Theorem mptresid 6071

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 6070	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5232	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2766	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {copab 5210   ↦ cmpt 5231   I cid 5582   ↾ cres 5691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-res 5701

This theorem is referenced by:  idref  7166  2fvcoidd  7317  pwfseqlem5  10701  restid2  17477  curf2ndf  18304  hofcl  18316  yonedainv  18338  smndex2dlinvh  18943  sylow1lem2  19632  sylow3lem1  19660  0frgp  19812  frgpcyg  21610  evpmodpmf1o  21632  cnmptid  23685  txswaphmeolem  23828  idnghm  24780  dvexp  26006  dvmptid  26010  mvth  26046  plyid  26263  coeidp  26318  dgrid  26319  plyremlem  26361  taylply2  26424  taylply2OLD  26425  wilthlem2  27127  ftalem7  27137  fusgrfis  29362  fzto1st1  33105  cycpm2tr  33122  zrhre  33982  qqhre  33983  fsovcnvlem  44003  fourierdlem60  46122  fourierdlem61  46123  itcoval0mpt  48516