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Theorem opabresid 4879
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 4870 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
2 equcom 1683 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
32opabbii 4002 . . . 4 {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
4 df-id 4222 . . . 4 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
53, 4eqtr4i 2164 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} = I
65reseq1i 4822 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥} ↾ 𝐴) = ( I ↾ 𝐴)
71, 6eqtr3i 2163 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ( I ↾ 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1332  wcel 1481  {copab 3995   I cid 4217  cres 4548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-res 4558
This theorem is referenced by:  mptresid  4880
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