Theorem mptresid 5921

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 5920	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5150	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2850	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 398   = wceq 1536   ∈ wcel 2113  {copab 5131   ↦ cmpt 5149   I cid 5462   ↾ cres 5560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-res 5570

This theorem is referenced by:  idref  6911  2fvcoidd  7056  pwfseqlem5  10088  restid2  16707  curf2ndf  17500  hofcl  17512  yonedainv  17534  smndex2dlinvh  18085  sylow1lem2  18727  sylow3lem1  18755  0frgp  18908  frgpcyg  20723  evpmodpmf1o  20743  cnmptid  22272  txswaphmeolem  22415  idnghm  23355  dvexp  24553  dvmptid  24557  mvth  24592  plyid  24802  coeidp  24856  dgrid  24857  plyremlem  24896  taylply2  24959  wilthlem2  25649  ftalem7  25659  fusgrfis  27115  fzto1st1  30748  cycpm2tr  30765  zrhre  31264  qqhre  31265  fsovcnvlem  40365  fourierdlem60  42458  fourierdlem61  42459