Theorem mptresid 5456

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 4730	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		opabresid 5455	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴)
3	1, 2	eqtri 2644	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴)

Syntax hints:   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {copab 4712   ↦ cmpt 4729   I cid 5023   ↾ cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-res 5126

This theorem is referenced by:  idref  6499  2fvcoidd  6552  pwfseqlem5  9485  restid2  16091  curf2ndf  16887  hofcl  16899  yonedainv  16921  sylow1lem2  18014  sylow3lem1  18042  0frgp  18192  frgpcyg  19922  evpmodpmf1o  19942  txswaphmeolem  21607  idnghm  22547  dvexp  23716  dvmptid  23720  mvth  23755  plyid  23965  coeidp  24019  dgrid  24020  plyremlem  24059  taylply2  24122  wilthlem2  24795  ftalem7  24805  fusgrfis  26222  fzto1st1  29852  zrhre  30063  qqhre  30064  fsovcnvlem  38307  fourierdlem60  40383  fourierdlem61  40384