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Theorem mptresid 4873
 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 3991 . 2 (𝑥𝐴𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
2 opabresid 4872 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ( I ↾ 𝐴)
31, 2eqtri 2160 1 (𝑥𝐴𝑥) = ( I ↾ 𝐴)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1331   ∈ wcel 1480  {copab 3988   ↦ cmpt 3989   I cid 4210   ↾ cres 4541 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-res 4551 This theorem is referenced by:  idref  5658  restid2  12129  txswaphmeolem  12489  dvexp  12844  dvmptidcn  12847
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