Theorem mptresid 5947

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 5946	. 2 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
2		df-mpt 5154	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	1, 2	eqtr4i 2769	1 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {copab 5132   ↦ cmpt 5153   I cid 5479   ↾ cres 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-res 5592

This theorem is referenced by:  idref  7000  2fvcoidd  7149  pwfseqlem5  10350  restid2  17058  curf2ndf  17881  hofcl  17893  yonedainv  17915  smndex2dlinvh  18471  sylow1lem2  19119  sylow3lem1  19147  0frgp  19300  frgpcyg  20693  evpmodpmf1o  20713  cnmptid  22720  txswaphmeolem  22863  idnghm  23813  dvexp  25022  dvmptid  25026  mvth  25061  plyid  25275  coeidp  25329  dgrid  25330  plyremlem  25369  taylply2  25432  wilthlem2  26123  ftalem7  26133  fusgrfis  27600  fzto1st1  31271  cycpm2tr  31288  zrhre  31869  qqhre  31870  fsovcnvlem  41510  fourierdlem60  43597  fourierdlem61  43598  itcoval0mpt  45900