Theorem mptresid 5700

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 4954	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		opabresid 5699	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴)
3	1, 2	eqtri 2850	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴)

Syntax hints:   ∧ wa 386   = wceq 1658   ∈ wcel 2166  {copab 4936   ↦ cmpt 4953   I cid 5250   ↾ cres 5345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-res 5355

This theorem is referenced by:  idref  6663  2fvcoidd  6808  pwfseqlem5  9801  restid2  16445  curf2ndf  17241  hofcl  17253  yonedainv  17275  sylow1lem2  18366  sylow3lem1  18394  0frgp  18546  frgpcyg  20282  evpmodpmf1o  20303  txswaphmeolem  21979  idnghm  22918  dvexp  24116  dvmptid  24120  mvth  24155  plyid  24365  coeidp  24419  dgrid  24420  plyremlem  24459  taylply2  24522  wilthlem2  25209  ftalem7  25219  fusgrfis  26628  fzto1st1  30398  zrhre  30609  qqhre  30610  fsovcnvlem  39148  fourierdlem60  41178  fourierdlem61  41179