Theorem mptresid 6009

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

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem mptresid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opabresid 6008	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5194	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2762	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {copab 5172   ↦ cmpt 5193   I cid 5535   ↾ cres 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-res 5650

This theorem is referenced by:  idref  7097  2fvcoidd  7248  pwfseqlem5  10608  restid2  17326  curf2ndf  18150  hofcl  18162  yonedainv  18184  smndex2dlinvh  18741  sylow1lem2  19395  sylow3lem1  19423  0frgp  19575  frgpcyg  21017  evpmodpmf1o  21037  cnmptid  23049  txswaphmeolem  23192  idnghm  24144  dvexp  25354  dvmptid  25358  mvth  25393  plyid  25607  coeidp  25661  dgrid  25662  plyremlem  25701  taylply2  25764  wilthlem2  26455  ftalem7  26465  fusgrfis  28341  fzto1st1  32021  cycpm2tr  32038  zrhre  32689  qqhre  32690  fsovcnvlem  42407  fourierdlem60  44527  fourierdlem61  44528  itcoval0mpt  46872