Theorem mptresid 5615

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 4883	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		opabresid 5614	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴)
3	1, 2	eqtri 2783	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴)

Syntax hints:   ∧ wa 383   = wceq 1632   ∈ wcel 2140  {copab 4865   ↦ cmpt 4882   I cid 5174   ↾ cres 5269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-res 5279

This theorem is referenced by:  idref  6664  2fvcoidd  6717  pwfseqlem5  9698  restid2  16314  curf2ndf  17109  hofcl  17121  yonedainv  17143  sylow1lem2  18235  sylow3lem1  18263  0frgp  18413  frgpcyg  20145  evpmodpmf1o  20165  txswaphmeolem  21830  idnghm  22769  dvexp  23936  dvmptid  23940  mvth  23975  plyid  24185  coeidp  24239  dgrid  24240  plyremlem  24279  taylply2  24342  wilthlem2  25016  ftalem7  25026  fusgrfis  26443  fzto1st1  30183  zrhre  30394  qqhre  30395  fsovcnvlem  38828  fourierdlem60  40905  fourierdlem61  40906