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Theorem opabresid 5893
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 5436 . . . 4 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2 equcom 2025 . . . . 5 (𝑥 = 𝑦𝑦 = 𝑥)
32opabbii 5109 . . . 4 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
41, 3eqtri 2843 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
54reseq1i 5825 . 2 ( I ↾ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴)
6 resopab 5878 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
75, 6eqtri 2843 1 ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1537  wcel 2114  {copab 5104   I cid 5435  cres 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-opab 5105  df-id 5436  df-xp 5537  df-rel 5538  df-res 5543
This theorem is referenced by:  mptresid  5894  iresn0n0  5899  pospo  17562  opsrtoslem1  20240  tgphaus  22701  relexp0eq  40181
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