32opabbii 5210 . . . 4 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
41, 3eqtri 2765 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
54reseq1i 5993 . 2 ( I ↾ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴)
6 resopab 6052 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
75, 6eqtri 2765 1 ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  {copab 5205   I cid 5577  cres 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-res 5697
This theorem is referenced by:  mptresid  6069  iresn0n0  6072  pospo  18390  eqg0subg  19214  opsrtoslem1  22079  tgphaus  24125  relexp0eq  43714
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