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Theorem mptresid 4690
 Description: The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
Assertion
Ref Expression
mptresid (𝑥𝐴𝑥) = ( I ↾ 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem mptresid
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-mpt 3849 . 2 (𝑥𝐴𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
2 opabresid 4689 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ( I ↾ 𝐴)
31, 2eqtri 2102 1 (𝑥𝐴𝑥) = ( I ↾ 𝐴)
 Colors of variables: wff set class Syntax hints:   ∧ wa 102   = wceq 1285   ∈ wcel 1434  {copab 3846   ↦ cmpt 3847   I cid 4051   ↾ cres 4373 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-res 4383 This theorem is referenced by:  idref  5428
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