Theorem mptresid 4938

Description: The restricted identity expressed with the maps-to notation. (Contributed by FL, 25-Apr-2012.)

Assertion

Ref	Expression
mptresid	⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem mptresid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 4045	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		opabresid 4937	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴)
3	1, 2	eqtri 2186	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴)

Syntax hints:   ∧ wa 103   = wceq 1343   ∈ wcel 2136  {copab 4042   ↦ cmpt 4043   I cid 4266   ↾ cres 4606

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-res 4616

This theorem is referenced by:  idref  5725  restid2  12565  txswaphmeolem  12960  dvexp  13315  dvmptidcn  13318