Theorem mptresid 5375

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 4645	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		opabresid 5374	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴)
3	1, 2	eqtri 2632	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴)

Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {copab 4642   ↦ cmpt 4643   I cid 4948   ↾ cres 5040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-res 5050

This theorem is referenced by:  idref  6403  2fvcoidd  6452  pwfseqlem5  9364  restid2  15914  curf2ndf  16710  hofcl  16722  yonedainv  16744  sylow1lem2  17837  sylow3lem1  17865  0frgp  18015  frgpcyg  19741  evpmodpmf1o  19761  txswaphmeolem  21417  idnghm  22357  dvexp  23522  dvmptid  23526  mvth  23559  plyid  23769  coeidp  23823  dgrid  23824  plyremlem  23863  taylply2  23926  wilthlem2  24595  ftalem7  24605  fzto1st1  29183  zrhre  29391  qqhre  29392  fsovcnvlem  37327  fourierdlem60  39059  fourierdlem61  39060  fusgrfis  40549